Switching amplifier design with S-functions, using a ZVA-24 network analyzer

Marc Vanden BosscheNMDG N.V., Fountain Business Center - Bld. 5,

Cesar van Kerckhovenstraat 110, 2880 Bornem, Belgium

AbstractIn this paper the S-function theory is explained and especially their assumptions are highlighted. Due to these assumptions, it is not evident that S-functions are applicable to switching amplifiers. The paper uses the term “switching amplifiers” in its broad sense, namely all classes of amplifiers that drive the transistor in pinch-off and that optimize in one or another way the behavior of the amplifier by tuning the harmonics. Some setups are illustrated to extract the S-functions and some possible pitffalls in these setups are mentioned. Finally, the extraction and verification of S-functions on a switching amplifier are demonstrated.

I. IntroductionIn the quest for the best power added efficiency, switching amplifiers are very important. Presently harmonic source- and load-pull are being used to shape the input waveforms and to provide the proper load impedances to improve different specifications. This process is quite tedious due to the large degree of freedom in optimizing different parameters. With these measurements it is neither possible to simulate. Therefore, good models would speed up the design using more automated search routines in simulation tools and would allow to understand the propagation of performance through the complete circuit. Also to improve the performance of amplifiers, pre-distortion circuits are being designed. This can be done in a more efficient way using a good model of the component. Of course, this model must be straightforward to extract, has to predict nonlinear behavior accurately and should take into account the interaction with other components, similar to S-parameters.

Lately, a lot of attention is given to different behavioral models, their extraction techniques for nonlinear high-frequency components and the related high-frequency instruments to extract these models.

S-functions is one of these behavioral models. They are finding their roots in the describing functions, a well-known approach in control theory ([1]-[3]). These S-functions can be extracted using commercial available network analyzers, equipped with the proper hardware and measurement software.

The S-functions are a natural extension of S-parameters into the nonlinear domain [4]. Together with the approaches described in [5]-[9], they belong to a class of frequency-domain behavioral models that describe the nonlinear behavior of a multi-port Device Under Test (DUT) by linearizing its response around a set of large-signal operating points. As a result, S-functions can predict accurately both the harmonic distortion due to the large-signal excitation, as well as linear interactions of small-signal reflections. Using appropriate impedance transformations, S-functions can be extracted under arbitrary

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ESA Microw ave Technology and Techniques Workshop 2010, 10-12 May 2010

impedance conditions [5]. Thanks to the linearization simplifications, the dimensionality of this type of models stays limited and can the model be extracted in a reasonable time.

But questions do rise about the applicability and validity range due to this linearization, especially applying it to switching amplifiers where the harmonics are determining a large part of the behavior. By nature, switching amplifiers are very nonlinear but the information must be transmitted with minimal distortion.

After explaining the S-function theory in section II, the extraction process is mentioned in Section III. In section IV the challenges are explained to apply S-functions to switching amplifiers and section V highlights the different setups, based on combining a network analyzer with tuners and active injection. Finally an example of S-function extraction and verification is given on a transistor in deep pinch-off. The goal is to determine the validity range of the model to be able to use it in confidence for designing switching amplifiers.

II. The S-function theoryA linear two-port device is completely described by its S-parameters as function of frequency. When this device is excited by a signal with spectral content within the frequency band of the S-parameters, the response is predicted accurately through linear theory.

Assume an amplifier as device under test, stimulated at one specific frequency. For low input power, the amplifier is fully characterized by its S-parameters at that frequency. Increasing the input power,

B1 and B2 will no longer be linearly related to A1 and A2 starting from a certain power level. As such the S-parameters are no longer valid. Additionally in many cases the generation of harmonics can be observed in B1 and B2 .

Assume a simple case where A1 and A2 consist of one spectral tone at the considered frequency (Fig. 1). It is possible to create a table that describes the amplitude and phase of fundamental and harmonics of B1 and B2 as function of the amplitude of the single tone of A1 and A2 and their phase relationship (1).

Furthermore, the amplitude and phase of Bi k i=1,2 are depending on the bias settings of the nonlinear component. Different ports of the device can be used to apply a combination of voltage- or current-forcing bias. Certain ports will combine RF signals and DC signals. The combination of

A11 , A2 1 and the DC bias is considered as the large signal operating point (LSOP) of the device that determines Bi k i=1,2 .

