Dr. Senad Bulja, PhD, FIET, SMIEEE https://drbulja.com Wed, 09 Oct 2024 09:29:51 +0000 en-US hourly 1 How to design an attenuator? https://drbulja.com/how-to-design-an-attenuator-1525/ https://drbulja.com/how-to-design-an-attenuator-1525/#respond Tue, 12 Mar 2024 09:32:39 +0000 https://drbulja.com/?p=1525 Variable attenuators, along with phase shifters, antennas and filters, are important RF devices widely used in modern telecommunication systems, such as in radar systems, point-to-point radio, smart antennas and Radio Frequency (RF) signal cancellation, to name a few. There exists a wide variety of attenuator types, depending on their architecture and implementation, however, one particular attenuator type has been dominant in the field of telecommunications. The Reflection-Type Attenuator (RTA) is a compact device and has been the mainstay in the telecommunications industry due to its compactness, low implementation cost, high dynamic range, and relatively low losses.

In the next sections, the principles of operation of RTAs are explained, however, it should be borne in mind that there exist many similarities with the design of RTPSs, as elaborated in depth in the previous article, entitled, “How to design a phase shifter?”.

RTA design

This section covers an in-depth design of RTA and is divided into two sections – RTA design with a single active device (such as PIN diode or cold FET) per reflective load and RTA design with multiple active devices per load. The section on RTA design with a single active device per reflective load covers the basics of RTA design, whereas the following section on RTA design with multiple active devices per reflective load presents more advanced techniques for increasing obtainable dynamic range.

Variable RTA and its reflective load
Fig.1 Generic circuit of variable RTA (left) and its reflective load (right)

RTA design with a single active device per reflective load

A generic circuit of an RTA based on a 3-dB coupler is shown in Fig.1. Under the assumption of an ideal 3-dB coupler and identical loads, Z = Zv + R, the expressions for the reflection and transmission coefficients of the RTA in Fig.1 can be written as [1, 2]:

(1)   \begin{equation*} S_{11} = 0 \(, and\ S_{21} = -j * \Gamma \end{equation*}

where 𝛤 is the reflection at the loads, given by:

(2)   \begin{equation*} \Gamma = \frac{Z_v + R - Z_0}{Z_v + R + Z_0} \end{equation*}

Just like in the case of RTPS, the structure of Fig.1 is inherently impedance matched and the reflection coefficient of the reflective loads becomes the transmission coefficient of the overall structure, i.e.

(3)   \begin{equation*} S_{21}=-j\frac{Z_v + R - Z_0}{Z_v + R + Z_0} \end{equation*}

The magnitude of (3) can be written as:

(4)   \begin{equation*} |S_{21}| = \sqrt{\frac{(real(Z_v) + R - Z_0)^2 + imag(Z_v)^2}{(real(Z_v) + R + Z_0)^2 + imag(Z_v)^2}} \end{equation*}

In other words, the amount of signal emanating from the output port is proportional to the value of impedance Zv. If this impedance is made variable, with imag(Zv) = 0, (4) simplifies to:

(5)   \begin{equation*} |S_{21}| = \frac{real(Z_v) + R - Z_0}{real(Z_v) + R + Z_0} \end{equation*}

If the value of the termination resistor R is made to be equal to Z0, i.e. R = Z0, and impedance Zv is purely real and variable from very low values (ideally 0 Ω) to very high values (ideally infinite), the attenuator of Fig.1 can provide an infinitely high dynamic ratio. In other words, it is capable of fully passing the RF signal through without any attenuation, when the real value of variable impedance Zv is infinite and fully attenuating the RF signal for the case when the real value of variable impedance Zv is 0.

As with any RF components, this is never the case in practice. For example, the 3-dB coupler has a non-zero loss, typically, around 0.5 dB, while the variable impedance device, Zv, is limited by the both the lowest and highest resistances it can acquire, hence dictating the dynamic range of the RTA. For the purpose of the variable impedance device, at RF one usually uses a PIN diode (current-controlled device) or a “cold” FET (voltage-controlled device), however other possibilities exist [3]. The choice of the active device is dictated by several factors, such as the ON resistance, OFF capacitance, package parasitics and Inter-Modulation (IM) performance, to name but a few.

As an example, let us have a look at the PIN diode. An important parameter for PIN diodes is the carrier/recombination lifetime, which determines the lower frequency of operation. Below this frequency the PIN diode behaves as a standard PN diode, whereas above this frequency the PIN diode behaves as an almost current controlled resistor [4]. The carrier lifetime of the most commercially available PIN is usually from 5 ns to 5 μs, inferring the critical frequencies of 32 MHz and 32 kHz, respectively. Above these frequencies, the PIN diode behaves like a resistor. The high frequency equivalent circuit of a PIN diode is shown in Fig.2. The extrinsic parameters of the diode, Cp, and Lp stand for the package capacitance and package inductance, respectively. The values of the extrinsic parameters are greatly reduced through improvements of package quality, which, in turn, improves the parameters of the diode, however, these usually come at a monetary cost. For plastic packages, the values of Cp are usually in the range from 0.25 pF to 1 pF, whereas Lp depends on the length of the leads and can usually be assumed to be lower than 1 nH, unless otherwise stated in a datasheet. RI stands for the junction resistance and is inversely proportional to forward diode current, Idc:

(6)   \begin{equation*} R_I = \frac{nkT}{qI_{dc}} \end{equation*}

Where, n, k and T stand for the diode coefficient, Boltzmann constant (1.380649x10^{(-23)}\frac{J}{K}) and absolute temperature, respectively. CI is the capacitance of the geometry dependent, intrinsic region of the diode, usually well below 1 pF. As can be seen, the PIN diode does not behave as an ideal variable resistor; instead, it contains a reactive part, which, in the circuit of the attenuator, depicted in Fig.1, limits the achievable dynamic range. A typical dynamic range of an RTA is, of course, dependent on the type diodes used and, but is usually in the range between 15 dB – 30 dB, with insertion losses being the range between 1 dB – 2 dB [5]. The bandwidth of operation of RTAs is limited by the bandwidth of the 3-dB couplers, which have traditionally experienced narrow percentage bandwidths, approximately 25%-30% when implemented in standard branch-line configurations. However, their bandwidth can be significantly increased by using impedance broad-banding techniques.

A question can now be asked if greater levels of dynamic range can be obtained than those presented using the structure in Fig.1? The answer is yes, and use will be made of the techniques already

Fig.2 Equivalent circuit of a PIN diode, Zv
Fig.2 Equivalent circuit of a PIN diode, Zv

elaborated in our previous article, entitled, “How to design a phase shifter?”. The next section addresses this point comprehensively.

RTA design with multiple varactor diodes per reflective load

The increase in the dynamic range in RTA with a single 3-dB coupler can be achieved in two ways – using either distributed circuits or lumped elements. RTA design using distributed circuits is described first.

RTA design with multiple varactor diodes per reflective load using distributed circuits

The generic RTA capable of doubling dynamic range provided by a single RTA as shown in Fig.1, is shown in Fig.3. Here, variable impedance Z, Z = Zv + R , as before represents a series connection of an active device (PIN diode, cold FET, etc) and a termination resistor, R, whose value is usually set to R = Z0. The condition for doubling the transmission zero is, as in the case with phase shifters, obtained by finding a double zero of the transmission coefficient obtained this way. The transmission coefficient of this circuit can be written as:

(7)   \begin{equation*} S_{21} = -j\Gamma = j\frac{-Zk_{11}^2 + Z_0k_{12}^2 + Z_0Z^2}{Zk_{11}^2 + Z_0k_{12}^2 + Z_0Z^2} \end{equation*}

Here, k11 and k12 stand for quarter-wave (λ⁄4) transformers. For the circuit of Fig.3 represented by (7) to yield two times the dynamic range provided by the RTA of Fig.1, the transmission coefficient of (7) needs to be representable in the following form:

(8)   \begin{equation*} S_{21.double} = -j\Gamma = j\frac{(Z - Z_0)(Z-Z_0)}{(Z + Z_0)(Z + Z_0)} \end{equation*}

This is achieved by setting (7) to zero and solving it for Z:

(9)   \begin{equation*} Z_{1,2} = \frac{k_{11}^2 \pm \sqrt{k_{11}^4 - 4Z_0^2k_{12}^2}}{2Z_0} \end{equation*}

The double zero condition is achieved when the discriminant in (9) is zero, which takes place when:

(10)   \begin{equation*} Z_0 = \frac{k_{11}^2}{2k_{12}} \end{equation*}

Inferring that the double zero of (7) occurs at:

(11)   \begin{equation*} Z_{1,2} = Z_{v1,2} + R = \frac{k_{11}^2}{2Z_0} \end{equation*}

Equating (11) to Z0, as required by (8), one obtains the value for k11:

Fig.3 RTA terminated in a circuit comprising two identical variable impedances
Fig.3 RTA terminated in a circuit comprising two identical variable impedances

(12)   \begin{equation*} k_{11} = Z_0\sqrt{2} = (Z_{v1,2} + R)\sqrt{2} \end{equation*}

Which upon substitution in (10), yields the value for k12:

(13)   \begin{equation*} k_{12} = Z_0 = Z_{v1,2} + R \end{equation*}

According to (8), if conditions (12) and (13) are satisfied, the circuit of Fig.3 will deliver two times greater dynamic range. Expressions (12) and (13) infers that the conditions are satisfied only when the variable impedance device is purely real and attains low values, since, usually, but not always Z0 = 50Ω.

It is now possible to generalize the RTA shown in Fig.3 by recognizing that its reflective load plays a vital role in “doubling” the number of transmission zeroes and, hence, the achievable phase shift. By successively substituting variable impedance Z with the reflective load in Fig.3, it becomes possible to increase the dynamic range by n times, where n stands for the number of variable impedance Z circuits. Mathematically, the transmission coefficient in that case becomes:

(14)   \begin{equation*} S_{21.n.tuple} = -j\Gamma = j\frac{(Z - Z_0)(Z - Z_0)\dots(Z-Z_0)}{(Z + Z_0)(Z + Z_0)\dots(Z + Z_0)} = |\frac{Z - Z_0)^n}{Z + Z_0)^n}| \end{equation*}

An example of the circuits containing 4 and an arbitrary number of reflective loads is shown in Fig.3. It is important to note that n cannot attain any integer number, but it is obtained in the following fashion: n = 2m, where m stands for the order of the RTA obtained in this way. For example, m = 0, n = 1, corresponds to the case with a single variable impedance Z, depicted in Fig.4 (left). The case with m = 1 and, hence, n = 2, corresponds to the case of Fig.3. The cases for m = 2, n = 4 and the general case of an arbitrary values of m and n, correspond to the reflective loads depicted in Fig.4.

