8/11/2019 Hot S22

1/22

Everything you've always wanted to know about Hot-S22

(but we're afraid to ask)

Jan Verspecht

Jan Verspecht bvba

Gertrudeveld 15

1840 Steenhuffel

Belgium

email: contact@janverspecht.com

web: http://www.janverspecht.com

2002 Agilent Technologies - Used with Permission

Presented at the Workshop

Introducing New Concepts in Nonlinear Network Design

(International Microwave Symposium 2002)

8/11/2019 Hot S22

2/22

1

Everything youve always

wanted to know about

Hot-S22

(but were afraid to ask)

Jan VerspechtAgilent Technologies

8/11/2019 Hot S22

3/22

2

2

Purpose

Convince people of a better Hot S22

Show that technology is fun (sometimes)

The purpose of this presentation is to convince people that one needs

to extend the classic concept of Hot S22 if one wants to describe

accurately how a device behaves under large-signal excitation with anon perfect termination.

A second purpose is to show that, when the proper set of

experiments are performed, Hot S22 can be fun.

8/11/2019 Hot S22

4/22

3

3

Outline

Introduction: What is Hot S22 ?

Getting and interpreting experimental data

Confront classic approaches with data

Derivation of the extended Hot S22 theory

Confront extended Hot S22 with data

Conclusion

Since there is a lot of confusion on Hot S22 I will first define what I

understand by it.

Next I will explain how to get and interpret practical experimentaldata.

Then I will confront classical approaches to Hot S22 with this

experimental data and I will show that the classic approach is

inaccurate.

An accurate and extended Hot S22 is then derived from the

gathered experimental data.

I will show that the new Hot S22 model can indeed accurately

describe the measured data, in contrast with the classic approach.

I will finally draw conclusions.

8/11/2019 Hot S22

5/22

4

4

What is Hot S22 ?

D.U.T. behavior is represented by

pseudo-waves (A1, B1, A2, B2)

Hot S22 describes the relationship betweenB2 and A2

Valid under Hot conditions (A1 significant)

D.U.T.A1 A2B1 B2

So what do I mean by Hot S22?

Or a better question is what problem do we want Hot S22 to solve?

The answer is that one wants to describe how a device-under-test

(D.U.T.) behaves under variable load conditions at the output, while a

large-signal excitation is present.

Equivalent to classic linear S-parameters, Hot S22 should solve this

problem by describing the relationship between the B2 and the A2

traveling voltage (pseudo-)waves.

The important difference with classic S-parameters is that the

mathematical model should be valid while a large input signal (A1) is

present.

This large signal at the input causes the DUT to start behaving in anonlinear way, such that a classical S-parameter, which is based on

the superposition principle, is no longer valid.

For the same reason this kind of behavior can not be characterized

by a standard commercial network analyzer. This measurement

instrument does not only uses the superposition principle for its

experiment design (one-tone applied at the same time at one of the

DUT ports), but also for its advanced and sophisticated calibration

procedures.

8/11/2019 Hot S22

6/22

5

5

Experimental investigation

Take a real life D.U.T. (CDMA RFIC amplifier)

Apply an A1 signal

Apply a set of A2s

Look at the corresponding B2s

Mathematically describe the relationship

between the A2s and B2s

Repeat for different A1s

The approach used in this presentation is to come up with an

accurate Hot S22 measurement method and mathematical model

based upon observations of DUT behavior. In order to do this thefollowing experiment is performed.

First one takes a real life DUT. In our case this is an RFIC power

amplifier aimed for CDMA applications. Next one applies a large-

signal at the input, this implies the application of a significant A1

spectral component at the input of the RFIC. Next one applies an

extended set of A2 waves. These are generated by a second

synthesizer and are injected towards the RFIC output. Then one

records for all applied A2s the corresponding B2s and one tries to

discover the mathematical relationship between the A2s and the

B2s. Finally one repeats this process for different A1s (typically oneperforms a power sweep). The mathematical relationship that is

discovered between the A2s and the B2s is what we call the Hot

S22-model of the DUT.

8/11/2019 Hot S22

7/22

6

6

Experimental set-up

Large-Signal Network Analyzer

(LSNA, formerly known as NNMS)

RFIC

A1 B1 B2 A2

synth2synth1

synth1 generates A1

synth2 generates a set of A2s

LSNA measures all A1s, B1s, A2s, B2s

The experimental set-up we used looks as follows. The set-up can be

viewed in some sense as an extended vector network analyzer.

