When an amplifier goes into compression, standard measurement methods produce wrong answers.

On visiting a high-performance GaAs fab, I was shown a problem with the measurement system. The on-wafer test system was calibrated using a power meter and on-wafer standards, using the built-in VNA-based Guided Power Calibration and both the low power gain and high power (near saturation power) output power were recorded. Then, the RF cable between the port 2 probe and the network analyzer was changed and the system recalibrated. The low-power results were identical, with gain measuring within 0.02 dB of the first test. For the high-drive level, however, the output power had shifted by 0.7 dB, making the part out of spec. The question from the test engineers: What’s wrong with the system?

The problem is in the way we measure the amplifier's S-parameter, S22. The amplifier's impedance at the output (S22) changes with drive level (as is the gain). As a result, normal, linear, 2-port error correction won't compensate for the changing S22 with drive power. You need to use Hot S-parameters (Hot S22).

If you need some background on S-parameter physics, read the rest of this page. If not, continue to: Finding Hot S22.

**Some physics**

For amplifiers operating in the linear range, the effect of the load impedance on the output power is completely described by the S-parameter definition from equation 1, in particular

In the case where the amplifier is terminated by some load impedance, Γ_{L}, this can be rewritten as:

The power delivered to the load is

Thus, the power delivered depends upon both the load impedance and the S22 of the amplifier.

At high powers, however, S21 and S22 can change with the drive power and with the load impedance, so that predicting the output power is not possible using normal techniques. In these cases, the most common method of evaluation is to drive an amplifier to the desired level while modifying the load to find which load provides the maximum output power and often, also the power added efficiency of the amplifier at the load. The load impedance is modified by using an impedance tuner at the output, or more recently by providing an active load. This measurement method is usually called Load-Pull. Unfortunately, this method isn't practical in an on-wafer production test system; the load pull measurements are simply too slow. The problem here is that S22 is normally measured with the input power turned off—port 1 is “cold.” We need to know the S22 when port 1 is turned on, as it seems the effective output impedance changes with drive level. This is sometimes called the Hot-S22.

**Finding Hot S22**

In some papers, a singular value for Hot-S22 was attempted to be discerned using a method of driving the amplifier into a nonlinear condition utilizing one source at the desired frequency, and then injecting another source into the output of the amplifier, at a slightly offset frequency, and measuring the S22 of this second signal [1,2]. I call this the “Traditional Hot-S22,” and while it provides essentially the correct result for a linear amplifier, *once the amplifier is driven into a nonlinear state, this traditional Hot-S22 is almost always completely wrong*. To understand the reason for this, we can look at the actual signals coming from the amplifier under these drive conditions.

Some VNA’s can be used as spectrum analyzers and utilize the directional couplers at port 2 to separate the reverse injected-signal (essentially the active load) from the signal reflected from the amplifier. For a first test, only the source at the input is turned on, and a source at port 2 is turned on (at 0 dBm) to drive a reverse signal, offset a little bit in frequency, shown on the left side of **Figure 1**.

**Figure 1. Traditional Hot-S22, with low power at the input (Left); Spectrum with a high power at the amplifier input (Right).**

This represents the traditional Hot-S22 measurement, and we can see by the change in the b2 signal (as represented by marker 2) that we are measuring the S22 of the amplifier, under the condition of forward drive, but here the input signal is low enough so that the amplifier is still in a linear state. If we use the standard vector-network analyzer (VNA) application to measure S22, it measures the same value. This is the traditional Hot-S22 measurement and works for linear amplifiers. But it provides an incomplete, and in most cases, totally wrong value for the effective output impedance of an amplifier when the amplifier is driven into a nonlinear state.

Now let's raise the input power so that the amplifier is near compression, and thus operating in a non-linear way. Figure 1 on the right shows the output spectral response when the amplifier input is driven with a sufficiently large input signal to cause the amplifier to operate in non-linear way. We now see a kind-of companion signal arise out of the noise floor on opposite (left) side of the port-2 drive frequency.

The amplitude of this signal rises in a nonlinear, nearly third-order response to changes in the port-1 input drive-signal amplitude but is linear with respect to changes in the port-2 reflected signal power. The three signals shown on the right side of Fig. 1 are directly related to the signal's X-parameters. The signal that changes in a nonlinear way with power, which we can think of as *Transposed* from the right-side port-2 drive signal, is associated directly with the *X ^{T}_{22}* term in equation 4, which describes the output wave from an amplifier, operating in a nonlinear response, using the X-parameter notation. [3]

Where *ΦA₁*-2*Φa₂* represents the conjugate of the *a₂* term.

The three signals can be directly assigned to the three X-parameters, as illustrated in **Figure 2**. The output *b ₂* wave consists of three parts, associated with the three X-parameters. Note that the effective Hot S22 is not single-valued but depends on the phase of the *a₂* wave.