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Fig. 1 One large input tone at input and output

A1

B2

A2

f0 3f

02f

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f0V

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i3

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Describing the nonlinear behavior, the self-biasing effect is as important as the high-frequency behavior. Therefore, one needs to describe also the DC behavior at the relevant ports as function of the LSOP. The nature of the component, e.g. BJT or FET, typically drives the selection of the DC forcing mode. This determines the independent variable while the dependent variable is the duality.

Bias depvar=Table ∣A11∣,∣A21∣, Phase A11 , A21 , Bias indep var Bi k =Tablei∣A11∣,∣A2 1∣, Phase A11 , A21 , Bias indep var with k≡k f 0

(1)

There are two main issues with this “simple” model. First of all, it is impossible to derive such model with any type of equipment. This is very similar to the extraction of S-parameters. No network analyzer exists that is perfectly 50 Ohm to extract immediately an S-parameter. A network analyzer always requires two independent measurements to extract the S-parameters. Similarly, it is impossible to find equipment that can generate the pure fundamental tones A11 and A2 1 at the component level. Even if a source would exist to generate a pure tone, one cannot avoid the presence of harmonics in

Ai . Indeed the harmonics of Bi k get reflected by the termination of the equipment and added to Ai . As such, one always has to consider the presence of harmonics in Ai , deviating from the

above simple case.

Secondly, the “simple” model is not practically useable. The purpose of the model is to predict the response of a combination of components and circuits. Predicting the harmonics in Bi and restricting

Ai to the fundamental tone does not make sense. Therefore the “simple” model must be expanded, taking the harmonics in Ai into account.

The naive approach is to simply extend the above model and describe Bi k as tables of all possible amplitudes and phase combinations of Ai k . This results in a vast amount of data and unacceptable measurement times.

In many practical cases, the harmonics in Ai are small compared to Ai 1 . As such the harmonics can be considered as perturbations to the “simple” model and one can linearize in the harmonics of

Ai . The linearization is similar to the conversion matrix theory [10]. One harmonic of Ai impacts all tones of Bi but in a linear relationship (Fig. 2). Thanks to this linearization the required amount of experiments is reduced meaningfully.

The region of validity of this model can be extended further by the linearization in the complex conjugate of the harmonics. This extension does not increase the complexity of the model, neither the extraction effort. The validity of this extension is easily proven using the Volterra theory [11]. When the incident waves get delayed, the reflected waves will also be delayed. This is expressed in the S-

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Fig. 2 Linear perturbation caused by small harmonics

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functions by applying the proper phase normalization against the main tone.

LSOP≡∣A11∣,∣A21∣, Phase A11 , A2 1 , Bias indep var =A11/∣A11∣Bi k =SF ik LSOP kSF ikjl LSOP A jl

k−lSF ikjlc LSOP A j

* l k−l with l1 (2)

It is possible to extend further the region of validity by including higher order terms related to the probing signal, like second and third order effects [12]. These equations can be derived by using the Volterra theory.

Of course, it is possible to extend the LSOP by adding other independent variables, like temperature, at the expense of larger measurement times.

III. S-function extractionTo extract an S-function, one needs a measurement setup that can apply the large-signal operating points in a controllable way. For each LSOP, one performs a set of independent experiments by varying

Ai k , to extract the S-function coefficients using one or another linear regression method. The variation of Ai k needs to be large enough to be properly measurable while the absolute amplitude needs to be small enough not to violate the linearity assumption.

To perform these independent experiments, the setup requires a synthesizer that can be switched sequentially to each DUT port of interest injecting a “small” tone (probing or tickling tone) sequentially at each harmonic (Fig. 3). At each harmonic, the phase of the tickle tone is rotated to create the independent experiments. Two phase values, separated by 90 degrees, are an optimal minimum [12]. To reduce the measurement uncertainty though, typically this number is increased at the expense of a slower extraction (e.g. 5 phase values equally distributed across 360 degrees).

IV. The switching amplifier and S-functionsIn the context of this article, the term “switching amplifiers” has been considered in its broadest sense. And for real switching amplifiers the article is most relevant.

Any type of amplifier beyond class A, drives the main transistor more or less into pinch-off by

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Fig. 3 Conceptual setup for S-function extraction

k f0

f0

f0

ZL

Large-SignalSource Tickling

Source

DC Bias

Large-SignalSource or Load

controlling the DC bias point. It is possible to shape the voltage and current waveform by adapting the harmonic impedances both at the output and at the input of the transistor. Also over-driving the amplifier at the input, switches the transistor faster between on- and off-states bringing it closer to an ideal switch behavior [14].

Usually it is the goal to maximize PAE, often combined with maximizing delivered output power with minimal input power at the fundamental frequency.

