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This post is a mix between an unpublished long article, an article published in IEEE MWCL¹, and further additions.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) ) :}

RFID coils should not be grounded.

22 Mar 2022

Putting a solid ground plane in a PCB is a good practice. This allows to have good interconnexions between the different grounds of the components, to have a proximity shielding of the lines, and to reduce the cross coupling between the lines. All these effects have one root cause: a ground plane reacts to an electric or magnetic field by generating induced currents which tend to reduce this incoming field.

Due to these currents, the ground plane tends to mirror the lines: when you have on one side a line in which circulates a current, all things behave like there was a symmetrical copy with an opposite current on the other side.

Of course, for various reasons, this effect is not perfect. However, still very efficient.

However, for the very same reasons, it’s not a good idea to put a ground plane straght below an NFC coil. An NFC coil is precisely designed to make coupling to circuits in proximity, and a ground plane reduces this coupling. Nobody would make this on purpose, but this error is easy during the rounting stage.

The following pictures produced with OpenEMS shows clearly this effect:

Magnetic field is shown in all figures. Left side are with a solid ground plane, right side is without. Line 1 is between the coil and the bottom of the PCB. Line 2 is at 10 mm height. Last line is a perpendicular cut.

The strange box around the loop which can be seen in the different pictures is a simulation artefact explained in footnote¹.

First line shows that even inside (!) the PCB, the grounded coil tends to guide the field just below it. Nothing would happen in the center. This behavior is the expected behavior of a transmission line: take a wave on one side, transport it towards the other sides, and radiate as little as possible. On the contraty, the coil behaves like a coil and the center has an high magnetic field.

Second line shows at a distance² of 10 mm, proximity shielding reduces the generated fields in the ground plane case (left). On the contrary, without it, the magnetic field is high and would easily couple an NFC tag.

Third line shows an X-cut, which allows to see the variation of the magnetic field in function of the height. Left pictures shows that it decreases quickly with height where a ground plane is used, while this decrease is lower when no ground plane is used.

The distance where the magnetic field is highly attenuated is proportionnal to the distance between the coil and the ground plane in the first case, while it is proportional to the coil size when no ground plane is used. In this case, it is possible to increase this distance simply by increasing the coil size.

In conclusion, by the proximity shielding effect, a ground plane defeats the whole purpose of a NFC coil, which is to couple nearby circuits. To avoid this problem, remove the ground plane below the coil, and let enough distance between the NFC coil and the ground plane of the circuits using it.

Miller effect and solutions.

22 Mar 2022

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

In Scientific Papers of the Bureau of Standards, Volume 15, 1919-1920, John M. Miller published a paper in titled “Dependence of the Input Impedance of a Three-Electrode Vacuum Tube Upon the Load in the Plate Circuit.” For this work, Mr. Miller is forever associated with the “Miller effect” which is still relevant a century later.

Mr. Miller was taking about vacuum tubes in his seminal paper, but the concept applies to all three-terminal amplifier devices. In terms of a more modern microwave field-effect transistor (FET), the Miller effect is an increase of the apparent gate to drain capacitance compared to the real one due to a feedback effect from the drain to the gate.

The gate to source capacitance, Cgs, sees at its terminals only the gate voltage, Vg. The drain to source capacitance, Cds, sees at its terminals the the drain voltage, Vl. The drain voltage is basically the gate voltage multiplied by the voltage gain (hey, why do you think it was called an amplifier?) And the gate to drain capacitor sees at its terminals the gate voltage multiplied by (1+A), A being the voltage current of the device in the schematic below. The voltage current being a direct function of the output load. And since the gate to drain capacitor sees a multiplied voltage, its effect is multiplied by the same factor.

Thus the apparent input capacity can become a number of times greater than the actual capacitance between the tube electrodes…

– Miller’s original article, http://www.mit.edu/~klund/papers/jmiller.pdf, page 374

This increase of the apparent capacitance is problematic in broad-band circuits because the bandwidth is reduced when the capacitance increases. In narrowband circuits, the Miller effect is less of a problem because capacitance can always be compensated for by the inductance of the bias circuits. However, keep in mind that the bandwidth of a circuit must be sufficient to keep a margin for process variations.

Some remedies to this problems are:

• Put in parallel to the gate to drain capacitor an inductor to resonate the capacitor. But this compensation has some serious drawbacks. It is narrowband. Nasty oscillations can occur. Yikes! And the inductor must be DC-decoupled by a capacitor, because gate and drain bias voltages are different. Double trouble!

