Excel stuff.

02 Oct 2023

This blog pages is a summary of some useful Excel spreadsheets I made:

• Hilbert transform demo [French]: Hilbert-demo.xlsx.
• Hilbert transform demo [English]: Hilbert-demo-EN.xlsx.

LC ladder impedance matching.

25 Jun 2023

This blog page is an English translation and adaptation of a part of my PhD thesis. Numbers in brackets refers to the original bibliography, they will be replaced in a future revision.

Impedance matching is performed by LC ladder networks. This method allows to synthesize low impedances (around 5 Ω) on the same PCB than the standard 50 Ω output (no need for a second PCB with high permittivity). Moreover, this method is more compact than quarter-wave transformer.

Exact value calculation was performed by numerical optimization. Manual calculation would be too difficule because the output impedance of the transistor is not a pure resistance¹. However, numerical optimization needs to know the number of components of the ladder, because the ADS optimizer is not able to add components when needed, but is only able to determine their value. Moreover, an initial estimate of the values of the components of the LC ladder is useful for the optimizer to converge quicker towards the solution. Calculation method is the one of [84], adapted for the needs of the PhD thesis.

Inductors and capacitors are assumed ideal and lossless, as well as the microstrip junctions. The effets of the polarization networks of the transistors are also ignored. Such effects are absolutely not negligeable, but will be easily corrected by the numerical optimizer in the final phase of the design.

A simple empirical method is commonly used [32], but it doesn’t allows a priori calculation of the order and of the mismatch of the matching network.

In [116] and [29], tables of [84] are used to calculate a low-pass matching network of Chebychev type. Unfortunately these tables does not provide values for very broadband impedance matching network (1:6 ratio for the amplifier module of the PhD thesis).

For these reasons this page describes in detail the calculation of such impedance matching networks. The calculation method is the one of [84], adapted for the needs of the PhD thesis.

As usual, f is the usual frequency in s^-1 and \omega the angular pulsation in rad \cdot s^-1. Calculations will use mainly \omega.

In a first time, the matching network is calculated for the center frequency \omega_m=1 and source impedance R_S=1. This normalization is not mandatory, but allows to compare intermediate results with those of [84] to test the good operation of the Python program which was written during the PhD thesis.

The reflexion coefficient of a LC ladder (output) matching network of type Chebychev, seen from the source, is²:

|\Gamma|^2 = (\epsilon^2\cdotT_n^2((\omega^2-\omega_0^2)/(\Delta\omega^2)))/(1+\epsilon^2\cdotT_n^2((\omega^2-\omega_0^2)/(\Delta\omega^2)))

with \omega_0=sqrt(\omega_a+\omega_b), \omega_a the beginning of the passband, and \omega_b the end of the passband.

Next figure shows the reflection coefficient seen from the source of an example of an (output) LC matching network going from 5 Ω towards 50 Ω from 1 to 2,5 GHz. These values are approximately those of the first wideband amplifier of the PhD thesis.

In previous expression, \epsilon is chosen such as:

|\Gamma(f=0)|^2=((Z_2-Z_1)/(Z_2+Z_1))^2

This last condition is needed because LC ladders have no effect in DC. So, the transfert function is entirely determined by the order and the &&Z_2/Z_1&& ratio.

In the passband, maximum reflection coefficient and maximum insertion losses are respectively:

|\Gamma_max|^2=(\epsilon^2)/(1+\epsilon^2) |S_(21,min)|^2=1/(1+\epsilon^2)

The first step of the calculation is to determine the first n such as |\Gamma_max|^2 is less than the requirements. This calcul is done numerically, by testing all the integers n from 1 until this requirement is met.

This n is half the number of elements of the final network [84].

Next, variable change p = j · ω is performed. This variable change enables to simplify greatly the calculations to come. Then, the square of the magnitude of the reflection coefficient is factored as such:

|\Gamma(p)|^2 = (a(p) \cdot a(-p)) / (b(p) \cdot b(-p))

with a and b two polynomials whose roots have negative real parts³.