Operating as an ideal switch, namely when the voltage and current waveforms do not overlap and are close to square waves, does not result in the best PAE due to the presence of the harmonics which dissipate across the broadband load.

Instead of dissipating power at the harmonics, it is better to reflect the power back to the transistor in such a way that it recombines through the nonlinearities with the fundamental power and optimizes the fundamental behavior. This occurs for properly selected values of amplitude and phase of the harmonics in combination with DC bias and input power levels.

Harmonic source- and load-pull systems are being used to find these optimal points in a multi-dimensional search space. These systems allow to sweep input power, DC bias, fundamental and harmonic impedances at the input and the output, measure the specifications of interest and display them in, amongst others, contour plots. This process is elaborate and one does not get feedback on the actual class of amplifier.

Large-signal network analyzers give a tool to the amplifier designer to make a shortcut in this multi-dimensional search space. By providing in almost real-time voltage- and current- waveforms, fundamental and harmonic impedances, dynamic load-lines etc., it is becoming possible to apply text-book design techniques for the different classes of amplifiers [14].

For small-signal designs or designs where the power is backed off meaningfully to maintain linearity, it is very efficient to perform amplifier design using S-parameters [15]. Nowadays this mode of operation is unacceptable in many cases. Pushing the amplifiers in their nonlinear mode of operation, S-parameters are not adequate and can be replaced by S-functions.

Nevertheless, the use of S-functions requires caution due to their assumptions.

First of all, the LSOP needs to stay constant while the tickle tone is applied. By using control loops on the LSOP during the tickling, it is possible to maintain the LSOP constant. Instead of this time consuming operation, it is possible to adapt the S-function equations to take into account small variations of the LSOP [12].

Secondly the tickling tones should be large enough to measure them with enough accuracy but even more important they should be small enough to satisfy the linearization assumption. Usually it is possible, whether or not assisted by dedicated S-function tools, to select proper amplitudes for the tickle tones. These amplitudes could be frequency dependent.

Once the S-functions are extracted, one can interpolate between the large-signal operating points and one can predict the component behavior properly as long as the applied harmonics will not violate the linearization. This is guaranteed as long as the harmonics in the simulation are smaller then the tickle tones during the extraction, assuming that they were selected properly. But as long as one does not violate the linearization, one can increase the amplitude further of the harmonics.

For switching amplifiers, for which S-function models would be a great advantage, the linearization assumption is not trivial. Due to the important role of the harmonics to achieve high PAE, it is not

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guaranteed that they can be considered as a perturbation, allowing to linearize.

Therefore, either one verifies before the extraction that the tickle tones can be increased in power relevant to the switching amplifier without violating the linearity principle [16]. Typically this means that one needs to realize reflection factors close to one. Another approach is to extract an S-function model and verifying it afterwards with measurements, which are relevant to the switching amplifier. The consequence is that the measurement setup needs to be capable of applying the signals and impedances which are relevant to the switching amplifier and cannot be limited to the setup to extract only S-functions.

V. Setups to measure S-functions for switching amplifier designTo extract S-functions in a near 50 Ohm environment, a minimal setup should contain a large-signal high-frequency source and a second tickle source, synchronized to the large-signal source, to create the tickle tone at the input and output using a switch and a set of couplers (Fig. 3). Nowadays most of this hardware is present in advanced network analyzers.

To support S-functions predicting load-pull conditions, this setup needs to be extended with a fundamental tuner at the output of the device under test. To apply the large-signal power efficiently to components with high reflection coefficients at the input it is advised to use a source tuner.

Because the tuners are in between the device under test and the measurement couplers, the error coefficients, characterizing the systematic errors of the setup, need to be updated continuously as a function of the S-parameters of the tuners, depending on their position. In this way, the measurement system gives continuously the correct voltage and currents at the device ports.

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Fig. 4 Setup for S-function extraction in non-50 Ohm environment

DUT

Variable loss Variable loss

Fig. 5 Setup with harmonic load tuner and probing coupler

DUT

f 0

k f 0

f 0 k f 0

Because the injection of the tickle tone is behind the tuners (Fig. 4), the resulting amplitude of the tickle tone at the device under test will change depending on the position of the tuner. Therefore it is important that the the S-function extraction software compensates for the losses across the tuners while applying the tickle tone. Otherwise, it is not guaranteed that adequate power reaches the device under test. Due to these losses in combination with the couplers, it is possible that an additional amplifier is needed to boost the tickle power.