Even though increase in dynamic ratio is enabled by the increase of the number of variable impedance Z sections, it is not possible to increase this number indefinitely. First, this is since circuits with a large number n can be impractical and, second, the addition of a large number of impedance transformers k11 and k12 lead to bandwidth reduction. For practical purposes, it is not recommended to have more than 4

Fig.4 Reflective loads using distributed elements for: m = 2, n = 4 (left) and generalized for arbitrary RTA order (values of m, right)
Fig.4 Reflective loads using distributed elements for: m = 2, n = 4 (left) and generalized for arbitrary RTA order (values of m, right)

variable impedance Z sections. A question could be posed now if, instead of the increased complexity of the reflective loads, the same result can be achieved using a simple replication of the RTAs shown in Fig.1. The answer to this question will entirely depend on the losses exhibited by these two RTA types.
The general expression the losses of a cascade connection of n number of single variable impedance Z RTAs can be expressed as:

(15)   \begin{equation*} \begin{split} |S_{21}|_{dB.n.cas.single} = 20log_{10}|\frac{(Z- Z_0)^n}{(Z + Z_0)^n}| + n*L_{coupler} + \\ + (n-1)*L_{int.conn.line} + n*L_{bias.circuit} \end{split} \end{equation*}

whereas the losses of the RTA with the generalized transmission zero duplicating reflective loads as shown in Fig.4 (right) can be expressed as:

(16)   \begin{equation*} \begin{split} |S_{21}|_{dB.n.tuple.dist} = 20log_{10}|\frac{(Z- Z_0)^n}{(Z+Z_0)^n}| + L_{coupler} + \\ + (n-1)L_{dupl.zero.network.dist} + L_{bias.circuit}) \end{split} \end{equation*}

In (15) and (16), Lcoupler, stand for RF losses in the 3-dB coupler, Lint.conn.line stands for the losses of the microstrip lines connecting n single variable impedance RTAs, Lbias.circuit stands for the losses in the dc bias network and Ldupl.zero.network.dist stands for the losses of the distributed “duplicating” network, as shown in dashed line in Fig.4 (right). The difference between (15) and (16) is:

(17)   \begin{equation*} \begin{split} \Delta|S_{21}|_{dB.dist} = (n-1)*(L_{coupler} + L_{int.conn.line} - \\ - L_{dupl.zero.network.dist} + L_{bias.circuit}) \end{split} \end{equation*}

This equation shows that the difference in the losses of (15) and (16) is dependent on the losses of the 3-dB couplers, interconnecting lines, losses in the DC biasing circuit and the losses of the transmission zero duplicating network. This implies that the benefits gained using the reflective load replicating circuits as generally shown in Fig.4 (right) will be fully dependent on the exact realization of the RTA.

RTA design with multiple varactor diodes per reflective load using lumped elements

The generic RTA capable of doubling the dynamic ratio provided by an RTA terminated in reflective loads containing variable impedance Z as shown in Fig.1 using lumped elements is shown in Fig.5 (left). Just like in the case with phase shifters, the condition to double the dynamic range is reached when:

(18)   \begin{equation*} X_1 = -X_2 \end{equation*}

Using this condition, the reflection coefficient of the circuit of Fig.5 (left) becomes:

(19)   \begin{equation*} S_{21} = -j\Gamma = j\frac{Z_{in}-Z_0}{Z_{in} + Z_0} = \frac{Z^2 + Z(-2Z_0 ) + X_1^2}{Z^2 + Z(2Z_0) + X_1^2} \end{equation*}

Fig.5 RTA terminated in lumped element transmission zero duplicating network: with generic lumped elements (left) and with inductors and capacitors (right)
Fig.5 RTA terminated in lumped element transmission zero duplicating network: with generic lumped elements (left) and with inductors and capacitors (right)

Just like in the previous case with distributed elements, the required condition for the doubling of dynamic range is achieved when (19) can be expressed as:

(20)   \begin{equation*} S_{21.double} = -j\Gamma = j\frac{(Z - Z_0)(Z - Z_0)}{(Z + Z_0)(Z + Z_0)} \end{equation*}

This is achieved by finding the roots of (19) i.e.:

(21)   \begin{equation*} Z_{1,2} = \frac{2Z_0 \pm \sqrt{4Z_0^2 - 4X_1^2}}{2} \end{equation*}

The necessary condition for the achievement of double zero of (20) is that the discriminant of (21) is zero, which yields:

(22)   \begin{equation*} X_1=Z_0 \end{equation*}

Which upon substitution into (21) provides:

(23)   \begin{equation*} Z_{1,2} = Z_v + R = Z_0 \end{equation*}

Combining (18), (22) and (23) finally provides:

(24)   \begin{equation*} Z_0 = Z_v + R = \pm X_1 =\mp X_2 \end{equation*}

In other words, the lumped elements X_1 and X_2 are reactances of identical magnitudes, but of opposite signs – one is an inductor, and the other one is a capacitor. If we assume that X_1 is a capacitor and X_2 is an inductor, their exact values are calculated from:

(25)   \begin{equation*} C = \frac{1}{\omega Z_0} \(, and\ L = \frac{Z_0}{\omega} \end{equation*}

Where ω stands for the angular frequency of operation. Just like in the case using distributed elements,
here, too, is also possible to generalize the RTA shown in Fig.4 for an arbitrary order of the reflective load network. Fig.6 (left) depicts the case when the reflective loads are terminated in 4 variable impedances, Z, and Fig.6 (right) depicts the general case when the reflective loads are terminated in an arbitrary number of variable impedances. In a similar manner as with distributed reflective loads in the previous section, it is instructive to compare the losses exhibited by the proposed RTA structure and the losses obtained by a cascade connection of n number of single variable impedance Z RTAs, given by (15).

Fig.6 Reflective loads using lumped elements for: m = 2, n = 4 (left) and generalized for arbitrary RTA order (values of m, right)
Fig.6 Reflective loads using lumped elements for: m = 2, n = 4 (left) and generalized for arbitrary RTA order (values of m, right)

The losses exhibited by the cascade connection of the proposed reflective loads can be written as:

(26)   \begin{equation*} \begin{split} |S_{21}|_{dB.n.tuple.lumped} = 20log_{10}|\frac{(Z - Z_0)^n}{(Z + Z_0)^n}| + L_{coupler} + \\ + (n-1)L_{dupl.zero.network.lumped} + L_{bias.circuit} \end{split} \end{equation*}

The difference between the losses given by (15) and (26) can now be written as:

(27)   \begin{equation*} \begin{split} \Delta|S_{21}|_{dB.lumped} = (n - 1)*(L_{coupler} + L_{int.conn.line} - \\ - L_{dupl.zero.network.lumped} + L_{bias.circuit}) \end{split} \end{equation*}

Just like in the previous case with the transmission zero “duplicating” network realized in the distributed form the difference of the losses given by (15) and (26) is dependent on the losses of the 3-dB couplers, interconnecting lines, losses in the DC biasing circuit and the losses of the lumped element transmission zero duplicating network. Since in the present case the transmission zero “duplicating” network is composed of lumped elements – one inductor and one capacitor, its losses are expected to be lower than the losses of the interconnecting lines, however, this is entirely dependent on the exact RTA realization. It would now be instructive to design an RTA and demonstrate achievable dynamic ranges. This is provided in the next section.

Example – RTA design

Let us design several RTAs using the theory in the previous sections operating at a centre frequency of f0 = 2.5 GHz . As the active device, assume a PIN diode from Skyworks [6] with a parasitic capacitance of Cp = 0.35 pF and parasitic inductance of Lp = 1 nH. The minimum series resistance of the diodes is 0.5 Ω. Assume all other components to be lossless.
Solution: The PIN diode of [6] achieves the variation of its resistance from 1000 Ω (zero current) to 0.5 Ω (100 mA current). Using this PIN diode in series with a 50 Ω resistor in the reflective load, as shown in Fig.1, one achieves the variation of attenuation as indicated in Fig.7. The results are obtained using Keysight’s Advanced Design System (ADS), [7]. As evident from this figure, the maximum attenuation is achieved when the PIN diode is fully ON, i.e. when its resistance is around 0.5 Ω to yield the attenuation of 15.5 dB. As the current through the PIN diode is reduced, (or its resistance increased), the power dissipated on the termination resistor reduces and more power is passed to the output. At a diode resistance of approximately 1000 Ω, the insertion loss becomes 1.8 dB. This infers that the dynamic range of the attenuator obtained in this is around 13.7 dB. As a next step, the RTA circuits enabling doubling and quadrupling of dynamic range using distributed, Fig.4, elements. In this case, considering Z0 = 50 Ω, the impedance transformers k11 and k12 from (12) and (13), respectively, become k_{11} = 50 \sqrt{2} = 70.7 \Omega and k12 = 50Ω. The attenuation as function of the diode resistance of the RTAs designed in this way are presented in Fig.8. As can be seen, the values of attenuation have increased correspondingly. Such circuits have increased the dynamic range in proportion to the number of PIN

Fig.7 Variation of transmission coefficient of a single PIN diode reflective load of RTA of Fig.1 as a function of diode resistance at 2.5 GHz
Fig.7 Variation of transmission coefficient of a single PIN diode reflective load of RTA of Fig.1 as a function of diode resistance at 2.5 GHz

diodes – 27.6 dB for the 2 PIN diode per load RTA and 55.5 dB for the 4 PIN diode RTA per load. In a similar manner, the values of the lumped element RTAs, L and C are calculated using (25) to yield C = 1.27 pF and L = 3.18 nH. The attenuation as a function of the diode resistance of the RTAs designed in this way is presented in Fig.9. Just like in the case depicted in Fig.8, the dynamic range has increased – in the case of the 2 PIN diode lumped element RTA the dynamic range is now 27.2 dB, while for the 4 PIN diode counterpart it stands at 55.9 dB, which are in line with the dynamic range values obtained for their distributed element counterparts.
In practice, the exact values of attenuation will be ultimately dependent on the exact realizations of such circuits. For example, the losses in the 3-dB coupler, PIN diode, lumped and distributed elements and PCB substrate will all influence the extent of achieved phase shift and, ultimately, the losses. The choice of the attenuator will, therefore, be entirely dependent on the application and frequency of operation and, in many cases, the real estate available on PCB board.