With this factorization, reflection coefficient (and not only his squared norm) can be calculted as such:

\Gamma(p) = (a(p)) / (b(p))

At the beginning of our work on the subject, factorization was performed numerically. This method was thereafter discarded due to numerical instability problems for high orders. This is why a semi-analytic method was taken, inspired by [47, 84]. Roots of the numerator and of the denominator are calculated analytically. Next, factorized polynomianls are calculated by taken only roots with negative real parts.

The calculation, more long than complex, won’t be detailed. The roots of the numerator and of the denominator are given by the following formulas:

{: ( +-j sqrt(Delta omega^2 \cdot cos((pi)/(2 \cdot n) \cdot (1 + 2 \cdot k))) , k in [1, n] ), ( +-j sqrt(Delta omega^2 \cdot cos(1/n \cdot arccos(j/epsilon))) , k in [1, 2 \cdot n] ) :}

The first equation give directly the set of needed roots, since the numerator has double imaginary roots. However, negative real part roots need to be selected from all the roots given by second equation. This point is easily done numerically.

A polynomial is defined by the set of its roots, but up to a multiplicative factor. The next step is to determine this multiplicative factor. Details of the calculation won’t be given here, but only the result:

{: ( a = a_1/(a_1(0)) \cdot abs(epsilon \cdot cos(n \cdot arccos(-omega_0^2/(Delta omega^2)))) ), ( b = b_1/(b_1(0)) \cdot sqrt(1 + (epsilon \cdot cos(n \cdot arccos(-omega_0^2/(Delta omega^2))))) ) :}

with &&a_1&& and &&b_1&& the polynomials initially determined.

Next, the input impedance, normalized⁴ with respect to Z1, is calculated as follows:

Z(p) = (b(p) + a(p))/(b(p) - a(p))

This impedance is then expanded into a continued fraction through successive divisions:

Z(p) = g_1 \cdot p + 1 / (g_2 \cdot p + 1/ (g_3 \cdot p + ... + 1 / (g_m \cdot p + g_(m+1))))

This expression immediately leads to an LC network. The odd gm values are the normalized values of the inductances, while the even gm values are the normalized values of the capacitances. This denormalization is performed according to the following equations⁵:

{: ( L = g / (2 pi f_0) \cdot Z_1 ), ( C = g / (2 pi f_0) \cdot 1 / Z_1 ) :}

The last &&g_m&& is the load resistance, which is also normalized. Its value has been known for a long time, but it can be interesting to recalculate it to verify that there is no significant error due to numerical inaccuracies.

On group delay of antennas.

10 Jul 2022

Many thanks, in the order of appearance in the LinkedIn discussion, to Dr. Pierre-Antoine Garcia, Theunis Beukman, Benoit Derat, Hüseyin Yiğit, Andreas Barchanski for the insightful LinkedIn discussion which led to this post.

Recently on LinkedIn, a fellow colleague asked whether the group delay of an antenna could be calculated by the simulated complex gain. Sure it can, but with the right precautions.

What is tried to be measured ?

A reasonable asumption when dealing with an antenna is that there will be an other antenna facing it. The transmission coefficient S21 between the two antennas can be written as follows, using the Friis transmission equation https://en.wikipedia.org/wiki/Friis_transmission_equation :

S_{21} = ubrace{G_a \cdot G_b}_{"Gains."} \cdot \lambda / {4 \cdot \pi} \cdot ubrace{1 / d}_{:("Free space"),("attenuation."):} \cdot ubrace{e^{-j \cdot 2 \cdot \pi \cdot d / \lambda}}_{:("Distance"),("phase shift."):}

Note this form is slightly different from the usual one, since the amplitude is taken instead of the power. Rearranging in function of the frequency gives:

S_{21} = G_a \cdot G_b \cdot {c} / {4 \cdot \pi \cdot f} \cdot 1 / d \cdot e^{-j \cdot 2 \cdot \pi \cdot d / c \cdot f}

For the group delay, only phase matters, hence:

"phase"(S_{21}) = "phase"(G_a) + "phase"(G_b) - 2 \cdot \pi \cdot d / c \cdot f

Group delay is calculated using the derivative of the phase by \tau_g = - 1 / (2 \cdot \pi) \cdot (d)/(df) "phase"(S_{21}) , which leads to ask which terms are constant with frequency and which terms changes.