The set-up contains two synthesizers synth1 and synth2.

Synth1 generates the large-signal input, noted A1. This ensures that

all measurements can be performed under hot conditions.

Synth2 generates a set of A2s. For a good experiment design it is

very important that A2 can be applied independent from A1.

The two synthesizers can put the DUT in all experimental conditions

that are needed to investigate Hot S22.

Note that it is possible to use a tuner in stead of a Synth2. In our case

the use of a synthesizer makes the set-up more flexible. The

synthesizer allows, for example, to emulate any given loadimpedance through active loadpull.

Next to the signal generation, there is of course the problem of the

accurate measurement of the A1s, the A2s and the B2s. A simple

vector network analyzer is of no use here since it can merely

measure ratios of waves (the S-parameters) rather than the absolute

waves.

For the purpose of absolutely measuring the waves we connect the

DUT to the test-set of a Large-Signal Network Analyzer, the

instrument formerly known as NNMS. This instrument allows to

accurately detect the incident and scattered (pseudo-)voltage wavesA1, B1, A2 and B2 which appear at both of the DUT signal ports.

8/11/2019 Hot S22

8/22

7

7

Interpretation of the data

0.3

0.2

0.1 0 0.1 0.2 0.3

0.3

0.2

0.1

0

0.1

0.2

0.3

0.75

0.5

0.25 0 0.25 0.5 0.75

0.75

0.5

0.25

0

0.25

0.5

0.75

A2 (V) B2 (V)

IQ-plots of the A2s and B2s for a constant A1(x-axis = real part, y-axis = imaginary part)

Next we do the following experiment. We choose one value for A1.

Then we apply a set of different A2s and we look at the relationship

between the B2s and the A2s for the particular value of A1.In the graph we have plotted the set of applied A2s and the

corresponding set of B2s in so-called IQ-plots. This means that we

plot the real part of each measured complex number on the x-axis

and the corresponding imaginary part on the y-axis. At a first glance it

is hard to interpret the data. In the past people have turned to

Volterra and other theories in order to get the mathematical

relationship between the two measured quantities A2 and B2. An

simple intuitive interpretation is possible, however, by introducing a

so-called phase normalization. The idea is to use the A1 wave as a

phase reference.This is done as follows. Suppose that one has one measurement of

the quantities A1, B2 and A2. One will then apply a phase shift to all

of these quantities which equals the opposite of the phase of A1. As

a result one gets a new set of A1, A2 and B2 where A1 has a zero

phase. In what follows this kind of phase normalization is always

used for representing A2 and B2.

8/11/2019 Hot S22

9/22

8

8

Phase normalization is good

Normalize the phases relative to A1

P = ej.arg(A1)

0.3 0.2 0.1 0 0.1 0.2 0.3

0.3

0.2

0.1

0

0.1

0.2

0.3

0.6

0.4

0.2 0

0

0.2

0.4

0.6

0.8A2.P

-1

(V) B2.P-1

(V)

When applied to the data previously shown, it is clear that the result

leans itself better for an intuitive understanding of the relationship

between the A2s and the B2s.Note that the phase normalization is mathematically expressed as a

division by the complex phasor P, which has a phase equal to the

phase of A1 but which has a unity amplitude.

8/11/2019 Hot S22

10/22

9

9

1.25 1 0.75 0.5 0.25 0

0

0.25

0.5

0.75

1

1.25

1.5

Varying the amplitude of A1

B2.P-1 (V)

A1 increas

es

Let us now do an experiment, using the same set of A2s, but varying

the amplitude of A1.

If we plot the set of corresponding B2s we note several things.

First of all the midpoint of the resulting B2s changes and gets a

larger amplitude with increasing A1. This makes a lot of sense. The

midpoint of the B2s can be interpreted as the output of the amplifier

when it is perfectly matched (A2 equal to zero). Applying the A2s can

be viewed as applying a deviation from the perfect match. We also

note that these midpoints are approximately located on a straight line.

This shows that the component that we are measuring has

insignificant AM-to-PM.

But the most striking observation is that the smiley gets distorted as

A1 increases. The question now is whether a classic Hot S22predicts this kind of behavior, and, if this is not the case, whether it is

possible to come up with an alternative solution.