Fig.8 Variation of transmission coefficient of a 2 (red line) and 4 (blue line) PIN diodes per reflective load of RTAs of Figs. 2 and 3 as a function of diode resistance at 2.5 GHz
Fig.8 Variation of transmission coefficient of a 2 (red line) and 4 (blue line) PIN diodes per reflective load of RTAs of Figs. 2 and 3 as a function of diode resistance at 2.5 GHz
Fig.9 Variation of transmission coefficient of a 2 (red line) and 4 (blue line) PIN diodes per reflective load of RTAs of Figs. 4 and 5 as a function of diode resistance at 2.5 GHz
Fig.9 Variation of transmission coefficient of a 2 (red line) and 4 (blue line) PIN diodes per reflective load of RTAs of Figs. 4 and 5 as a function of diode resistance at 2.5 GHz

Conclusion

In this article a comprehensive study on how to design Reflective Type Attenuators (RTAs) is presented. The article concludes with an example demonstrating agreement between the developed theory and simulations.

References

[1] S. Bulja, A. Grebennikov and P. Rulikowski, “Theory, analysis and design of high order reflective, absorptive filters,” in IET Microwaves, Antennas and Propagation, vol. 11, issue 6, pp.787-795, 2017.
[2] V. Kirillov, D. Kozlov and S. Bulja, “Series vs Parallel Reflection-type Phase Shifters”, IEEE Access, vol. 8, Oct. 2020, doi:10.1109/ACCESS.2020.3030463.
[3] S. Bulja and A. Grebennikov, “Variable Reflection-Type Attenuators Based on Varactor Diodes”, IEEE Trans. Microwave Theory and Tech., vol. 60, issue 12, 3719-3727, 2012.
[4] https://www.qsl.net/n9zia/an922.pdf
[5] https://www.skyworksinc.com/-/media/SkyWorks/Documents/Products/2001-2100/SKY12236_11_202529H.pdf
[6] https://www.skyworksinc.com/-/media/SkyWorks/Documents/Products/2201-2300/SMP1331_087LF_203589B.pdf
[7] https://www.keysight.com/us/en/products/software/pathwave-design-software/pathwave-advanced-design-system.html

]]> https://drbulja.com/how-to-design-an-attenuator-1525/feed/ 0 How to design a phase shifter? https://drbulja.com/how-to-design-a-phase-shifter-1491/ https://drbulja.com/how-to-design-a-phase-shifter-1491/#respond Tue, 20 Feb 2024 07:24:20 +0000 https://drbulja.com/?p=1491 Phase shifters, together with filters and antennas, form one of the basic building blocks of modern telecommunication systems. They are extensively used in radar, phased-arrays, receiver systems and many other types of communication networks. There exists a wide variety of phase shifter types, depending on their architecture and implementation, however, one particular phase shifter type has been dominant in the field of telecommunications. The Reflection-Type Phase Shifter (RTPS) is a compact device and has been the mainstay in the telecommunications industry due to its compactness, low implementation cost, high values of achievable phase shift and relatively low losses. Due to its importance to the telecommunications industry, it is pertinent to examine its history in more detail.

The first ever RTPS was proposed by Hardin in 1960 [1]. His experimental set-up, comprising of a circulator and a varactor diode, is shown in Fig. 1. The principle of operation is rather simple – the input signal at Port 1 is reflected by the reactance of the varactor diode at Port 2 and, by virtue of circulator operation, emerges from Port 3. The extent of phase shift is directly influenced by the capacitance ratio of the varactor diode and in the case of Fig. 1, it attains rather modest values. The use of a varactor diode resonated with an inductor, as a means to increase low values of phase shift was proposed by Searing in 1961 [2]. However, it was not until 1969, Garver, [3] when the first thorough treatment of RTPSs was provided. In that classical paper, Garver presented the conditions for achieving a 360o phase shift in a linear fashion. From this point in time, rapid development of phase shifting circuits followed. The bulky circulators were replaced with lightweight and cost-effective 3-dB couplers and a wide variety of reflective loads were investigated. These focused on either increasing the phase shift tuning range, bandwidth, power handing capability or miniaturisation – Terrio [4], Glance, [5], White, [6], Dawson, [7] and Chen, [8], Malczewski, [9]. Even to this day, RTPSs draw considerable attention. Due to this, in the next section, the exact operation of the RTPS will be explained, together with ways of increasing their phase shift range.

Fig. 1 Circulator based RTPS – Hardin, 1960 [1]
Fig. 1 Circulator based RTPS – Hardin, 1960 [1]

RTPS design

This section covers an in-depth design of RTPS and is divided into two sections – RTPS design with a single active device (such as varactor diode) per reflective load and RTPS design with multiple varactor diodes per load. The section on RTPS design with a single varactor diode per reflective load covers the basics of

RTPS design and details how phase shift can be maximized. The following section on RTPS design with multiple varactor diodes per reflective load builds upon the previous section to present more advanced techniques for increasing obtainable values of phase shift.

Fig. 2 Generic circuit of phase shifter
Fig. 2 Generic circuit of phase shifter

RTPS design with a single varactor diode per reflective load

A generic circuit of an RTPS based on a 3-dB coupler is shown in Fig. 2.

Under the assumption of an ideal 3-dB coupler, the expressions for the reflection and transmission coefficients of the structure of Fig. 2 become [10, 11]:

S11=0.5*(Γ1-Γ2) and S21=-j*0.5*(Γ1-Γ2) (1)

If the reflective loads are the same, i.e.,Γ1=Γ2=Γ, implying that Z1=Z2=Z, (1) simplifies to S11=0 and S21=-j*Γ . In other words, the structure of Fig. 2 is inherently impedance matched and the reflection coefficient of the reflective loads becomes the transmission coefficient of the overall structure. 𝛤 is then equal to:

Γ=(Z-Z0)/(Z+Z0) (2)

Where Z0 is the characteristic impedance of the interconnecting line and Z is the impedance of reflective loads. The general expression for the phase component of (2) can be written as:

For the case when the reflective loads are composed of a varactor diode (or any other device capable of variable reactance) with a capacitance ratio r=Cmax/Cmin  ,  the magnitude of the reflective coefficient at the loads and hence the overall structure becomes, inferring no losses, while the exact value of phase shift becomes:

Finding the first derivative of (4) reveals that its maximum occurs when  to yield the maximum phase shift of 180o.

As mentioned in the previous section, phase shift can be increased by introducing an inductor in the circuit of the varactor diode. The inductor can be connected both in series and in parallel with some interesting implications with regards to the exact values of obtainable phase shift. Both configurations are shown in Fig. 3. The value for the inductor, as shown in Fig.3, needed for the maximum obtainable phase shift is found by obtaining the first derivative of:

Fig. 3 Single inductor-varactor diode RTPS: series circuit inductor-varactor diode (left) and parallel circuit inductor-varactor diode (left)
Fig. 3 Single inductor-varactor diode RTPS: series circuit inductor-varactor diode (left) and parallel circuit inductor-varactor diode (left)

(12) indicates that the parallel connection as shown in Fig. 3 (right) can achieve a full 360o phase shift using only one varactor diode, whereas for the case of the series connection, the maximum achievable phase shift is a function of the characteristic impedance, frequency and the minimum diode capacitance. Also, such a circuit is not theoretically possible to achieve a full 360o phase shift unlike with the case of the parallel connection. An excellent examination of the series vs parallel reflective load configurations is presented in [11], which, in addition to examining the linear parameters (phase shift and insertion losses), also examines their non-linear characteristics. The article concludes that the parallel inductor-varactor combination is superior in every aspect to its series connection counterpart. However, in practice, the series inductor-varactor combination is more widely used – one of the possible reasons lies with the ease of DC biasing of the varactor diode in such a configuration. In this aspect and with reference to Fig. 3 (left), the application of DC bias on varactor diode can be done directly as the series inductor, ,  provides a short to the ground. One the other hand, the parallel combination requires addition of a decoupling capacitor in series with inductor , so that the applied DC bias voltage is not shorted through inductor .

A question can now be asked if phase shifts greater than those presented in this section are obtainable? The answer is yes, and the next section addresses this point comprehensively.

RTPS design with multiple varactor diodes per reflective load

The increase in the values of obtainable phase shift using an RTPS with a single 3-dB coupler can be achieved in two ways – using either distributed circuits or lumped elements. RTPS design using distributed circuits is described first.

 RTPS design with multiple varactor diodes per reflective load using distributed circuits

The generic RTPS capable of doubling phase shift provided by an RTPS terminated in reflective loads containing variable impedance Z as shown in Fig. 4 (left) is shown in Fig. 4, (right). Here, variable impedance Z can take many forms – it could be a simple varactor diode or series/parallel connection or a varactor diode and an inductor. The transmission coefficient of this circuit (Fig. 4, right) can be written as:

Here,  and  stand for quarter-wave ((λ⁄4)) transformers. For the circuit of Fig. 4 (right) represented by (13) to yield two times the amount of phase shift provided by the RTPS of Fig. 4 (left), the transmission coefficient of (13) needs to be representable in the following form:

In other words, (14) depicts the situation that is equivalent to having two circuits of Fig. 4 (left) connected in series. Mathematically, the phase component of (14) now becomes:

To have (13) representable by (14) its two transmission zeroes need to be identical. This is achieved by setting (13) to zero and solving it for Z:

Fig. 4 RTPS terminated in a single variable impedance, Z (left) and RTPS terminated in a circuit comprising two identical variable impedances, Z (right)
Fig. 4 RTPS terminated in a single variable impedance, Z (left) and RTPS terminated in a circuit comprising two identical variable impedances, Z (right)

According to (14), if conditions (19) and (20) are satisfied, the circuit of Fig. 4 (right) will deliver two times greater phase shift than the one provided in Fig. 4 (left).

It is now possible to generalize the RTPS shown in Fig. 4 (right) by recognizing that its reflective load plays a vital role in “doubling” the number of transmission zeroes and, hence, the achievable phase shift. By successively substituting variable impedance Z with the reflective load in Fig. 4 (right), it becomes possible to increase obtainable phase shift by n times, where n stands for the number of variable impedance Z circuits. Mathematically, the transmission coefficient in that case becomes:

An example of the circuits containing 4 and an arbitrary number of reflective loads is shown in Fig. 5. It is important to note that n cannot attain any integer number, but it is obtained in the following fashion: n = 2m, where m stands for the order of the RTPS obtained in this way. For example, m = 0, n = 1, corresponds to the case with a single variable impedance Z, depicted in Fig. 4 (left). The case with m = 1 and, hence, n = 2, corresponds to the case of Fig. 4 (right). The cases for m = 2, n = 4 and the general case of an arbitrary values of m and n, correspond to the reflective loads depicted in Fig. 5.

Even though increase in phase shift is enabled by the increase of the number of variable impedance Z sections, it is not possible to increase this number indefinitely. First, this is since circuits with a large number n can be impractical and, second, the addition of a large number of impedance transformers  and  lead to bandwidth reduction. For practical purposes, it is not recommended to have more than 4 variable impedance Z sections.