Phase center of antennas

A non-trivial varying term is the distance between the antennas. Antennas are often big enough compared to the wavelength to forbid the use of a random point to calculate the phase shift from the distance. The right point to take is the phase center, which is the center of the spherical wavefronts at infinity. The phase center depends on the direction, but most importantly for group delay calculations, it depends on frequency¹.

The group delay formula can be rewritten taking this into account:

"phase"(S_{21}) = "phase"(G_a) + "phase"(G_b) - 2 \cdot \pi \cdot (d_"geom" - \Delta_1 - \Delta_2) / c \cdot f

were d_"geom" is the geometrical distance between the antennas and \Delta_1 and \Delta_2 are the positions of the phase centers relative to the geometrical reference point. This reference can be any point. The sign of \Delta_1 and \Delta_2 is taken positive when the phase center gets closer to the other antenna.

Since the interest in only in the first antenna and not in the second, this formula can be normalized by taken all parameters of the second antenna to reference values: unit gain, phase center coinciding with geometrical center, and zero distance. The meaning of the last hypothesis is important to emphasize: the physical distance cannot be zero, it’s just a way to separate the effects depending of the antenna from the rest. So:

"phase"(S_{21}) = "phase"(G_a) + 2 \cdot \pi \cdot \Delta_1 / c \cdot f

Knowing which terms are constant and which are not, the derivative can be calculated:

\tau_g = - 1 / (2 \cdot \pi) \cdot (d)/(df) [ "phase"(G_a) + 2 \cdot \pi \cdot \Delta_1 / c \cdot f ]

This last equation shows that not only the phase of the gain must be taken into account, but also the move of the phase center.

Conclusion

The group delay of an antenna can be calculated using simulation results, but the total phase shift change must be taken into account, including the move of the phase center.

Design of a coaxial power combiner with low-impedance inputs and increased isolation.

11 Apr 2022

Hadrien Theveneau, Christophe Gaquière, Romain Lenglet, Matthieu Werquin, Jean-Christophe Joly, and Stéphane Tortel

This post is a mix between an unpublished long article, an article published in IEEE MWCL¹, and further additions. Original articles by the authors in front of this post, revisions by the first author.

This article describes the design, fabrication and measurement of a 2.5 Ω, 8-way, 1 – 6 GHz spatial power combiner using an absorbing material to increase the isolation. Insertion losses are lower than 1.8 dB in the 1 – 6 GHz band, except for a few peaks. Isolation is at least 10 dB for 45° input pairs and better than -15 dB for other pairs. This is the first power combiner to provide wide bandwidth, high isolation, and low input impedances at the same time.

Introduction

More and more applications need to generate very high power pulsed microwave signals, in the order of tens of kilowatts: radars, EM warfare, and so on. These powers are traditionally generated with hyperfrequency tubes: magnetrons, klystrons, TWT, and so on. However, these techniques can have several major drawbacks: poor reliability, short lifespan, fragility, complex waveforms generation difficult and difficult power supplies.

Gallium nitride (GaN) transistors can provide a solution to these problems. However, the output power of a single transistor unit is insufficient for such applications. Efficient power combining schemes must be used to provide needed output power.

However, current solutions have limitations. Spatium^® power combiners^2,3 confine the amplifiers inside the structure, which makes thermal cooling difficult. Combiners using probes in linear waveguides⁴, coaxial waveguides^5,6, or radial waveguides⁷ have low losses but their bandwidth is lower than 50 % and are not practical for lower frequencies like 1 GHz. These solutions also lack isolation between the inputs, which can lead to instability and failure propagation problems. Waveguide combiners⁸ have low losses and isolate their inputs, but they also have bandwidth limitations, and are bulky for lower frequencies. All of the previously mentioned solutions have high-impedance inputs (50 Ω), difficult to match on the low impedances (≈ 2.5 Ω) of GaN power transistors on large bandwidths

In this article, an innovative spatial power combiner with high bandwidth, isolation between its inputs and low input impedance is designed, manufactured, and measured. Section I presents a theoretical analysis of the lack of isolation of traditional power combiners, section II presents the general architecture of this new combiner, section III presents equivalents schematics in even and odd modes, section IV shows the structure of impedance prematching used to have low-impedances inputs, section V gives the complete procedure to calculate the shape of the impedance transition, and section VI discusses measurement and simulation results.