It is possible to extract S-functions for a switching amplifier with a setup with fundamental source and load tuner. The disadvantage is that it is impossible to verify the S-functions whether they properly describe the device behavior under high harmonic reflection factors. By adding the necessary amplification to the tickle source it is possible to synthesize actively high reflection factors and to verify the model. Though the more power, generated by the device, the more difficult it gets to overcome the losses in the system.

For verification purpose of the S-functions, a different setup is being preferred (Fig. 5). This setup replaces the fundamental load tuner with a harmonic tuner. This tuner can create high reflection factors at fundamental and harmonics and will be used for the S-function verification.

Even during the S-function extractions, this setup has favorable advantages. As the fundamental load is being swept, the harmonic loads can be kept constant, in contrast with a fundamental tuner only. This is an advantage in case of potential instabilities. Additionally the tickle tone is presently only compensated in power for the losses across the tuner. There is no compensation for the fact that there is an initial reflection, caused by the tuner. As the phase rotates, the load impedance will rotate around this initial reflection factor. This reflection will change in an uncontrolled way when the fundamental load is changed. Of course, as long the linearisation is valid, there is no problem. But with switching amplifiers, this linearisation is actually questionned.

Because the harmonic tuner is used to synthesize large reflection factors, especially for the harmonics, the coupler has to be moved between the device under test and the tuner. Ideally the coupler should not reduce the maximum reflection factor that can be synthesized at the device under test. Therefore, the coupler has to have minimal loss. Amongst others, this can be realized by probing structures [17-18].

V. S-function extraction of a FET for the design of switching amplifiersFor this case study, a high efficiency heterojunction power FET, EPA120B-100P from Excelics, is used in a Focus Microwaves fixture. This device has a typical power output of 30 dBm and power gain of 11 dB at 12 GHz.

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Fig. 6 Setup for S-function extraction in non-50 Ohm environment

Fig. 7 Class A - Dynamic Loadline

The measurement system is based on a Rohde&Schwarz ZVA 24 with the proper options and the NMDG ZVxPlus add-on kit. As fundamental source and harmonic load tuner respectively a CCMT 1808 and MPT 1818 from Focus Microwaves were used. As coupler structure at the output, the low-loss VI probes from Focus Microwaves were selected (Fig. 6). The measurement system is calibrated at 2 GHz with 5 harmonics. The transistor is being de-embedded including the package. A package model is provided by the transistor manufacturer.

Based on the maximum current and voltage excursion at the drain, the fundamental load was determined and the bias settings were optimized (Error: Reference source not found). Then the gate bias was reduced to pinch off the transistor and halve the peak drain current (Vg = -0.93V and Vd = 6.3 V). The second and third harmonic impedances were shorted as good as possible such that the drain voltage waveform would be as close as possible to a sine wave. The source tuner was matched to the input impedance of the transistor, averaged for small and large input powers. Then the input power was increased again to achieve the original maximal drain current of the class A mode of operation (Fig. 8).

The goal is to extract and validate an S-function for the given fundamental load impedance and bias settings such that the model can be used to design different types of switching amplifiers. The concern is the validity of the linearization assumption for switching amplifiers. When valid, the S-function should predict properly the behavior of the transistor under different high harmonic reflective conditions.

The S-function is extracted for a sweep of incident power, going from -5 dBm up to 9.5 dBm with the second and third harmonic load terminated in 50 Ohm. Before starting the S-function extraction, the level of the tickle tones must be determined. As said, the tickle signal must be large enough to be

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Fig. 8 Class B - Voltage and current behavior at gate and drain + load impedance

gate drain

f02f0 3f0

Dynam ic Loadline

measured properly but may not violate the linearization assumption. A simple approach to determine the maximum tickle level is to offset the tickle tone in frequency from the harmonic frequency such that it can be considered a negligible offset for the transistor but that it is out of the resolution bandwidth of the IF filter of the network analyzer. In this way, the measurement setup measures only the harmonic response. The power of the tickle tone is gradually increased until a change in harmonic response and / or DC currents are observed. This procedure resulted in a tickle power at the input of -20 dBm and at the output of -5 dBm. At the output the tickle power was not sufficient to influence the harmonic behavior and / or DC currents. This is due to the available power of the source, the coupling factor of the coupler and the tuner insertion loss.

The S-functions were extracted using control loops to keep the LSOP constant during tickling (Fig. 9). The constancy of the LSOP was verified and confirmed for input power and bias settings. Also the interpolation capability of the S-function was verified by predicting the reflected waves and comparing with the measurements for in-between input powers (Fig. 10).