A question could be posed now if, instead of the increased complexity of the reflective loads the same result can be achieved using by a simple replication of the RTPSs shown in Fig. 4 (left). The answer to this question will entirely depend on the losses exhibited by these two RTPS types.

The general expression the losses of a cascade connection of n number of single variable impedance Z RTPSs can be expressed as:

whereas the losses of the RTPS with the generalized transmission zero duplicating reflective loads as shown in Fig. 5 (right) can be expressed as:

Fig. 5 Reflective loads using distributed elements for: m=2, n = 4 (left) and generalized for arbitrary RTPS order (values of m, right)
Fig. 5 Reflective loads using distributed elements for: m=2, n = 4 (left) and generalized for arbitrary RTPS order (values of m, right)

In (22) and (23), Lcoupler, stand for RF losses in the 3-dB coupler, L(int_conn_line) stands for the losses of the microstrip lines connecting n single variable impedance RTPSs, L(bias_circuit) stands for the losses in the dc bias network and L(dupl_zero_network_dist) stands for the losses of the distributed “duplicating” network, as shown in dashed line in Fig. 4 (right). The difference between (22) and (23) is:

This equation shows that the difference in the losses of (22) and (23) is dependent on the losses of the 3-dB couplers, interconnecting lines, losses in the DC biasing circuit and the losses of the transmission zero duplicating network. This implies that the benefits gained using the reflective load replicating circuits as generally shown in Fig. 5 (right) will be fully dependent on the exact realization of the RTPS.

RTPS design with multiple varactor diodes per reflective load using lumped elements

The generic RTPS capable of doubling phase shift provided by an RTPS terminated in reflective loads containing variable impedance Z as shown in Fig. 4 (left) using lumped elements is shown in Fig. 6 (left). Just like in the previous section, variable impedance Z can take many forms – it could be a simple varactor diode or series/parallel connection or a varactor diode and an inductor. The input impedance of the reflective loads, as depicted in Fig. 6 (right) can be written as:

Just like in the previous case with distributed elements, the required condition for the doubling of phase shift is achieved when (27) can be expressed as:

Fig. 6 RTPS terminated in lumped element transmission zero duplicating network: with generic lumped elements (left) and with inductors and capacitors (right)
Fig. 6 RTPS terminated in lumped element transmission zero duplicating network: with generic lumped elements (left) and with inductors and capacitors (right)

Where ω stands for the angular frequency of operation. Just like in the case using distributed elements, here, too, is also possible to generalize the RTPS shown in Fig. 6 for an arbitrary order of the reflective load network. Fig. 7 (left) depicts the case when the reflective loads are terminated in 4 variable impedances, Z, and Fig. 7 (right) depicts the general case when the reflective loads are terminated in an arbitrary number of variable impedances.

In a similar manner as with distributed reflective loads in the previous section, it is instructive to compare the losses exhibited by the proposed RTPS structure and the losses obtained by a cascade connection of n number of single variable impedance Z RTPs, given by (22). The losses exhibited by the cascade connection of the proposed reflective loads can be written as:

Fig. 7 Reflective loads using lumped elements for: m=2, n = 4 (left) and generalized for arbitrary RTPS order (values of m, right)
Fig. 7 Reflective loads using lumped elements for: m=2, n = 4 (left) and generalized for arbitrary RTPS order (values of m, right)

The difference between the losses given by (22) and (34) can now be written as:

Just like in the previous case with the transmission zero “duplicating” network realized in the distributed form the difference of the losses given by (22) and (35) is dependent on the losses of the 3-dB couplers, interconnecting lines, losses in the DC biasing circuit and the losses of the lumped element transmission zero duplicating network. Since in the present case the transmission zero “duplicating” network is composed of lumped elements – one inductor and one capacitor, its losses are expected to be lower than the losses of the interconnecting lines, however, this is entirely dependent on the exact RTPS realization.  It would now be instructive to design an RTPS and demonstrate achievable phase shifts. This is provided in the next section.

Example – RTPS design 

Let us design several RTPSs using the theory in the previous sections operating at a centre frequency of f0=2.5 GHz . As the active device, assume a varactor diode with a minimum capacitance of Cmin=1 pF and capacitance ratio of r=5. The parasitic resistance of the diode is 1 Ω. Assume all other components to be lossless.

Solution: Let assume that the basic variable impedance will be a series configuration of a varactor diode and an inductor, as shown in Fig. 3 (left). For the given parameters of the varactor diode, the value of the series inductor, Ls, is calculated using (7) to yield Ls = 2.53 nH. In conjunction with (9), this value yields the maximum, theoretically achievable phase shift of 107.9o, obtained using Keysight’s Advanced Design System (ADS), [12]. The phase and magnitude of the transmission coefficient as a function of capacitance ratio r are shown in Fig. 8. As evident, the obtained phase shift is in full alignment with predictions.

As a next step and based on the optimal series reflective load comprising the series inductor, Ls = 2.43 nH, the RTPS circuits enabling doubling of the amount of phase shift using distributed, Fig. 4 (right), and lumped elements, Fig. 6 (right), are designed.

For the distributed element circuit and considering Z0 = 50 Ω, the impedance transformers  and  from (19) and (20), respectively, become  and . In a similar manner, the values of the lumped elements, L and C are calculated using (34) to yield  and . The insertion phase and loss as function of capacitance ratio, r, of the RTPS designed in this way are presented

in Fig. 9. As can be seen, the values of insertion phase and transmission loss are identical in both cases, and both have resulted in doubling of the achievable phase shift, at the expense of doubling the insertion losses, as predicted. The maximum achieved phase shift achieved in this case is 215o. This is the direct consequence of the fact that no losses are assumed for both the distributed and lumped elements of the RTPS circuits.

Fig. 8 Insertion phase and loss of a single, series inductor-varactor diode reflective load RTPS of Fig. 3 (left) as a function of capacitance ratio, r, at 2.5 GHz
Fig. 8 Insertion phase and loss of a single, series inductor-varactor diode reflective load RTPS of Fig. 3 (left) as a function of capacitance ratio, r, at 2.5 GHz

Lastly, the response of the RTPS realized using 4 optimal variable impedance loads in both the distributed, Fig. 5 (left), and lumped, Fig. 7 (left). The responses of such circuits are shown in Fig. 10. As predicted, the

insertion phase is now 4 times greater than the insertion phase of a single load RTPS and it stands, for both distributed and lumped element realizations, at 432o. Also, the insertion loss has also increased 4-fold compared to the single load RTPS realization.

However, the exact values of phase shift and losses of RTPSs will be ultimately dependent on the exact realizations of such circuits. For example, the losses in the 3-dB coupler, varactor diode, lumped and distributed elements and PCB substrate will all influence the extent of achieved phase shift and, ultimately, the losses. The choice of the phase shifter will, therefore, be entirely dependent on the application and frequency of operation and, in many cases, the real estate available on PCB board.

Conclusion

In this article a comprehensive study on how to design Reflective Type Phase Shifters (RTPS) is presented.  The study was initiated with a brief history of RTPS and basic principles in order to explain the maximum, theoretical values of achievable phase shifts that can be expected using simple varactor diode circuits. The study then proceeded to systematically present ways of increasing the amount of phase shift by modifying reflective load circuits. The article concludes with an example demonstrating agreement between the developed theory and simulations.

Fig. 9 Left: Insertion phase and loss of a distributed, Fig. 4 (right) double load RTPS as a function of capacitance ratio, r, at 2.5 GHz and Right: Insertion phase and loss of a lumped, Fig. 6 (right) double load RTPS as a function of capacitance ratio, r, at 2.5 GHz
Fig. 9 Left: Insertion phase and loss of a distributed, Fig. 4 (right) double load RTPS as a function of capacitance ratio, r, at 2.5 GHz and Right: Insertion phase and loss of a lumped, Fig. 6 (right) double load RTPS as a function of capacitance ratio, r, at 2.5 GHz
Fig. 10 Left: Insertion phase and loss of a distributed quadruple load RTPS as a function of capacitance ratio, r, at 2.5 GHz and Right: Insertion phase and loss of a lumped double load RTPS as a function of capacitance ratio, r, at 2.5 GHz
Fig. 10 Left: Insertion phase and loss of a distributed quadruple load RTPS as a function of capacitance ratio, r, at 2.5 GHz and Right: Insertion phase and loss of a lumped double load RTPS as a function of capacitance ratio, r, at 2.5 GHz

References:

[1] R. N. Hardin, E. J. Downey and J. Munushian, ‘Electronically variable phase shifter utilizing variable capacitance diodes’, Proc. IRE, vol. 48, pp. 944-945, May 1960.

[2] R. M. Searing, ‘Variable capacitance diodes used as phase-shift devices’, Proc. IRE, vol. 49, pp. 640-641, March 1961.

[3] R. V. Garver, ‘360o Varactor linear phase modulator’, IEEE Trans. Microwave Theory Tech., vol. MTT-17, no. 3, pp. 137-147, March 1969.

[4] F. G. Terrio, R. J. Stockton and W. D. Sato, ‘A low-cost P-I-N diode phase shifter for airborne phased-array antennas’, IEEE Trans. Microwave Theory and Tech., vol. MTT-22, no. 6, pp. 688-692, June 1974.

[5] B. Glance, ‘A fast low-loss microstrip p-i-n phase shifter’, IEEE Trans. Microwave Theory and Tech., vol. MTT-27, no.1, pp. 14-16, January 1979.

[6] J. F. White, ‘Diode phase shifters for array antennas’, IEEE Trans. Microwave Theory and Tech., vol. MTT-22, no. 6, pp. 658-674, June 1974.

[7] D. E. Dawson et al., ‘An analog X-band phase shifter’, IEEE MTT-S Int. Microwave Symp. Dig., pp. 6-10, 1984.

[8] C. L. Chen et al., ‘A low loss Ku-band monolithic analog phase shifter’, IEEE Trans. Microwave Theory Tech., vol. MTT-35, no.3., pp. 315-320, March 1987.

[9] A. Malczewski et al., ‘X-band RF MEMS phase shifters for phased array applications’, IEEE Microwave and Guided Letters, vol. 9, no. 12, pp. 517-519, December 1999.

[10] S. Bulja, A. Grebennikov and P. Rulikowski, “Theory, analysis and design of high order reflective, absorptive filters,” in IET Microwaves, Antennas and Propagation, vol. 11, issue 6, pp.787-795, 2017.

[11] V. Kirillov, D. Kozlov and S. Bulja, “Series vs Parallel Reflection-type Phase Shifters”, IEEE Access, vol. 8, Oct. 2020, doi:10.1109/ACCESS.2020.3030463.