I. Even and odd modes analysis of power combiners

We will show in this section that the lack of isolation of power combiners without an isolation mechanism is linked to the reflection of the odd modes.

Fig. 1. shows the even and odd mode excitations of a power combiner without isolation mechanism. In the even mode, the power of the inputs goes to the output. If the impedance matching is correct, no output is reflected to the inputs and they don’t see each other. However, in symmetrical odd modes, due to symmetry, the output is connected to a virtual ground. The input waves are totally reflected. This reflection is the cause of the lack of isolation between the inputs. To provide isolation between the inputs, we need to find a way to absorb this reflection. Similar analysis is done in [⁹] and [¹⁰] for combiners using several discrete resistors.

Mathematically, for a combiner with rotational symmetry, the input modes matrix is given by:

M = \begin{pmatrix} \; 1 & 1 & 1 & \dots & 1 \\ \; 1 & e^{\frac{j 2 \pi}{N}} & e^{\frac{j 4 \pi}{N}} & \dots & e^{\frac{j 2 (N - 1) \pi}{N}} \\ \; 1 & e^{\frac{j 4 \pi}{N}} & e^{\frac{j 8 \pi}{N}} & \dots & e^{\frac{j 4 (N - 1) \pi}{N}} \\ \; \vdots & \vdots & \vdots & \ddots & \vdots \\ \; 1 & e^{\frac{j 2 (N - 1) \pi}{N}} & e^{\frac{j 4 (N - 1) \pi}{N}} & \dots & e^{\frac{j 2 (N - 1)^2 \pi}{N}} \\ \end{pmatrix}

where N is the number of inputs. The first column is the normal combining mode, the even mode, while the other columns are the odd modes.

For example, for 2 and 4 inputs, M is given by:

M_2 = \begin{pmatrix} \; 1 & \phantom{-}1 \\ \; 1 & -1 \\ \end{pmatrix} \qquad M_4 = \begin{pmatrix} \; 1 & \phantom{-}1 & \phantom{-}1 & \phantom{-}1 \\ \; 1 & \phantom{-}j & -j & -1 \\ \; 1 & -1 & -1 & \phantom{-}1 \\ \; 1 & -j & \phantom{-}j & -1 \\ \end{pmatrix}

The inverse of M is given by:

M^{-1} = \begin{pmatrix} \; 1 & 1 & 1 & \dots & 1 \\ \; 1 & e^{\frac{- j 2 \pi}{N}} & e^{\frac{- j 4 \pi}{N}} & \dots & e^{\frac{- j 2 (N - 1) \pi}{N}} \\ \; 1 & e^{\frac{- j 4 \pi}{N}} & e^{\frac{- j 8 \pi}{N}} & \dots & e^{\frac{- j 4 (N - 1) \pi}{N}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \; 1 & e^{\frac{-j 2 (N - 1) \pi}{N}} & e^{\frac{-j 4 (N - 1) \pi}{N}} & \dots & e^{\frac{-j 2 (N - 1)^2 \pi}{N}} \; \\ \end{pmatrix}

By splitting the input excitation in even and odd modes, the isolation can be calculated as following :

S_{l,k} = \sum_{m = 1}^{N}{\frac{1}{N} e^{\frac{j 2 (l - 1) (m - 1) \pi}{N}} \Gamma_m e^{\frac{- j 2 (k - 1) \pi}{N}}}, \qquad l \neq k

where Gamma_m is the reflexion coefficient for the odd mode number m. This equation shows that, when the odd mode matching gets better, the isolation also does. It should be noted that this equation has the same rotational symmetry than the power combiner.