• Use a balanced amplifier and compensate Cgd by another capacitor of the same value connected to the opposite voltage, like explained in following picture. We let the derivation to the reader. However, two problems limits that cool scheme. First, oscillation can occur. Second, the layout needs an RF-RF cross-over, which is not practical.

• Decrease the load impedance seen by the transistor, to reduce the voltage gain. Remember that Miller effect is an effect of the voltage gain. Two schemes for that are the cascode and the Cherry-Hooper amplifier.

The cascode

The image below illustrates a common way to alleviate the Miller effect: the cascode. The load impedance seen by the first transistor is dramatically reduced by the second transistor. In such a configuration, the voltage gain of the first transistor is approximately 1¹. Since the voltage gain is low, the Miller effect is drastically reduced.

The second transistor has voltage gain but no current gain. And the Miller effect is eliminated because the gate is grounded for high frequencies. This scheme is usable only up to ft, not fmax. In general, ft is lower than fmax, at least for FET transistors. This should be kept in mind when designing really high frequency amplifiers, for example in the millimeter wave range. Also, an other problem of this scheme is an huge tendency to oscillate. A damping RC network is almost always added to the gate of the second transistor stage to compensate for this problem. Finally, the DC power consumption of the whole amplifier is doubled. Each transistor has a similar VDS voltage across it and the same current flowing into it. Additionally, for low voltage circuits, this scheme doubles the needed bias voltage so it’s problematic. Lot of solutions have been developed to solve this problem but, hey, it’s not an IEEE article here.

Cherry-Hooper amplifiers

Professor Rodwell (UCSB) has some notes on this topic: https://www.ece.ucsb.edu/Faculty/rodwell/Classes/mixed_signal/mixed_signal_notes_set_3.pdf.

References

John M. Miller, “Dependence of the Input Impedance of a Three-Electrode Vacuum Tube Upon the Load in the Plate Circuit”, Scientific Papers of the Bureau of Standards, Volume 15, 1919-1920.

Why RFID antennas should be called neither RFID nor antennas ?

07 Jul 2021

A friend of mine told me that he was looking for an “expert antenna engineer” to design RFID antennas, but he was not sure he searches well, because the last antenna expert he interview told him that he had designed hundreds of antennas, but never an rfid antenna. Never. Nada.

He tried to search on antenna books some information on rfid antenna to orient his search. He was quickly disappointed. He opened the excellent “Electromagnetic Waves and Antennas” ([https://web.archive.org/web/20240528230718/www.ece.rutgers.edu/~orfanidi/ewa/)) from Orfanidis, and search for RFID and NFC. Nothing! Same thing with “Antenna Theory: Analysis and Design” from Balanis. This starts bad.

The coup de grâce comes from Antenna Theory (https://www.antenna-theory.com/definitions/nfc-antenna.php):

Therefore, NFC antennas are not really antennas, in that no one cares about typical antenna parameters, such as the radiation pattern or the antenna gain

Here we go! Now, time for more explanations.

What are antennas ?

That’s probably the best question to ask to an antenna expect if you want to be sure that not only he knows his stuff but that he will also be able to explain to you the job you pay him for in terms you are able to understand.

Most science magazines would tell that an antenna is something which “converts a current to a wave”. Nothing can be more misleading! First, between the conductors who carry a current, there is already a wave. Second, antennas can also be fed by metallic waveguides, who are not well described by their currents, or by dielectric waveguides, who have NO conduction current at all. Third, it does not explain how the wave emitted or received by an antenna is different from the wave in a cable.

A much better definition would be: “an antenna is a device which converts a guided wave to or from a freely propagating wave”.

The wave in the wire which connects a transmitter to an antenna is guided by the wire. Same thing for a metallic or a dielectric waveguide. For the output wave, freely propagating is a very important point: an antenna emits a wave even if there is nothing and just the free space in front of it. In this case, the emitted wave will simply travel during eternity until it meets something. The antenna emits equally well whether there is or not an antenna in front of it.

It’s actually a difficulty to measure antenna: to make measurements of an antenna. On the one side, we must put the measured antenna in an enclosure to avoid both disturbance from outside transmitters and to disturb outside receivers. On the other side, the antenna must not see the enclosure walls and only something which look like free space. Metallic enclosures reflect waves and look like a mirror from the antenna point of view. Therefore it must be covered by special materials absorbing electromagnetic waves in order to look like free space.