[12] https://www.keysight.com/us/en/products/software/pathwave-design-software/pathwave-advanced-design-system.html

 

 

]]> https://drbulja.com/how-to-design-a-phase-shifter-1491/feed/ 0 How to design and tune a microstrip antenna? https://drbulja.com/how-to-design-and-tune-a-microstrip-antenna-1463/ https://drbulja.com/how-to-design-and-tune-a-microstrip-antenna-1463/#respond Mon, 19 Feb 2024 10:38:12 +0000 https://drbulja.com/?p=1463 Microstrip antennas trace their history back to the 1950s, when the concept of microstrip radiator was introduced by Deschamps [1]. The development of the microstrip patch antenna, however, lay dormant until the early 1970, when their use became more widespread, primarily due to the improved availability of good substrates, with low-loss tangents and better theoretical models capable of better prediction of their performance. The very first practical microstrip antennas have been demonstrated by Howell [2] and Munson [3]. Since then, microstrip antennas have been the subject of intensive research and development, due to their attractive features, such as their low cost, low weight, low and conformable   profiles and compatibility with integrated circuits.

However, practical microstrip antenna design still draws attention both from academia and industry, where the main focus is aimed at impedance bandwidth increase and desirable radiation characteristics (polarization, cross-polarization ratios, to name but a couple). Even though the design of such antennas has been greatly aided by the advent of full-wave simulation software packages, such as Computer Studio, CST [4] and HFSS [5], practical patch antenna design is still a matter of interest and is, particularly for the case of stacked patches, done, primarily, using trial and error. The particular problem relates to input coupling, coupling among stacked patches and their relative position with respect to each other in order to yield a broad impedance bandwidth and desirable radiation characteristics. In essence the problem relates to how a stacked antenna can be tuned. The literature is sparse on how this should be done in a systematic and repeatable fashion.

In this article we approach antenna design and tuning methodically using a combination of full-wave simulations, circuit simulations and filter theory. Such an approach allows us to tune the antenna for the broadest bandwidth possible using the frequency domain techniques in a prompt and straightforward fashion.

General principle

Despite their superficial differences, a great deal of work on filter theory can be readily applied to microstrip antenna design. For example, filters are composed of individual resonators which in the antenna case corresponds to individual patches. Further, just like in filter design, input coupling determines the extent of achievable bandwidth. What is different, though, is that the number of stacked patch antennas is usually smaller than the number of resonators in a filter. Stacked patch antennas usually have a maximum of three patches, whereas filter orders up to 10 are not unheard of. In a way, this makes stacked patch antennas easier to tune, but it needs to be understood that unlike in filters where the input and output impedances are usually, but not necessarily, 50 Ω, in antennas only the input is terminated in 50 Ω, whereas the output impedance of the antenna is given by Z0=Rout=√((μ0 μr)/(ε0 εr )), which in the case of air becomes Z0=Rout= √(μ0/ε0) ≈377 Ω. .  This infers that if one were to utilize filter theory to design stacked antennas, such an analysis would only be valid in the first order approximation and optimization procedures will need to be used to address the impedance differential.

The general principle of stacked patch antenna tuning, as pursued in this article, using the frequency domain technique will be explained with reference to an ideal bandpass filter with no cross coupling, as depicted in Fig. 1. This follows up on our previous article, entitled, “How to design and tune an RF filter”, which provided steps with regards how filters can be tuned. With reference to this figure, the equivalent

circuit of a stacked patch antenna having a number of patches equal to n is represented using n resonators, with each resonator representing a patch in the antenna stack. The coupling among the patches and input

Fig. 1 Equivalent circuit realization of a stacked microstrip patch antenna with a number of stacked patches equal to n
Fig. 1 Equivalent circuit realization of a stacked microstrip patch antenna with a number of stacked patches equal to n

and output is represented using n+1 coupling (admittance transformation) sections. The resonators/patches in Fig. 1 are depicted using the parallel connection of capacitors (C1, C2, …Cn) and inductors (L1, L2,….Ln) and the coupling sections with admittance transformers Y1, Y2,…Yn.

Just like in the case with filters, the correct input and inter-resonator coupling coefficients can be calculated using either filter tables [6] or software packages such as Guided Wave Technology’s Filter & Coupling Matrix Synthesis Software [7]. The coupling coefficient at the input is calculated in a similar way. However, as mentioned earlier, the coupling coefficient values obtained in this way as applied to antennas would be only valid as the first order approximation, due to the existence of an impedance mismatch at the output. The best way to learn how to design and tune an antenna in the frequency domain is via an example.

Example – stacked 2-patch antenna

Let us design a microstrip antenna with 2 stacked patches, operating at a centre frequency of 11.72 GHz. The antenna is to be designed with a view of obtaining as wide bandwidth as possible, with a minimum return loss of 10 dB.

Solution: Since the bandwidth of the antenna is not specifically stated, the first step in the design relates to examining the maximum bandwidth the antenna can support. As we have learnt from the previous article on filter tuning, excitation (external coupling) plays an important role in determining the extent of the bandwidth the antenna can support.

External coupling

Antenna excitation provides the means for RF energy to enter the device. The level of excitation is primarily determined by the percentage bandwidth (BW/f0)  and the values of the coupling coefficient/admittance transformer, as evident from Fig. 1. In mathematical terms, the level of external coupling is given by:

However, apart from the frequency of operation, f0=11.72 GHz , we have no knowledge of other parameters that would help us determine the right level of external coupling. This means that we will need to examine the maximum bandwidth that the antenna can provide. This is done through the parametric investigation of Qext of the first antenna patch. Just like with the case with filters, it is not necessary to simulate the entire stacked 2-patch antenna – the first patch is all that is required to infer the values of the Qext.

To this end, let us assume that the antenna is to be fed using a probe, as shown in Fig. 2, however, its exact position is to be determined with a view of maximizing the bandwidth. In addition, let us also assume that

Fig. 2 First antenna patch used in determination of external coupling, Qext
Fig. 2 First antenna patch used in determination of external coupling, Qext

the substrate thickness of the probe-fed patch antenna is also a parameter, which we will use to determine the optimum height to support wide bandwidth.

The structure of Fig. 2 was simulated using a full-wave simulator, CST, for its reflection coefficient, S11. Qext .  was then calculated from the reflection coefficient using an equation alternative to that of (1):

The patch of Fig. 2 has the dimensions of Xpatch = Ypatch = 6.63 mm and is mounted on Rogers RO3003 [8] substrate, with εr = 3 and tan(δ) = 0.001. The values of Qext simulated for different feed offset distances, D, and substrate thicknesses, H, are shown in Fig. 3. For the cases depicted in Fig. 3 (left), the distance from the probe feed to the patch centre is kept at D = 3 mm and, for the cases shown in Fig. 3 (right), the substrate height was kept at H = 0.75 mm.

For broad bandwidths, the values of Qext need to be as low as possible, as can be inferred from (1). From Fig. 3, it follows that the lowest values of Qext are recorded when the feed probe is positioned at the edge of the patch (Fig. 3 right) and when the substrate thickness is approximately 0.75 mm (Fig. 3 left). It is further obvious from Fig. 3 (left) that Qext is not a monotonous function of substrate thickness and that a range of substrate thicknesses exists for which the input impedance bandwidth is at a maximum; in the present case this corresponds substrate heights in the region from 0.5 mm – 1 mm. This point requires further elaboration.

Choosing the lowest substrate height is beneficial for several reasons. First, the impedance bandwidth of

Fig. 3 Qext as function of: Rogers RO3003 substrate thickness (left) for the case when the probe-feed is located at D = 3 mm away from the patch centre; and feed probe distance from the centre of the patch (right) for the case when substrate thickness is H = 0.75 mm
Fig. 3 Qext as function of: Rogers RO3003 substrate thickness (left) for the case when the probe-feed is located at D = 3 mm away from the patch centre; and feed probe distance from the centre of the patch (right) for the case when substrate thickness is H = 0.75 mm
Fig. 4 Dual-probe fed antenna for dual/circular polarizations
Fig. 4 Dual-probe fed antenna for dual/circular polarizations

the antenna is maximized, while retaining a low profile. Second, lower substrate heights offer improved radiation characteristics as the cut-off frequency of surface waves is inversely proportional to substrate height. Thirdly, when used in configurations requiring dual or circular polarization, utilizing two probe feed structures as shown in Fig. 4, probe-fed patch antennas tend to suffer from a high cross-polarization ratio, compared to, for example, aperture fed patch antennas. This is primarily attributed to the fact that parasitic coupling exists between the two probe feeds. This is greatly reduced by reducing the profile of the antenna. This topic will be, however, covered in depth in another article.

For the broadest possible bandwidths, according to Fig. 3, the probe feed needs to be placed at 3 mm away from the patch centre and substrate thickness of 0.75 mm needs to be chosen. In this set-up, Qext becomes 6.657. Using CMS software [7] in conjunction with (1), the required frequency of operation of 11.72 GHz and Qext = 6.657, one obtains the coupling matrix for the double patch antenna:

Inter-resonator (patch) coupling

The actual inter-resonator (patch) coupling, as implied from (3), is equal to k12 = 0.2002016. Effectively, the actual coefficient infers how much energy is coupled from one patch to another. In standard filter settings it is determined using a system of two resonators in conjunction with an Eigenmode solver, which can calculate the natural frequencies of such a system. In the present case actual coupling between patches is calculated to a first order approximation using the two-patch structure shown in Fig. 5.  In a manner similar to inter-resonator coupling with filters, the coupling between the two patches is obtained by monitoring the difference between the two resonant frequencies of the system of Fig. 5 as a function of displacement along the Y-axis:

Fig. 5 Two antenna structure used for the calculation of coupling coefficients
Fig. 5 Two antenna structure used for the calculation of coupling coefficients

However, unlike in the calculations of the inter-resonator coupling in filter design where the Eigenmode solver can be used to infer the frequencies of the two-resonator system, here this is not possible. The primary reason for this lies with the fact that the Eigenmode solver is only applicable to closed form (bounded) structures, without the need for external excitation. In the case of antennas examined here, this is performed using a Time domain solver. To perform this task adequately, the antennas need to be very weakly excited, effectively mimicking the operation using the Eigenmode solver, with high Qext to avoid unnecessarily loading the antenna. This is accomplished by setting the positions of the probe-feed near to the centre of the bottom patch. In the present case, the probe-feed is located at a displacement distance of 1 mm from the centre of the patch. The sizes of both the bottom and top patches are kept the same – 6.63 mm in both the X and Y directions and the substrate thickness for the top patch is 1.25 mm. A typical response of the two-antenna system is shown in Fig. 6 (left). The coupling coefficients as a function of top patch displacement are shown in Fig. 6 (right) and are calculated using (5). The coupling among the patches is fully magnetic. From the coupling matrix (3), the necessary actual coupling coefficient of 0.20036 is obtained when the top patch is displaced with respect to the bottom patch by approximately 1.3 mm. This, in conjunction with the exact feed position obtained in the previous section enables us to proceed towards the design of the broadband antenna. This is elaborated in the next section.