II. Overview of the combiner

The mechanical layout of the power combiner is presented in Fig. 2. and Fig. 3. Standard 50 Ω connectors are present for measurement purposes, but this combiner is designed for 2.5 Ω inputs. These connectors will be eventually replaced by amplifier modules with low impedance outputs.

Impedance prematching networks raise the input impedance from 2.5 Ω to 50 Ω. This provides a sufficiently high impedance at the start of the combining section to avoid short distances between the inner part of this section and the outer part. Without the prematching, this distance would be on the order of 120 µm, which is very difficult to manufacture. With the prematching, this distance is between 0.8 and 1.9 mm, which is much easier to manufacture.

The actual combining¹¹ begins after the prematching. The first part of the combiner is a coaxial structure containing several copper lines around an absorber material. This provides isolation between the inputs by adding some loss mechanism for the odd modes. Without this, the odd modes would be reflected back to the inputs, and this would cause the poor isolation found in lossless combiners. This absorber is similar to the resistor in Wilkinson combiners^12,13,14,9, but its distributed operation provides an higher bandwidth.

Inside the absorber, there is a metal core connected to the body of the combiner. This provides a return path for the heat generated into the combiner. This metal core increases insersion losses of the combiner, particularly in the low frequency range, but this effect is low. Without this return path, the combiner would not be able to sustain high powers. Additionally, this structure allows mechanical sustaining.

The shape of the combining section is precisely calculated to ensure impedance matching between the inputs and the output. This calculation will be described in section V

At the exit of the combining section, there is one transition between air and PEHD, and one transition in diameter. No impedance matching is performed in these transitions: they have no electric function and are purely mechanical. The PEHD is here to ensure the positioning of the core inside the coaxial structure. The output coaxial connector alone would be insufficient to provide enough mechanical strength. Both transitions were optimized in separate EM simulations.

A CAD of the combiner with the power amplifier modules is shown in Fig. 4. A dummy circuit is shown in the figure to make mechanical CAD easier. Special care was taken with the cooling of the modules because the average lifespan of a GaN transistor is divided by 2 for each 6~°C increase in temperature. Unlike Spatium^® power combiners^2,3, where the power amplifiers MMICs are trapped inside the structure, the transistors are mounted on the outer part and thus can be more easily cooled. The overall structure is contained in a 333×322×286×mm³ volume.

III. Equivalent schematics in even and odd modes

The simplest method to analyse this power combiner is to use two separate schematics for the even mode and the odd modes, as shown in Fig.~\ref{fig-sch-even-odd-combiner}.

The prematching section behaves in exactly the same way in the even and the odd modes. Its equivalent schematic is a transmission line of continuously variable impedance. Its structure will be detailed in Fig. 7.

In the even mode, as shown in Fig. 5a, the strips have the same potential and almost no field goes into the absorber. The lines have low losses and the even mode is transmitted from point A to point B. The rest of the combiner, made with a full metal core, completes the impedance transformation, which was started in the prematch section. The heat spreader behaves approximately like a shunt inductor. This mode is the normal combing mode.

In contrast, in the odd modes, as shown in Fig. 5b and Fig. 5c, the strips do not have the same potential and lots of electric field goes into the absorber. The lines have high losses and the odd modes are highly attenuated from point A to point B.

In the odd modes, due to symmetry, point B behaves like a virtual ground. The heat spreader can be removed from the odd modes’ equivalent schematic because it is connected to the virtual ground. Without the absorber, the odd mode would be reflected towards the inputs. This reflection would cause a lack of isolation. With the absorber, the reflection of the odd modes is strongly attenuated, which increases isolation between the inputs. This absorber has the same function as the resistor in Wilkinson combiners^12,13,14,9, but its distributed operation provides higher bandwidths.

IV. Impedance prematch

This combiner has an impedance preadaptation from 2.5 Ω to 50 Ω to make the manufacturing easier. This adaptation cannot be performed with a microstrip line on substrate with constant thickness. The 2.5 Ω low impedance side has a limited width because, otherwise, higher order modes would propagate. Using high-k substrates is not a solution because they would reduce the maximal width relative to the propagation of higher order modes. This forces the use of a thin substrate. However, a thin substrate forces the use of a narrow strip on the 50 Ω high impedance side, limiting the power handling and increasing the losses.