8/11/2019 Hot S22

11/22

10

10

Varying the amplitude of A2

Linear dependency versus A2

0.5 0.25 0 0.25

0.6

0.4

0.2

0

0.2

0.4

0.5 0.25 0 0.25

0.5 0.25 0 0.25

0.6

0.4

0.2

0

0.2

0.4

0.5 0.25 0 0.25

0.5 0.25 0 0.25

0.6

0.4

0.2

0

0.2

0.4

0.5 0.25 0 0.25

1.3 1.2 1.1

1.4

1.45

1.5

1.55

1.6

1.65

1.3

1.2

1.1

1.3 1.2 1.1

1.4

1.45

1.5

1.55

1.6

1.65

1.3

1.2

1.1

1.3 1.2 1.1

1.4

1.45

1.5

1.55

1.6

1.65

1.3

1.2

1.1

B2.P-1 (V)

A2.P-1 (V)

Next we will do another experiment. We will now keep A1 constant

and we will vary the amplitude of A2.

From the plots above we can conclude that, in a good approximationand despite the distortion, the B2 deviation from the midpoint is

proportional to A2. The brings us to the conclusion that the

relationship is almost linear.

8/11/2019 Hot S22

12/22

11

11

Data interpretation as loadpull

Increasing amplitude of A1

Just out of curiosity, the experimental data can also be viewed in a

loadpull mode.

The graph shown corresponds to the experiment with increasingamplitude of A1 (the set of A2s is kept constant). For a loadpull

graph one plots the ratio of A2 and B2 on a Smith chart in order to

see the corresponding reflection coefficients and impedances. Note

that the area covered by the smiley decreases for an increasing level

of A1. This is because the corresponding B2 have a higher amplitude.

8/11/2019 Hot S22

13/22

12

12

Classic S-parameter description

B2 = S21 (|A1|).A1+ S22.A2

1.25 1 0.75 0.5 0.25 0 0.25

0

0.5

1

1.5

B2.P-1 (V)

Let us know check how good existing approaches can explain the

experimental data corresponding to the increasing amplitude of A1.

As a first approach I consider a classic S22-parameter, combinedwith a large-signal S21, which in fact corresponds to a compression

and AM-PM characteristic.

The mathematical relationship between the B2s and the A2s and

A1s in this case is given by a simple S-parameter relationship, where

the S21 has become a function of the amplitude of A1.

The experimental data is depicted in orange, the S22-model is

depicted in yellow.

As we expected the S22 is not capable of describing the fact that the

relationship between the A2s and B2s is clearly a function of theamplitude of A1.

This S22-model is also not capable of describing the kind of distortion

that we see in the experimental data.

8/11/2019 Hot S22

14/22

13

13

Simple Hot S22 description

B2 = S21(|A1|).A1+ S22(|A1|).A2

1.25 1 0.75 0.5 0.25 0 0.25

0

0.5

1

1.5B2.P

-1 (V)

In the past, people improved the accuracy of the previous model by

also making S22 a function of the amplitude of A1.

The resulting mathematical equation is shown above. This Hot S22can actually be measured by modified vector network analyzer set-

ups. It is clear that this model is linear in A2 (what we want it to be),

and can handle a varying amplitude of A1. Unfortunately this kind of

model can never describe the kind of distortion that we see in the

experimental data. It still looks like something is missing.

8/11/2019 Hot S22

15/22

14

14

Model linearity & squeezing

We look for a mathematical model which

is linear (superposition valid)

squeezes

Squeezing implies that the phase of A2.P-1

matters

We need different coefficients for the real and

the imaginary part of A2.P-1

More elegant expression results when using

A2.P-1 and its conjugate

So we are looking for an alternative model which:

-Is linear in A2

-Describes the squeezing of the smiley

-Is a general function of the amplitude of A1

The fact that we see a flattening of the smiley implies that the phase

of A2 relative to A1 matters. This implies that we should use a

different coefficient for the real and imaginary part of the phase

normalized A2. This is in fact equivalent to using a different

coefficient for the phase normalized A2 and its conjugate (this results

in a more elegant expression).

8/11/2019 Hot S22

16/22

15

15

Mathematical expression

B2.P-1 = S21(|A1|).A1 .P

-1 +

S22(|A1|).A2 .P-1 +

R22(|A1|).conjugate(A2 .P-1)

B2 = S21(|A1|).A1+

S22

(|A1|).A

2+

R22(|A1|).P2.conjugate(A2)

The resulting expression is shown above.