Fig. 6 Typical response of two-antenna system of Fig. 4 (left); Coupling coefficient as function of top patch displacement along the Y-axis (right).
Fig. 6 Typical response of two-antenna system of Fig. 4 (left); Coupling coefficient as function of top patch displacement along the Y-axis (right).

Antenna design and tuning

The designed antenna is shown in Fig. 7. As per design requirements, the antenna consists of 2 stacked patches. The antenna is also fed using a probe, positioned 3 mm away from the centre of the bottom patch.  The height of the bottom patch is 0.75 mm, and the height of the top patch is 1.25 mm. The top patch is displaced 1.3 mm relative to the bottom patch since, according to Fig. 6 (right), this gives rise to

Fig. 7 Designed stacked 2-patch antenna
Fig. 7 Designed stacked 2-patch antenna

the required coupling coefficient of 0.20036. In order to allow antenna tuning in a post-processing step, discrete ports are added to each patch. It should be noted that unlike the case with filters, where each resonator has one discrete port attached, each antenna patch in this case has 4 discrete ports attached. This is needed to adequately ground the patch which will be required during the post-processing step. The reflection coefficient of the antenna as designed in Fig. 7 is shown in Fig. 8. As can be seen, the antenna is not fully tuned for wide bandwidth (-10 dB) as required by the specifications. This situation is similar to that of filter tuning addressed in our previous article, where the initial response needs refinement in order to fully satisfy requirements. In essence, the sizes of the patches and their relative displacement need to be tuned in order to yield an optimum response. This is performed using frequency domain tuning.

The first step towards antenna tuning lies with full-wave simulation of the filter of Fig. 7 with dummy ports attached, using commercially available software such as CST [4] or HFSS [5]. The port impedance of the dummy ports is, just like in the case of filter tuning, kept very high – in the present case, the port impedance is set to 100 GΩ. After the full wave simulation, the resultant s9p files are imported into a circuit simulator, such as Keysight’s Advanced Design System, (ADS), [9], as schematically depicted in Fig.9

Fig. 8 Initial response of two-patch antenna of Fig. 7
Fig. 8 Initial response of two-patch antenna of Fig. 7
Fig. 9 Circuit representation of designed antenna of Fig. 7 with dummy ports
Fig. 9 Circuit representation of designed antenna of Fig. 7 with dummy ports

The waveguide port or port 1 is terminated in 50 Ω impedance, whereas the dummy ports are terminated in sets of ideal capacitors, C1, and C2. The C12 capacitor set cross-connects sets of capacitors C1, and C2 connected through dummy ports to the same sides of the antenna patches. These capacitors are referred to as correction capacitors as their values infer what type of correction (if any) needs to be applied to the physical antenna of Fig. 7. A set of identical correction capacitors, termed C1 are attached to the bottom patch, and, similarly, another set of identical capacitors, C2, are attached top the top patch. These capacitors are connected to each other using cross-capacitors, C12. Just like in the case with filter tuning, frequency correction capacitors C1 and C2 provide information with regards to the frequency or operation of the patch to which they are attached. For example, if these capacitors are positive, this infers that the frequency of operation of the patch needs to be reduced. Similarly, if their values are negative, the frequency of operation of the patch needs to be increased. With regards to the cross capacitor, C12, its positive values infer that the coupling between the two patches needs to be increased and as evident from Fig. 6 (right), this infers that the relative displacement between the patches needs to be increased. Similarly, negative values of C12 dictate that the displacement between the patches needs to be reduced.

Frequency domain tuning is initiated by shorting all patches expect the first (bottom), as shown in Fig. 10 (top). Its circuit implementation is shown in Fig. 10 (bottom). Capacitor C1 is tuned until the dip of the reflection coefficient corresponds to the centre frequency of operation of the antenna – 11.72 GHz in this case. This is achieved then C1 = 28 fF. The response of the circuit of Fig. 10 is shown in Fig. 11. Before proceeding, just like in the case of filter tuning, we need to record the value of the phase of the reflection coefficient at the centre frequency, 11.72 GHz of the antenna. From Fig. 11, the phase of the reflection coefficient at 11.72 GHz is equal to -114o. Any subsequent patches, with the exception of the last patch, are tuned in an identical way as it was done with filters, i.e. by removing the short circuit of the next patch and tuning the phase of the reflection coefficient until it is altered by +180o.  Since in the present case, the antenna consists of 2 patches and the second (top patch) is last, the procedure consists of varying the value of capacitor C2 in a circuit simulator until the optimal reflection coefficient has been achieved over the broadest bandwidth. In the present case this is achieved when C2 = -56 fF, with the corresponding

Fig. 10 Shorted top patch of antenna of Fig. 7: Top: implementation in circuit simulator and bottom: equivalent circuit implementation with top patch shorted
Fig. 10 Shorted top patch of antenna of Fig. 7: Top: implementation in circuit simulator and bottom: equivalent circuit implementation with top patch shorted

response shown in Fig. 12. As can be seen, the response is much improved compared to the return loss in Fig. 8. However, a closer inspection reveals that the -10 dB bandwidth is moved towards lower frequencies: from 10.47 GHz to 12.67 GHz. This infers that further optimization is likely to yield better results.  Using

Fig. 11 Response of circuit of Fig. 10
Fig. 11 Response of circuit of Fig. 10
Fig. 12 Filter response of Fig. 7 after applying correction to capacitors and C1 (C1 = 28 fF) and C2 (C2 =-35 fF). The remaining set of C12 are set to zero
Fig. 12 Filter response of Fig. 7 after applying correction to capacitors and C1 (C1 = 28 fF) and C2 (C2 =-35 fF). The remaining set of C12 are set to zero

the starting point with C1 = 28 fF, C2=-35 fF and C12 = 0 fF, the return loss was optimized for broadest possible bandwidth (10.6 GHz-12.82 GHz) using correction capacitors as independent parameters. The optimized antenna response is shown in Fig. 13 and yields the following values for the correction capacitors: C1 = 25 fF, C2=-56 fF and C12 = -6.53 fF.  As seen, the response of this antenna is excellent, however, the level of the correction capacitors needs to be reduced. By iteratively applying modifications to the designed antenna of Fig. 7 in accordance with the correction capacitor values, one arrives at the Final tuned antenna design once the values of the correction capacitors become either zero or attain very low values. The exact tolerable low values of these capacitors are dependent on the antenna’s impedance bandwidth, however, it is accepted that for C1….Cn capacitors this is usually reached when the magnitude

Fig. 13 Filter response of Fig. 7 after applying correction to capacitors C1 = 28 fF, C2=-56 fF and C12 = -6.53 fF
Fig. 13 Filter response of Fig. 7 after applying correction to capacitors C1 = 28 fF, C2=-56 fF and C12 = -6.53 fF
Fig. 14 Final antenna response of Fig. 7 after applying correction to capacitors C1 = -1.39 fF, C2=--0.39 fF and C12 = -0.53 fF (red) and without correction capacitors (blue)
Fig. 14 Final antenna response of Fig. 7 after applying correction to capacitors C1 = -1.39 fF, C2=–0.39 fF and C12 = -0.53 fF (red) and without correction capacitors (blue)

of their values is lower than 5 fF and for C12….Cn-,n when the magnitude of their values is lower than 2 fF. The response of the antenna after several iterations to reduce the levels of correction capacitors against the antenna with no correction capacitors is shown in Fig. 14. The values of correction capacitors are C1 = -1.39 fF, C2=-0.39 fF and C12 = -0.53 fF and the exact dimensions of the bottom and top patches are: Xbottom = Ybottom = 6.56 mm, Xto = Ytop = 6.08 mm, with a relative patch displacement of Ydisp = 1.02 mm. As can be seen, the two responses are almost identical. Even though in principle it is possible to continue antenna tuning until the correction capacitors have reached even lower values, such a case is of limited use as it does not provide any additional benefits. The antenna exhibits the -10 dB impedance bandwidth from 10.63 GHz to 12.82 GHz (BW = 2.19 GHz), which is slightly lower (0.1%) than the theoretically predicted impedance bandwidth of 2.21 GHz in (4).

In summary, the design and tuning of a wideband microstrip patch antenna consists of the following steps:

  1. Decide on the type of excitation to be used and fix its position to yield the lowest Qext. This part is dependent on the dielectric characteristics and height of the substrate. Refer to section “External coupling” for details.
  2. Decide on the number of patches – usually going beyond 3 stacked patches is impractical and designs using 2 stacked patches are most common.
  3. Using filter theory [6,7] and the values of Qext found in 1., calculate the coupling coefficients required for broad bandwidth and the desired number of antenna patches. Refer to section “External coupling” for details.
  4. Using the structure of Fig. 5 in conjunction with the time domain solver, infer first order patch displacement necessary for realization coupling coefficients for broad bandwidth. Refer to section “Inter-resonator (patch) coupling” for details.
  5. Simulate the antenna with waveguide and dummy ports attached. For stacked patch antennas one can use either the Momentum engine in ADS [9] or full-wave simulators, such as CST [4] or HFSS [5]. Use at least 4 dummy ports for each patch, ensuring that at least one dummy port is connected to a side of the individual patch.
  6. Export the obtained snp file into a circuit simulator.
  7. Connect ideal capacitors to the dummy ports and standard RF port to the input of the antenna. Also, introduce “cross-connect” capacitors as in Figs. 9 and 10, ensuring that the “cross-connect capacitors” connect frequency tuning capacitors on the side of the respective patches. The capacitors connected directly to the dummy ports control the frequency of operation of the resonator to which they are connected on and “cross-connect” capacitors control the extent of coupling. Set all capacitors to zero.
  8. Short all patches apart from the bottom (fed) patch and run the circuit simulator. Using capacitor C1 tune the reflection coefficient until the dip in the reflection coefficient coincides with the required centre frequency of the antenna. Record the phase of the reflection coefficient at this frequency point.
  9. Remove the short circuit from patch 2 and tune capacitor C2 until the phase of the reflection coefficient changes by +180o.
  10. Repeat steps 8 and 9 until the top patch is reached. Once the top patch is reached, remove its short circuit, and tune the last capacitor, Cn, until the minimum reflection coefficient is obtained across the bandwidth of interest. This concludes the tuning of the directly connected capacitors, C1…. Cn and forms the starting point for the next step.
  11. Using the optimization feature in the circuit simulator optimize the response of the antenna using the capacitors C1…. Cn and C12…. Cn,n. Optimization needs be carried out with regards to the reflection coefficient.
  12. Upon completion of the optimization, use the values of the capacitors to influence the main antenna design, i.e. change its physical characteristics – such as increase or reduce frequency of operation or coupling. If correction capacitors C1…. Cn are greater than zero, reduce the frequency of operation of the concerned patches. Alternatively, increase the frequency of operation. In a similar manner, if capacitors C12…. Cn,n are positive, increase the extent of displacement between the respective patches. Alternatively, reduce the extent of displacement.
  13. Repeat steps 11 and 13 until the correction capacitors become zero or attain very low values. Usually, the criterion is satisfied when the absolute values of C1…. Cn are smaller than 5 fF and when the absolute values of C12…. Cn,n are smaller than 2 fF.