We need some way to change the thickness of the substrate. Using multiple substrates can be difficult for the manufacturing. A continuous variation of the thickness of the substrate would be difficult to manufacture. This is why we use a defective ground plane^16,17,18,19. Fig. 7. shows the structure of the impedance preadaptation. The ground plane is progressively opened to increase the “effective substrate height”. The ground plane is opened before the strip width is reduced to have minimal losses. Impedance in function of dimensions is computed using the mode solver of CST Microwave Studio^®.

V. Calculation of impedance transition

After the manufacturing of this combiner, the author learned that instead of a Klopfenstein transform followed by a postprocessing to remove the discontinuities, an Hecken transform should have been used instead. Hence, the first section presents what was done and the seconde section presents what should have been done instead.

V.I. Current calculation with Klopfenstein

A Klopfenstein impedance transition^{20,21,22,23,24,25} is used for this power combiner. The impedance preadaptation and the combining sections together make a single Klopfenstein impedance transition. Putting two Klopfenstein transitions in series would have been easier to calculate, but this would have increased the length and the losses of the overall structure.

The preadaptation section is the 2.5 Ω to 50 Ω part of a Klopfenstein transition going from 2.5 Ω to 8×50=400 Ω. The combining section is the 50/8=6.25 Ω to 50 Ω part of a 2.5/8=0.31 Ω to 50 Ω Klopfenstein transition. Both transitions are identical besides a multiplication of the impedance by 8, the number of the lines. This is because, in the prematching section, the impedance is the impedance of a single line, while in the combining section the impedance is the common-mode impedance of all the lines in parallel. Klopfenstein transitions are much discussed in literature^{20,21,22,23,24,25}, but some points deserve special attention. The following procedure is used for the calculation:

1) Impedances Z(y) are calculated in function of the normalized position y in range [-1;1] using standard formulas for Klopfenstein transitions^{20,20,21,22,23,24,25}.

2) An affine transformation was performed on ln(Z_0) to remove start and end discontinuities.

3) The electrical length of the transition, theta_min is calculated. Usual formulas^{20,21,22,23,24} in literature are no longer valid because the small-reflection hypothesis^20,21 is not valid outside the passband due to the high impedance transformation ratio. This ratio is 160 for this combiner, while it is only 32 in Spatium^® power combiners and 1.5 in original Klopfenstein paper.

theta_min is calculated by a numerical search of the smallest theta such as rho(theta) < rho_max, where rho(theta) is calculated by numerical integration for y from 1 to -1 of the following differential equation:

\begin{equation*} \frac{\mathrm{d}\rho}{\mathrm{d}y} = j \cdot \Theta \cdot \rho - \frac{1}{2} \cdot \left(1 - \rho\right) \cdot \frac{\mathrm{d}\ln\left(Z_0\right)}{\mathrm{d}y} \end{equation*}

4) The calculated impedances Z_0(y) are cut from 2.5 Ω (0.31 Ω) to 50 Ω (0.63 Ω) for the prematching and from 50 Ω (0.63 Ω) to 400 Ω (50 Ω) for the remaining of the combiner.

5) A table of the Z_0 and K_{eff} in function of the transverse dimensions is calculated by EM simulation. The transverse dimensions is the ground plane opening for the first part of the preadaptation section, the microstrip line width for its second part, and the inner part diameter for the coaxial part. CST Microwave Studio^® is used for this calculation.

6) Profile transverse dimensions are calculated from Z_0(y) by numerical interpolation of the table calculated in step 6. Effective}dielectric constant K_{eff}, which depends on the profile dimensions, is calculated in the same step.

7) Position z is calculated from K_{eff}(y) with:

z= \int\limits_{y=-1}^{1}{\frac{\Theta_\text{min} \cdot c}{4 \cdot \pi \cdot f_\text{min} \cdot \sqrt{K_\text{eff}(y)}} \cdot \mathrm{d}y} \label{eqn-z}

by numerical integration. Fig. 8. shows the variation of the different parameters of the transition.