Something important for the remaining of this article is that the field in the near vicinity of the antenna does not depend whether there is an antenna or nothing far before the antenna.

What are transformers ?

Transformers are a device made from two coupled coils. In transformers used for power supplies, the coupling is made as strong as possible. The details on the various ways to ensure strong coupling are outside of the scope of this article. When a voltage is applied to the first coil, some current flow in it, just enough to produce a “magnetic field”¹ which would in turn induce in the coil a voltage equal to the applied voltage, as well as a voltage in the second coil. When nothing is connected to the second coil, no current flows into the second coil. When something is connected to the second coil, it draws a current which changes the magnetic field, which in turn cause more current to flow in the first coil².

For a well designed transformer, when no current in drawn from the second coil, the current in the first coil is low and mainly inductive. This is an important point: the inductive current is out of phase with the voltage so the average power consumed is null: it periodically takes and give back power. And inductive current stores energy but don’t consumes it.

Of course, there are always somme losses, but they are low. There is no power which escapes and travels to infinity.

What is NFC ?

NFC means near field communication. This system allows for a reader device with its power supply to communicate with and to provide enough power to process the requested informations to an electronic chip embedded, for example, in a card. Some modulated high frequency voltage is applied to the coil of the reader devices. This produced in turn a voltage to the coil of the tag which allows it to be powered. The modulation of this high frequency voltages allow to transmit information. The tag transmits information by modulating the current it draws to the coil. Which can be sensed by the reader by monitoring the current in its own coil.

Exactly a transformer action: increase of output current, increase of input current.

There are some differences between such transformers and the typical 50/60 Hz mains electricity transformer: the NFC coils are designed for high frequencies and the coupling is lower because the two coils are separated by a few centimeters.

Coils in a transformer are described by both both self-inductances, quantifying the inductive behavior of the coils if they were separated, and a mutual inductance, quantifying the coupling between the two coils. In the same way, an NFC coil is described by its inductance. Mutual inductance changes a lot depending on the application. Secondary coil inductance belongs to the other system, so it’s interesting mainly for the designer of the other system.

When looking for nfc coil formulas, all formulas specifies the inductance of the coil. Resistance is often ignored. And when it is calculated or measured, this resistance is mainly ohmic³: the radiation resistance is negligible. On the contrary, radiation resistance of antennas cannot be neglected because it determines the efficiency⁴.

Why RFID is not a good term ?

“Radio frequency identification”. Radio frequency is a loose term designated almost any alternating thing starting from 100 kHz (old amplitude modulated broadcast radios). Identification is a precise term but applies to lots of situations. For instance, have identifiers for various reasons Bluetooth devices, airplanes transponders and so on.

If RFID is a term applicable for lots of things, its almost a term for nothing.

What are NFC “antennas” ?

Good question. We have seen that their mechanism is not an antenna mechanism but a transformer mechanism. So, antenna is not the proper term. Transformer would apply to the two coils. Half-transformer would be perfect but a bit strange. Coil is perfect.

The only drawback is that coil focuses too much on how it is built than what is does. But it’s still ok because there is not so much ways to design NFC coils than with coils.

How should I search NFC coil designers ?

Good question. For the terms, the usual terms are bad, but highly common, so stick to it. But keywords more specific to NFC must be added to find candidates coming from the NFC domain. Searching among antenna designers is not a bad starting point, but most people in this domain will be used to radiating antennas. Microwaves conferences are not the best place to search for this reason. Searching among transformer designers might be interesting because NFC coils are half-transformers.

Don’t start searching an “antenna expert” and tell him later than it’s antenna will be an RFID/NFC antenna. RFID/NFC should be among the first words. “Antenna” should be here just for Google: “coil” is the right term.

An other answer is “don’t search NFC coil designers”. NFC coil design is easy, provided one has some minimum knowledge in electromagnetism. Application notes like https://www.nxp.com/docs/en/application-note/AN11564.pdf explains all what is needed to do this job. However, matching it, properly select the capacitors, designing the filter, the components, the rest of the system is not. Search RFID experts, who will know these not so easy topics. Search analog electronics designers who will know how to use the NFC coil. Both will learn quickly how to design an NFC coil.

Sum up

NFC coils should not be called antennas because they operate in a different ways. Antennas radiate a far field in free space while NFC coils tend to keep energy in their near field. Keep this in mind when searching for NFC coil designer.