Conclusion

In this article a comprehensive study on how to design and tune stacked microstrip patch antennas is presented. The study shows how one can maximize the impedance bandwidth of a stacked patch antenna in a systematic way by borrowing the theory developed for the design and tuning of RF filters. The study is initiated using basic filter principles applied to a practical antenna design (stacked 2 patch antenna). However, the principles outlined can be equally well applied to almost any antenna design.

References:

[1] Deschamp, G. A., “Microstrip Microwave Antennas,” 3rd USAF Symposium on Antennas, 1953.

[2] Howell, J. Q., “Microstrip antennas,” IEEE AP-S Int. Symp. Digest, 1972, pp. 177-180.

[3] Munson, R. E., “Conformal Microstrip Antennas and Microstrip Phased Arrays,” IEEE Trans. On Antennas and Propagation, vol. AP-22, 1974, pp. 74-78.

[4] https://www.3ds.com/products/simulia/cst-studio-suite

[5] https://www.ansys.com/products/electronics/ansys-hfss

[6] Handbook of filter synthesis, A. Zverev, ISBN 978-0-471-74942-4.

[7] https://www.gwtsoft.com/

[8] https://www.rogerscorp.com/advanced-electronics-solutions/ro3000-series-laminates/ro3003-laminates

[9] https://www.keysight.com/us/en/products/software/pathwave-design-software/pathwave-advanced-design-system.html

 

]]> https://drbulja.com/how-to-design-and-tune-a-microstrip-antenna-1463/feed/ 0 How to design and tune an RF filter? https://drbulja.com/how-to-design-and-tune-an-rf-filter-1440/ https://drbulja.com/how-to-design-and-tune-an-rf-filter-1440/#respond Fri, 16 Feb 2024 11:09:56 +0000 https://drbulja.com/?p=1440 Radio Frequency (RF) filters are very important in all types of telecommunications equipment. They help filter out the unwanted frequencies and only let the frequencies that the communication system is designed for pass through. They are used both in Time Division Duplex (TDD) and Frequency Division Duplex (FDD) systems, however there are some slight differences between them. For example, filters in FDD systems usually need to meet stricter electrical requirements compared to their TDD counterparts – this is simply since in FDD systems the transmit (TX) and receive (RX) channels are frequency separated whereas in TDD systems they occupy the same frequency band but operate at interleaved time intervals. As such, in FDD systems, filters need to be carefully designed to avoid signal leakage between its TX and RX branches. Usually, the leakage from TX to RX is of greater concern due to much higher powers involved with the TX channel. Such problems do not exist with TDD systems as the entire filter’s passband is shared between the TX and RX channels.

Depending on the type of telecommunications system and its filtering requirements, filters come in a variety of flavours. Examples include cavity filters, ceramic filters and Printed Circuit Board (PCB) filters, to name but a few. The choice of the filter for a particular application depends on the electrical and space requirements. Usually, the highest performing filters are ceramic based with Quality (Q) factors of individual resonators up to 5,000. They are closely followed by silver-plated metal cavity filters with individual resonator Q-factors of up to 3,000-3,500. PCB resonators are among the lowest performing resonators with individual resonator Q-factors rarely exceeding 200-300, however, this is strongly dependent on the losses of the PCB substrate.

Any type of RF filter consists of suitably coupled resonators designed and tuned to operate within a certain frequency band, however, a very often posed question is how are they designed and tuned?  There are, of course, several answers to this question and the answer fully depends on the exact filter structure, such as the number of resonators and the number of cross coupling sections. In general, the higher the order of the filter the more challenging filter tuning becomes.

Even though a filter can be tuned using either the time or frequency domain techniques, frequency domain tuning is easier to implement both in commercial circuit simulators, such as ADS [1] and AWR [2] and in practice using Vector Network Analyzers (VNAs), as they may not come equipped with the time domain options. As such, the present article will focus on the frequency domain technique.

General principle

The general principle of filter tuning using the frequency domain technique will be explained with reference to an ideal bandpass filter with no cross coupling, as depicted in Fig. 1. A standard nth order lossless bandpass filter consists of n resonators and n+1 coupling (admittance transformation) sections. The resonators in Fig. 1 are depicted using the parallel connection of capacitors (C1, C2, …Cn) and inductors (L1, L2,….Ln) and the coupling sections with admittance transformers Y1, Y2,…Yn. The exact topology of individual resonators are determined through the filter requirements and as mentioned earlier, can attain the shape of a metal cavity, ceramics or be in the PCB form. Once the resonator topology is determined, it is required to calculate the coupling coefficients or the values of admittance transformers necessary for the correct operation of the filter. The correct inter-resonator coupling coefficients can be calculated using either filter tables [3] or software packages such as Guided Wave Technology’s Filter & Coupling Matrix Synthesis Software [4]. The coupling coefficients at the input/output are calculated in a similar way. The best way to learn how to tune a filter in the frequency domain is best described via an example.

Fig. 1 Realization of a standard nth order bandpass filter
Fig. 1 Realization of a standard nth order bandpass filter

Example – 3-pole filter

Let us design a 3-pole filter with a centre frequency of f0 = 1.8 GHz and with a bandwidth of BW = 40 MHz.

The actual inter-resonator coupling matrix for such a filter obtained using the CMS software [4] is:

Coupling matrices given by (1) and (2) are all that is required to proceed to design a filter. As mentioned earlier, the design of a filter is initiated using an appropriate resonator. For demonstration, let us use a concentric distributed resonator, as described in [5], Fig. 2. The resonator has dimensions of 40 x 40 x 5 mm3, operates at a frequency of 1.8 GHz and has an unloaded Q factor of 1,800. The resonator is assumed to be silver plated. We will start designing the filter by calculating the required coupling coefficients as required by (1) and (2). Here, we distinguish between inter-resonator coupling and external coupling.

Inter-resonator coupling

The actual inter-resonator coupling, as from (1), is equal to k12 = k23 = 0.02016. Effectively, the actual coefficient infers how much energy is coupled from one resonator to another. It is usually calculated using a system of two resonators, such as the one shown in Fig. 3. The resonator structure shown in Fig. 3 has 2 distinct resonant frequencies, which depend on the level of resonator coupling. This was well-explained in [6], however, it will be briefly repeated here. With reference to Fig. 4 (a), let us assume that a resonator in isolation operates at a resonant frequency . The resonant frequency is obtained by setting the input admittance,  to zero. In a similar way, the zeroes of the coupled resonator system of Fig. 4 (b) are obtained by setting its input admittance, , Yin to zero. With reference to this figure,  Yt represents the the admittance transformer with the characteristic admittance of. By setting Yin  to zero, two distinct resonant frequencies are obtained, f1 and f2:

Fig. 2 Concentric distributed resonator
Fig. 2 Concentric distributed resonator
Fig. 3 Two resonator structure used for the calculation of coupling coefficients
Fig. 3 Two resonator structure used for the calculation of coupling coefficients

In this way, the relationship between the equivalent circuit of a filter (Fig. 1) and coupled resonators (Fig. 4 (b)), respectively, with coupling coefficients of (1) and (2) is established.

The resonant frequencies of the two-resonator structure of Fig. 3 can be obtained using an eigenmode solver of any commercially available full-wave resonator, such as Computer Studio, (CST) [7] or Ansys’ HFSS [8].  The coupling between the resonators in the present case is obtained using CST and is presented.

Fig. 4 Equivalent circuit of (a) single resonator and (b) two coupled resonators
Fig. 5 Coupling coefficient as function of wall opening between two resonators of Fig. 3
Fig. 5 Coupling coefficient as function of wall opening between two resonators of Fig. 3

in Fig. 5 as a function of iris opening, W. Given the fact that inter-resonator coupling between two resonators, as inferred from the coupling matrix (1) is 0.02162, implies that the iris opening of, approximately, 8.2 mm should suffice, however, the exact value will be determined later, as elaborated in the section of filter tuning.

External coupling

The excitation of the input and output resonators provides the means for RF energy to enter and exit the filter. The level of excitation is primarily determined by the percentage bandwidth of the filter ()  and the values of the coupling coefficient/admittance transformer. In mathematical terms, the level of external coupling is given by:

Using filter specifications fo=1800 MHz, BW=40 MHz and M01=0.9728 or K01=0.02162, Qext  is determined to be 47.55. This is the value of external coupling that will need to be exhibited at a centre frequency of a filter, f0=1800 MHz).

A filter can be excited in many ways and addressing different ways of excitation is the subject for another topic. For the present design, the first/last resonator is excited using a grounded pin, as shown in Fig. 6. For the calculation of Qext it is not required to simulate the entire filter – only the input/output resonators should suffice. In cases when the input and output coupling coefficients are not the same, the procedure will need to be repeated both for the input Qext_in  and output Qext_out. The resonator of Fig. 6 is excited using a grounded pin, whose proximity to the resonator dictates the correct excitation bandwidth.

Fig. 6 External filter excitation
Fig. 6 External filter excitation
Fig. 7 Calculated values of Q_ext as function of distance, R, as indicated in Fig. 6 (left); and as function of frequency for R = 15.5 mm (right)
Fig. 7 Calculated values of Qext as function of distance, R, as indicated in Fig. 6 (left); and as function of frequency for R = 15.5 mm (right)

In general, the greater the distance between the feed pin and the resonator centre, the higher the loaded . Higher values of the  result in lower excitation bandwidths.

The structure of Fig. 6 was simulated using a full-wave simulator, CST, for its reflection coefficient, .  can be calculated from the reflection coefficient using the following equation:

Qext =fπ(delay (S_11 ))/2 (8)

The values of  simulated for different separation values of R are plotted in Fig. 7.

For the present filter, the required value of external coupling is Qext =47.55 ,which, according to Fig. 7, is achieved by placing the feed pin at approximately R = 15.5 mm from the resonator centre. Having determined all the required coupling coefficients, we can now proceed to design the 3-pole filter.