8) The number of points is reduced from 2001 to fewer than 20 points using an iterative end-point fit algorithm. This simple step is very important. Without this step, the mesher would not be able to mesh the structure for numerical simulation. This also helps manufacturing a lot.

V.I. Correct calculation with Hecken method

Instread of a modified Klopfenstein transition, the calculation method can be updates to use an Hecken transition. For this, at step 1, Z(y) should be calculated using the Hecken method.

VI. Simulation and measurement results

To validate the performance of the design before manufacturing, we perform a global electromagnetic simulation. The model includes the initial microstrip lines, the impedance prematching, the combiner itself and the final diameter reduction. SMA input connectors are not considered in this simulation because they are used for measurements only and should be removed for use with 2.5 Ω power amplifier modules. Their effect is removed from the measurement results. The output connector is also not considered because its effect is low compared to the other losses.

Separate CAD models are used for the mechanical manufacturing and the electromagnetic simulation. The use of two separate models is mandatory because the electromagnetic model and mechanical model have different requirements. For example, the screws must be included in the mechanical model, but they are ignored in the electromagnetic model.

The simulation is performed with CST Microwave Studio^®. We use the frequency domain solver with tetrahedral meshing, automatic meshing and meshing adaptation. This simulation and meshing method are used because they allow both small and large features to be considered in the same simulation. The smallest features are present in the input microstrip lines, made on 127 µm substrate, and largest are present at the end of the combining, whose diameter is 30 mm.

Measurements of this power combiner are very difficult because the inputs are wide microtrip lines with low-impedances. Neither commercial connectors nor measurement instruments can directly connect to such inputs. For the measurements, we manufactured and measured a test version with standard 50 Ω SMA female connectors as inputs.

This test version is measured with a standard 50 Ω vector network analyser. The measurement setup is shown in Fig. 9. It is made two ports at a time with the remaining ports loaded with matched 50 Ω loads. The N to SMA transition at the 50 Ω output is simply ignored. A Python script using scikit-rf library is used to combine the multiple Touchstone 2-port files into a single 9-port file. The &&S_{i,j}&& coefficients measured several times, e.g. &&S_{1,1}&&, are averaged between all measurements. De-embedding and impedance renormalization are performed with Keysight Advanced Design System.

The transitions from the 50 Ω SMA coaxial connectors to the 2.5 Ω wide microstrip lines, present in the test version of the combiner, need to be desembedded. This desembedding is done using the simulated EM model of the transition shown in Fig. 10.

Measurement results after desembedding and impedance renormalization are shown in Fig. 11, Fig. 11, Fig. 12, and Fig. 13, along with the simulation results. Combining losses (Fig. 11) are lower than 1.8 dB on a 1 – 6 GHz band, except a few peaks. These peaks are due to mechanical tolerances in the start of the combining section, where minimal distance between parts is only 0.8 mm, and can be removed by a redesign of this combiner, by increasing this small dimension. This figure is good because it includes the impedance preadaptation from 2.5 Ω. RMS phase error of the inputs (Fig. 12) is lower than 15° on the full band. These two parameters, insertion loss and RMS phase error, are very promising for power combining.

The worst case isolation between inputs is at least 10 dB for inputs close to each other (45° pair) and at least 15 dB for other pairs. This isolation enables instability problems to be avoided, and enables the propagation of a failure from a device to another one to be avoided. If a power amplifier module fails, the system can be operated at a reduced power with the other power modules, compared to TWTA-based solutions.

Power handling of this combiner is difficult to measure, because the spatial distribution of thermal losses is not the same in splitting and combining modes, and the inputs cannot be directly connected to standard coaxial 50 Ω amplifiers. The power handling is estimated by an electro-thermal simulation. Fig. 15a. shows the temperature of the device for a combined input power of 500 W CW at 1 GHz. This frequency is the worst case for thermal effects. Output power is 330 W CW and maximal temperature is 225 °C in the absorber. The current version uses an MF124 absorber, able to withstand a maximum temperature of 180 °C, which gives a maximal output power of 260 W at 1 GHz. This power can be easily increased to 380 W using a MF500-124 absorber, withstanding 260~ °C.