We start by writing a linear relationship between the phase

normalized quantities B2, A2, the conjugate of A2, and A1, whereeach of the coefficients is a general function of the amplitude of A1.

Next we multiply both sides by the phasor P, which results in the

lower expression. This expression is actually an extension of the

previous Hot S22, where a linear term is added including the

conjugate of A2. Note that this variable is always multiplied by the

square of P. As such this term is not only a function of the amplitude

of A1 but also of the phase between A2 and A1.

8/11/2019 Hot S22

17/22

16

16

Extended Hot S22

B2 = S21(|A1|).A1+ S22(|A1|).A2+ R22(|A1|).P2.conjugate(A2)

1.25 1 0.75 0.5 0.25 0 0.25

0

0.5

1

1.5

B2.P-1 (V)

We will now confront the so-called extended Hot S22 with the

experimental data.

We immediately see that the introduction of the conjugate term isprecisely what is needed in order to describe the squeezing that

takes place.

Note that the resulting model is still linear in A2!

8/11/2019 Hot S22

18/22

17

17

Quadratic Hot S22

Further improvement is possible by using a

polynomial in A2 and conj(A2)

E.g.: quadratic Hot S22

B2 = F.P + G.A2+ H.P2.conj(A2) +

K.P-1.A22 + L.P3.conj(A2)

2 + M.P.A2.conj(A2)

Note the presence of the P factors(theory of describing functions)

For those who want even more accuracy, especially for an increasing

amplitude of A2, the idea can easily be extended to describe

nonlinear dependencies on A2.The result is shown above. Although not explicitly written for clarity,

all coefficients F, G, H, K, L and M are general functions of the

amplitude of A1. Also note the presence of the P factors raised to a

certain power (factors given by the theory of describing functions).

8/11/2019 Hot S22

19/22

18

18

Comparison (highest A1 amplitude)

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

1.4 1.3 1.2

1.4

1.5

1.6

1.7

B2.P-1 (V)

Classic

S22

Simple

Hot S22

Extended

Hot S22Quadratic

Hot S22

The figure above shows the performance of the 4 Hot S22

approaches when confronted with the experimental data.

A classic S22 is clearly inaccurate since it is not a function of A1.

The simple Hot S22 includes the dependency on the amplitude of

A1.

The extended Hot S22 also includes the dependency on the phase

of A2 versus A1 (the squeeze), resulting in a significantly better

match between experiment and model.

An even better match results with the quadratic Hot S22, which

describes a 2nd order nonlinear dependency on A2.

8/11/2019 Hot S22

20/22

19

19

Residuals

25 20 15 10 5 0 5

50

45

40

35

30Relative Error

(dB) S22

Simple

Hot S22

Extended

Hot S22

QuadraticHot S22

Amplitude of A1 (dBm)

A more qualitative measure for the performance of the models is

seen when looking at their residuals (relative to the total energy in the

signal) as a function of the amplitude of A1.It is clear from the graph that the addition of the conjugate term

significantly reduces the value of the residual (up to 15 dB), except

for the lowest value of A1. This makes a lot of sense since for low

amplitudes of A1 the Hot S22 reduces to a perfectly linear S-

parameter model which contains no conjugate term (the same is true

for the quadratic Hot S22).

We also see that the quadratic Hot S22, for our measurements, only

results in a marginal improvement (3 dB) , and this only for the

highest input powers of A1.

Note that the relative residual becomes lower than 50 dB with theproposed extended Hot S22, which approaches the dynamic range

of our measurement system (about 60 dB for this set of

measurements).

8/11/2019 Hot S22

21/22

20

20

Conclusion

An accurate Hot S22 exists

It has a coefficient for the conjugate of A2.P-1

It can accurately be measured

It describes the relationship between A2 and

B2 under large-signal excitation

There exists a more accurate Hot S22 concept.

It contains a linear term in the conjugate of the phase normalized A2.

It can be measured with a Large-Signal Network Analyzer and

accurately describes the relationship between B2 and A2 under a

large signal excitation.

8/11/2019 Hot S22

22/22

21

More information

More detailed information on this kind of

measuring and modeling techniques:

http://users.skynet.be/jan.verspecht

http://www.agilent.com/find/lsna