Filter design and tuning

The designed filter is shown in Fig. 8. As per design requirements, the filter has 3 resonators and 2 coupling irises. The widths of the coupling irises is set to 8.2 mm, as inferred from Fig. 5 to obtain the coupling coefficient of 0.02016. In a similar way, the feed pins are positioned at a distance of R = 15.5 mm in order to obtain the loaded Qext =47.55 as required to correctly excite the filter. The filter is excited using 2 waveguide ports, positioned at the input/output. In addition to the waveguide ports, at the top of each resonator high impedance discrete ports are added. These ports play a major role in the tuning of the filter at a post-processing (tuning) step.

The initial simulated results of the filter as shown in Fig. 8 are shown in Fig. 9.  As can be seen from Fig. 9, the initial response of the filter seems very close to the required characteristics, however, a closer.

Fig. 8 Designed three-pole filter
Fig. 8 Designed three-pole filter
Fig. 9 Initial response of three-pole filter of Fig. 8
Fig. 9 Initial response of three-pole filter of Fig. 8

Inspection reveals that the reflection coefficient at the lower passband edge is higher than required -16 dB. This infers that the filter is not fully tuned, requiring fine tuning of both the resonant frequencies of the individual resonators and, also, the coupling coefficient. However, the question is which resonators and which coupling irises require tuning? This is impossible to predict by looking at the response of Fig. 9 alone. At this point, we introduce frequency domain tuning.

It was mentioned earlier in the text that dummy ports are introduced at the top of each resonator. These dummy ports will need to be assigned a high impedance in order not to interfere with the distribution of the Electro-Magnetic field inside the resonator.

Frequency domain tuning

The first step towards frequency domain tuning lies with full-wave simulation of the filter of Fig. 8 with dummy ports attached, using commercially available software such as CST [7] or HFSS [8]. The port impedance of the dummy ports needs to be kept very high – in the present case, the port impedance is set to 100 GΩ. In the present case, the filter was simulated using CST. After the full wave simulation, the resultant s5p files are imported into a circuit simulator, as schematically depicted in Fig. 10. Waveguide (input/output) ports or ports 1 and 2 are terminated in 50 Ω impedance, whereas the dummy ports are terminated in ideal capacitors, C1, and C2. The C12 capacitor cross-connects capacitors C1, and C2. These capacitors are referred to as correction capacitors as their values are used to infer what type of correction (if any) needs to be applied to the physical filter of Fig. 8. Here, filter symmetry has been utilized to yield identical values for the correction capacitors attached to dummy ports 1 and 3 (C1). Correction capacitor C2 is attached to resonator 2. Correction capacitors C1 and C2 provide information with regards to the frequency or operation of the resonator to which they are attached. For example, if these capacitors are positive, this infers that the frequency of operation of the resonator needs to be reduced. Similarly, if their values are negative, the frequency of operation of the resonator needs to be increased. With regards to the cross capacitor, C12, its positive values infer that the coupling iris width needs to be reduced, while its negative values require the coupling iris to be widened. The coupling between resonators 1 and 2 and 2 and 3 is identical (as also seen from the coupling matrix (1) and (2)) and this is reflected in the identical value for the correction coupling capacitor, C12.

Frequency domain tuning is initiated by shorting all resonators expect the first resonator in the filter, Fig. 11 (top). Its circuit implementation is shown in Fig. 11 (bottom).

Fig. 10 Circuit representation of design filter of Fig. 8 with dummy ports
Fig. 10 Circuit representation of design filter of Fig. 8 with dummy ports

The response of the circuit of Fig. 11 is shown in Fig. 12. Since the dip of the reflection coefficient corresponds to 1.8 GHz, it can be inferred that the frequency of the first resonator (due to filter symmetry,

the third resonator also) is correctly tuned. We can now proceed to tune the second resonator. Before proceeding, we need to record the value of the phase of the reflection coefficient at the centre frequency of the filter. From Fig. 12, the phase of the reflection coefficient is equal to -88o. The tuning of the second resonator is initiated by removing the short circuit and adjusting the value of capacitor C2 until the phase of the reflection coefficient has been altered by + 180o. Given the fact that the phase of the reflection coefficient when the second resonator is shorted is – 88o, capacitor C2 is adjusted until the phase of the reflection coefficient becomes – 88o + 180o = 92o. This is achieved when the correction capacitor C2 attains the value of -3.6 fF. Removing the short circuit from resonator 3 (dummy capacitor value at this resonator

Fig. 11 Shorted resonators 2 and 3 of filter of Fig. 8: Top: implementation in circuit simulator and bottom: equivalent circuit implementation with resonators 2 and 3 shorted
Fig. 11 Shorted resonators 2 and 3 of filter of Fig. 8: Top: implementation in circuit simulator and bottom: equivalent circuit implementation with resonators 2 and 3 shorted
Fig. 12 Response of circuit of Fig. 11
Fig. 12 Response of circuit of Fig. 11

is identical to the dummy capacitor value on resonator 1) and leaving C12 = 0, the filter response using a circuit simulator of Fig. 13 is obtained. As can be seen, the response is much improved as the return loss now stands below -16 dB across the entire passband. However, a closer inspection reveals that a more symmetrical response is possible. Using the starting point with C1 = 0 fF, C2=-3.6 fF and C12 = 0 fF, the return loss was optimized using correction capacitors as independent parameters. The optimized filter response is shown in Fig. 14 and yielded the following values for the correction capacitors: C1 = 0.18 fF, C2=-4.2 fF and C12 = -0.31 fF. As can be seen, the of this filter is excellent. By iteratively applying the modifications to the designed filter of Fig. 8 in accordance with the correction capacitor values, one arrives at the final, tuned filter design once the values of the correction capacitors become either zero or very low values. For C1….Cn capacitors, this is reached when the magnitude of their values is lower than 1 fF and for C12….Cn-,n when the magnitude of their values is lower than 0.5 fF. The response of the filter after several iterations to reduce the levels of correction capacitors vs the filter with no correction capacitors is shown in Fig. 15. The values of correction capacitors are C1 = 0.35 fF, C2=-0.91 fF and C12 = 0.23 fF.  As can be seen, the two responses are almost identical. Even though in principle it is possible to continue tuning until the

Fig. 13 Filter response of Fig. 8 after applying correction to capacitor C2 (C2 =-3.6 fF). The remaining capacitors C1 and C12 are set to zero
Fig. 13 Filter response of Fig. 8 after applying correction to capacitor C2 (C2 =-3.6 fF). The remaining capacitors C1 and C12 are set to zero
Fig. 14 Filter response of Fig. 8 after applying correction to capacitors C1 = 0.18 fF, C2=-4.2 fF and C12 = -0.31 fF
Fig. 14 Filter response of Fig. 8 after applying correction to capacitors C1 = 0.18 fF, C2=-4.2 fF and C12 = -0.31 fF
Fig. 15 Final filter response of Fig. 8 after applying correction to capacitors C1 = 0.35 fF, C2=-0.91 fF and C12 = 0.23 fF (red and black) and without correction capacitors (green and purple)
Fig. 15 Final filter response of Fig. 8 after applying correction to capacitors C1 = 0.35 fF, C2=-0.91 fF and C12 = 0.23 fF (red and black) and without correction capacitors (green and purple)

correction capacitors have reached even lower values, such a case is of limited use as it does not provide any additional insights. Furthermore, in practice tuning will almost always be performed using tuning and coupling screws, which have practical limits with regards to tuning accuracy.

In summary, frequency domain filter tuning consists of the following steps:

 

  1. Simulate a filter with waveguide and dummy ports attached. For planar filters one can use the Momentum engine in ADS and for 3D filters one can use full-wave simulators, such as CST [7] or HFSS [8].
  2. Export the obtained snp file into a circuit simulator
  3. Connect ideal capacitors to the dummy ports and standard RF ports to the input/output of the filter. Also, introduce “cross-connect” capacitors as in Figs. 10 and 11. The capacitors connected directly to the dummy ports control the frequency of operation of the resonator to which they are connected on and “cross-connect” capacitors control the extent of coupling. Set all capacitors to zero.
  4. Short all resonators apart from resonator 1 and run the circuit simulator. Using capacitor C1 tune the reflection coefficient until the dip in the reflection coefficient coincides with the centre frequency of the filter. Record the phase of the reflection coefficient at this frequency point.
  5. Remove the short circuit from resonator 2 and tune capacitor C2 until the phase of the reflection coefficient changes by +180o.
  6. Repeat steps 4 and 5 until the last resonator is reached. Once the last resonator is reached, remove its short circuit, and tune the last capacitor, Cn, until the maximum transmission coefficient is obtained. This concludes the tuning of the directly connected capacitors, C1….Cn and forms the starting point for the next step.
  7. Using the optimization feature in the circuit simulator optimize the response of the filter using the capacitors C1….Cn and C12….Cn-,n. Optimisation can be carried out with regards to the reflection or transmission coefficients and, sometimes, depending on the filter requirements, both.
  8. Upon completion of the optimization, use the values of the capacitors to influence the main filter design, i.e. change its physical characteristics – such as increase or reduce frequency of operation or coupling. If correction capacitors C1….Cn are greater than zero, reduce the frequency of operation of the concerned resonators. Alternatively, increase the frequency of operation. In a similar manner, if capacitors C12….Cn-,n are positive reduce the extent of coupling (usually by decreasing the size of the irises). Alternatively, increase the size of the irises.
  9. Repeat steps 7 and 8 until the correction capacitors become zero or attain very low values. Usually, the criterion is satisfied when the absolute values of C1….Cn are smaller than 1 fF and when the absolute values of C12….Cn-,n are smaller than 0.5 fF.

    Conclusion

    In this article a comprehensive study on how to design and tune an RF filter is presented. The study was initiated using basic filter principles applied to a practical filter design (three-pole filter). However, the principles outlined can be equally well applied to almost any filter design.

     

    References:

    [1] https://www.keysight.com/us/en/products/software/pathwave-design-software/pathwave-advanced-design-system.html.

    [2]  https://www.cadence.com/en_US/home/tools/system-analysis/rf-microwave-design/awr-microwave-office.html.

    [3] Handbook of filter synthesis, A. Zverev, ISBN 978-0-471-74942-4.

    [4] https://www.gwtsoft.com/

    [5] S. Bulja, E. Doumanis and D. Kozlov, “Concentric distributed resonators and filters”, IEEE Radio and Wireless Symposium, Anaheim, California, US, 2018, pp. 267-269.

    [6] S. Bulja and M. Gimersky, “Low profile distributed cavity resonators and filters,” in IEEE Trans. Microwave Theory and Tech., vol. 65, issue 10, pp.3769-3779, 2017.

    [7] https://www.3ds.com/products/simulia/cst-studio-suite

    [8] https://www.ansys.com/products/electronics/ansys-hfss

     

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