**Figure 2. The three signals described in Eq. 4 are X-Parameters.**

Here, the amplifier is driven with a port-1 input drive of 0 dBm and resulting in an output power of 26.53 dBm. Thus, it's nearly 4 dB in compression compared with the previous linear measurements.

The genesis of this *X ^{T}_{22}* signal is a mixing of the nonlinear fundamental-output large-signal with the offset frequency port-2 signal, to create a kind-of intermodulation product at exactly the negative of the offset frequency. The mixing of these signals is realized through the time-variant nonlinear output impedance of the amplifier operating at the fundamental-drive signal, and commutating the port-2 injected signal at that offset frequency rate. It is analogous to a mixer where the fundamental frequency is acting like the local oscillator, causing a time-varying impedance in the mixer, operating on the RF input signal.

**Computing true Hot-S22**

Recent work [4] allowed an improved concept of a Hot-S22 value. In this work a partial derivative of the power delivered to a load is computed, with respect to the wave reflected from a load, to find the optimum reflection that maximizes the power delivered to a load as:

The solution to equation 5 was found to be:

Where the subscripts for ports are not shown for the *X ^{F}*,

*X*, and

^{S}*X*terms, for clarity’s sake.

^{T}From this result, one can find the *B₂* wave for the maximum power as:

And find the reflection that produces the maximum power delivered to the load as

This remarkable result indicates that the load for maximum power transfer depends only on the X^{S} and X^{T} parameters of the amplifier, and if one evaluates equation 8 under the condition of linear operation, where X^{T} approaches zero as:

It is well established that for a linear device, maximum power occurs when the load has the conjugate of the output impedance of the device.

**Hot S-parameter examples**

**Figure 3** shows the result of sweeping an amplifier with both low and high-power drives. The plot shows 4 measurement results, two at low power, where the amplifier operates in a linear manner, and two at a higher power. The normal S-parameters are characterized under full 2-port error correction. The Hot S-parameters are derived from fundamental X-parameter date, as described above.

This is one of the chief benefits of utilizing Hot S-parameters: only Hot S-parameters properly corrects for mismatch errors associated with a non-ideal load match of a test system. In this case, we can infer that the effective S22 of the amplifier is changing with drive level (as is the gain) and the normal, linear, 2-port error correction does not compensate for the changing S22 with drive power.

Consider the results of performing a power sweep on an amplifier, driving it into a non-linear condition while monitoring the gain. **Figure 4** shows the non-linear response of an amplifier with two different measurement methods. One uses traditional VNA 2-port error correction, where the S22 is measured with the port-1 turned off. The other shows the Hot-S21 and computed in equation 11. For this frequency, the gain shows a greater peaking and a higher 1 dB compression point when measured with Hot S-parameters (utilizing X-parameter representation) compared with the traditional 2-port error correction.

**Figure 4. Gain vs. Power for an amplifier, comparing traditional 2-port correction with the Hot-S21 result.**

The reason for this improvement becomes clear when one looks at the plot of Hot-S22, as a function of drive power, as shown below on a Smith Chart in **Figure 5**. Clearly the Hot S22 is becoming closer to 50 Ω, and thus the mismatch loss at this frequency is becoming less lossy, and the gain is higher than would be reported by a traditional VNA using 2-port error correction. In this case, the low power impedance is on the order of 62 Ω (marker 1), but at higher power, the impedance improves to about 46 Ω (marker 2). Another way of saying this is that the linear error correction algorithm is correcting for the low-power S22, when we see the Hot S22 is completely different.

**Figure 5. Hot-S22 for a power sweep.**

Hot S-parameters (sometimes called active parameters) are the only correct method for compensating for the system mismatch effects when devices are operating in the non-linear region. Besides improving measurements, they show how gain and output match change with drive level. Hot S-parameters have a final benefit of producing a fundamental X-parameter file that can be used in simulation or modelling to understand in a complete way how an amplifier’s non-linear behavior can affect system performance.

—Dr. Joel Dunsmore is a Keysight Technologies R&D Fellow.

**References**

- Dunsmore, Joel and Wayne Smith, "Predicting out-of-band nonlinear power amplifier stability using hot S-parameters",
*Microwave Engineering Europe*, pp. 23-28, March 2005. - "Four hot S parameters for the study of nonlinear parametric behaviors of microwave devices,"
*IEEE MTT-S International Microwave Symposium Digest*, 2003, Philadelphia, PA, USA, 2003, pp. 1663-1666 vol.3 - Root, David E., Jan Verspecht, Jason Horn, and Mihai Marcu.
*X-Parameters: Characterization, Modeling, and Design of Nonlinear RF and Microwave Components*. Cambridge University Press, 2013. - Root, David E., J. Verspecht and J. Xu, "Closed-form solutions to large-signal PA problems: Wirtinger calculus applied to X-parameter,"
*2017 12th European Microwave Integrated Circuits Conference (EuMIC)*, Nuremberg, 2017, pp. 212-215.doi: 10.23919/EuMIC.2017.8230697

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