Using the S-function for switching amplifier design, the linearity assumption must be validated. Two verifications were performed with the harmonic tuner. First, different second harmonic impedances with increasing reflection factor were synthesized, keeping the third harmonic impedance at 50 Ohm (Fig. 11).

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b2

Fig. 9 S-functions, related to S11, S12, S21 and S22 as function of power sweep

The incident and reflected waves are measured for each impedance setting, after making sure that the incident power and the bias did not change. Using the S-functions, the fundamental and harmonics of the reflected waves, Bi k , are predicted. The complex error is calculated and compared to the measured power level. Hereby for each amplitude of the reflection factor, the absolute power and the error prediction is plotted as function of the phase index of the reflection factor (Fig. 12). It can be observed clearly that with increasing harmonic reflection factor the prediction error increases. This is due to the fact that the linearization principle is violated more and more. To use waveform engineering, the error is still acceptable. For other purposes, maybe the error level is not acceptable anymore. It can also be seen that for certain load conditions, the transistor is oscillating. Of course, the power levels cannot be predicted by the model in this case. Another way of representing the prediction error on B2, is using countour plots (Fig. 13). It can be clearly seen that the error increases with increasing reflection factor.

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Fig. 11: Coverage of 2 f0 (or 3f0) load impedance with 3 f0 (or 2f0) matched

0.75 0.5 0.25 0.25 0.5 0.75

0.75

0.5

0.25

0.25

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0.75

Fig. 10 Interpolation verification predicting b2 for different tickling experiments (x-axis)

f0 2 f0

3 f0

Complex prediction error

Increase refl factor

16 phase values

The same was repeated stepping the third harmonic impedance while keeping the second harmonic impedance at 50 Ohm. It can be seen that the linearity assumption stays longer valid with increasing reflection factors but the device starts to oscillate more often (Fig. 14).

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Fig. 12 B1,2 and prediction error as function of phase index for increasing reflection factor at 2 f0

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dBm b2 f0 and complex error

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oscillation oscillation oscillation

oscillation

oscillation oscillation

Fig. 13 Prediction error of b2(f0,2f0 and 3f0) for 2f0 tuning

R0

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Usually for switching amplifiers both harmonic impedance are high reflective. Therefore, a verification test is done where the third harmonic reflection is kept at its largest value and is rotated 360 degrees. For each position of the third harmonic reflection, the second harmonic reflection is rotated 360 degrees for its highest reflection coefficient. For a given high reflection factor of the third harmonic, Fig. 16 shows B21 and its prediction error rotating the second harmonic reflection factor at its highest value. The power of B21 is around 20 dBm while the error is around -5 dBm. For three phase values the time waveforms are shown for the measured and predicted B2. Oscillation can also be observed.

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oscillation

oscillationoscillation oscillation

Fig. 15 b2(f0) and prediction error for given high refl at 3 f0 with tuning 2f0

4 6 8 10 12 14 16

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b1f0

Fig. 14 B1,2 and prediction error as function of phase index for increasing reflection factor at 3 f0

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dBm b1 f0 and complex error

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oscillation

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oscillation2f03f0

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f0oscillation

oscillationoscillation

oscillation

Increase refl factor

Fig. 16 b2(f0) and error for given high refl at 3 f0 with tuning 2f0 + corresponding time waveforms

4 6 8 10 12 14 16

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As can be seen, the linearity assumption is even more violated due to the accumulation of the high reflection at the second and third harmonic. Of course, the points of oscillation do not need to be considered. As can be seen in the time waveforms, the error is still acceptable in this case for waveform engineering, as long as the device is not oscillating. But caution is necessary and the S-functions cannot be applied blindly.

IV. ConclusionsIn this article, the S-function theory and the extraction process has been explained. Nevertheless, the aim of S-functions is to predict nonlinear behavior, a linearity assumption is needed to make it practically useable. On overview of different setups has been given, focusing on the extraction and verification of S-functions for switching amplifiers.

A case study was done on a commercial FET, were it is clearly demonstrated that care needs to be taken with the linearization assumption under high reflective conditions. These high reflective conditions just occure for switching amplifiers.

Therefore, S-functions are very useful and speed up the design process but can not be applied blindly for switching amplifiers and need to be verified with independent data sets with stimuli close to the realistic conditions. Without explicit verification it is not possible to predict the quality of the S-functions in advance. The advantage is that the verification data set is much more smaller than the data that would be needed to extract a model without linearization. The disadvantage is that a more extended setup is required.

AcknowledgmentThe use of the S-function models in ADS™ was made possible thanks to the support from Agilent Technologies.

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