Fig. 15b. shows the temperature of the device when the top input is turned off. Maximal temperature is 259 °C in the absorber.

Table 1. shows comparison with related power combiners. It is the only one which has low impedance inputs. Insertion losses are high, but such high losses are due to the combination of the high impedance transformation ratio and the low lower frequency.

BW	N	Zin	IL	Isolations	Refs
[GHz]			[dB]	[dB]
1 - 6	8	2.5	1.7	11.5 20.1 18.9 17.2	This work
2 - 8	8	50	0.4	9.5 17.0 16.3 12.7	26
2 - 17	8	50	1	9.5 17.0 16.3 12.7	26
7.6 - 10.4	12	50	1	-	27,28
5 - 20	64	50	1.5	-	29
6 - 18	20	50	0.97	-	25
2 - 16	32	50	1.2	-	30
0.52 - 1.86	8	50	0.2	-	31
12.1 - 15.7	8	50	0.17	7.0 12.0 11.0 7.0	32
8 - 18	8	50	0.5	9.0 7.2 10.2 11.0	33,7
28 - 36	20	377	1.0	12.0 15.0 17.0 18.0	8
11.5 - 16	8	50	0.5	9.0 14.6 22.6 16.3 14.6	8

Conclusion

A 1 – 6 GHz spatial power combiner with 2.5 Ω low-impedance inputs and increased isolation had been designed, manufactured and measured. Total losses, including impedance prematch, are lower than 1.8 dB, except on a few peaks. These peaks are due to mechanical tolerances and can be removed by increasing some critical dimensions.

This new power combiner is very promising for high power (kW) combining in a wide frequency range. Thanks to the low insertion losses, the impedance pre-matching, the ability to evacuate calories of the active devices, and the isolation between the inputs.

This design is the subject of the European patent “Spatial power combiner” number EP3171451A1.

Acknowledgments

The authors would like to thank Sylvie Lepilliet of the IEMN for her precious help during the measurements, Pierre Bruguière of the CEA for reviewing, CST for its cooperation, Steve Huettner and Terry Cisco of Microwaves101 for good advice, and Steve Arscott of the IEMN for help with English language.

Transfer S parameters.

23 Mar 2022

This content was originally published on Microwaves 101 (https://www.microwaves101.com/encyclopedias/transfer-s-parameters). Many thanks to Steve for hosting the original version. Have a look on his website for more interesting content.

Transfer S parameters are a convenient way to express S parameters in a way that allows to easily cascade blocks. They have the same principle as ABCD parameters: they express all relevant input quantities in function of all relevant output quantities, contrary to normal S parameters which express all scattered waves in function of all incident waves, and are messy when cascading blocks. They are sometimes more convenient than ABCD parameters, because they work with wave quantities instead of voltages and current, which are very difficult to measure at high frequencies.

It is very counter-intuitive, but expressing input in function of output and not the inverse allows to deal with unilateral blocks, what the other convention doesn’t allow. They are most often defined in the following way:

[[b_1],[a_1]] = T \ [[a_2],[b_2]]

Be careful! Another convention exists, with a and b inverted. Some people even express output in function of the input. So, pay attention to the used convention when reading calculations from other people!

With the definition used in this page, the transfer parameter matrix of a chain of elements can be calculated as follows:

T = T_1 \ T_2 \ cdots \ T_N

And, be careful, this is the inverse order as one expects. Yes, matrix multiplication is very convenient but sometimes crazy!

The following formulas can be used to pass from regular to transfer S parameter:

{: ( T_(11) = S_(12) - (S_(11)S_(22))/(S_(21)) , S_(11) = T_(12)/T_(22) ), ( T_(12) = S_(11)/S_(21) , S_(21) = 1/T_(22) ), ( T_(21) = - S_(22)/S_(21) , S_(12) = T_(11) - (T_(12)T_(21))/(T_(22)) ), ( T_(22) = 1/S_(21) , S_(22) = - T_(21)/T_(22) ) :}