### 4.1 Introduction and Overview

In this chapter, we consider the rigorous generalization of S-parameters to large-signal conditions. In the first part of the chapter we restrict the treatment to steady-state stimuli and responses or, equivalently, continuous wave (CW) conditions. Dynamics are introduced toward the end.

We saw in Chapter 1 that the behavior of real components generally violates linear superposition. Superposition fails even when the response frequency is always the same as the stimulus frequency. Additionally, real devices typically generate signal components at frequencies that are not present in the excitation spectra. Both of these cases are outside the simple and idealized realm of linear analysis and linear modeling, and require a more general treatment such as we consider here.

Like S-parameters, the general framework presented here is based on the notion of *time-invariant spectral maps*. (Recall time invariance was defined in Chapter 3 section 9). We saw, in Chapter 3, that linear scattering models automatically satisfy time invariance. We will see, in Section 4.3, that most nonlinear spectral maps do not satisfy the time-invariance condition. This means that to properly model nonlinear components (e.g. diodes and transistors) under large-signal stimulus conditions, we must impose the time-invariance property as a constraint on the mathematical formalism. Like S-parameters, composition rules for nonlinear system design follow from the behavioral models of constituent components based on the rules of nonlinear circuit theory. The models treated here can be combined with linear and nonlinear behavioral models, and also standard “compact” transistor models for large-signal frequency domain analysis such as harmonic balance, as described in detail in earlier chapters.

These large-signal behavioral models have a use model very similar to S-parameters but are much more powerful. They are rigorous supersets of S-parameters, to which they reduce in the small-signal limit. A conceptual diagram for this paradigm is shown in Figure 4.1.

Figure 4.1 Large-signal frequency-domain behavioral modeling paradigm.

Like S-parameters, the models considered here can be identified from measurements, derived from analytical models^{1}, or computed from numerical models in a circuit simulator. The types of required measurements are made on modern RF and microwave instruments called nonlinear vector network analyzers (NVNA) or large-signal network analyzers (LSNA) [1] [2] [3]. Based on good measurements, the resulting large-signal models of components, such as transistors, can be extremely accurate. Simulation-based nonlinear models can be created in the nonlinear simulator in analogy with how an S-parameter analysis produces a linear file-based behavioral model [4].

A major benefit of such high fidelity nonlinear frequency-domain behavioral models is that they can be shared between component vendors and system integrators freely, without the possibility that the component implementation can be reverse engineered. This feature protects vendor intellectual property (IP) and promotes its sharing and reuse. This is quite analogous to the case where S-parameters – measured or computed – are provided by manufacturers to represent the performance of a linear component, instead of a detailed equivalent circuit model that could betray a proprietary implementation or design approach.

The systematic set of approximations to the general theory, based on spectral linearization, is introduced in the chapter in a number of places, to enable practical engineering applications to important classes of problems. These simplifications can result in significant savings of measurement time and file-size, reduce hardware system complexity, and enable faster simulations at a minimal cost in accuracy. Multiple approaches to motivating, deriving, and interpreting the formalism are presented.

Finally, the important topic of dynamic memory is introduced, including its importance, some of its symptoms, and several major causes. A powerful extension beyond the static theory is presented in the guise of *dynamic X-parameters.* It is shown how this general approach can successfully model many important nonlinear dynamic systems showing a wide range of dynamic-to-the-envelope properties when stimulated with wideband modulated signals.

### 4.2 Signals and Spectral Maps on a Simple Harmonic Frequency Grid

An important application for nonlinear circuit simulation is the analysis of harmonic distortion generated by a nonlinear component, such as a transistor, in response to an excitation at a single fundamental frequency. These response signals, which generally have spectral energy at multiple harmonics of the fundamental frequency, propagate to other components in the circuit and can ultimately get reflected back into any or all of the ports of the device that created them, modifying the outputs in turn. An excitation class that covers most of these cases includes signals incident simultaneously at different ports with frequency components that fall on a grid of discrete non-negative integer harmonic multiples of the fundamental frequency. The DC component is included. A graphical representation is given in Figure 4.2.

We also assume that in response to signals with components at multiple harmonics of the fundamental frequency, the DUT produces scattered waves with nonzero components only on the same harmonic frequency grid. This is usually the case for microwave and RF applications. However, we know from Chapter 1 that it is possible for the DUT to be driven into a chaotic regime where this assumption fails. We also know that parasitic oscillations can be generated and appear in the output spectrum, falling outside the harmonic grid. A phenomenon that is not that uncommon in the PA context is the generation of oscillations at a frequency *half* that of the fundamental component of the driving frequency excitation, which also does not fall on the grid. We do not consider these cases further here – although the last phenomenon can be handled by minor adaptations of the methods presented here.

In Figure 4.2, the signal magnitudes are indicated by the lengths of the arrows. Each harmonic component of the signal also has a phase that can be well-defined relative to that of the fundamental frequency. This is a consequence of the fact that the frequency components are all commensurate. A set of *K* frequencies, {*f*_{k}}, *k* = 1, …, *K*fk,k=1,…,K, are commensurate if there exist nonzero integers, *n*_{k}nk, such that (4.1) is satisfied. For more on commensurate signals see [5].

#### 4.2.1 Notation for Wave Variables (Phasors) on Harmonic Grid

The complex-valued (for magnitude and phase) incident and scattered wave amplitudes (phasors) now require two indices. The first index refers to the port, and the second to the harmonic index relative to the fundamental. Incident and scattered waves will therefore be labeled according to (4.2):

For example, *A*_{2, 3}A2,3 is an incident sinusoidal wave at port two at a frequency three times the fundamental, while *B*_{1, 2}B1,2 is the scattered sinusoidal wave component at port 1 at the second harmonic of the fundamental.

By varying the complex amplitudes of the incident waves’ phasor components at each harmonic of the fundamental frequency and measuring the complex amplitudes of the scattered waves at each port, *p*, and harmonic frequency index*, k*, we can build up the set of complex-valued nonlinear algebraic maps of the form (4.3). This set of maps^{2} defines the behavior of the DUT under all steady-state conditions consistent with our previous assumptions.

*B*

_{p, k}=

*F*

_{p, k}(

*A*

_{1, 1},

*A*

_{1, 2}, …,

*A*

_{2, 1},

*A*

_{2, 2}, …)

To be more complete we need to treat the DC components consistently with the RF components. In the end, it is the actual signals – waveforms – that matter, and the DC components are as important as the RF. Adding the bias stimulus response interactions to those of the RF, we obtain a system of nonlinear functions that define the steady-state behavior of the component. A simple example is given in (4.4). The superscripts in (4.4) simply label the type of value returned by the respective functions (e.g., complex-valued scattered wave, real-valued current, and real-valued voltage, respectively). This notation is standardized in [6].^{3}

In (4.4) there are many independent variables. These include the DC applied voltage at port one, *V*_{1}; the incident waves at port one at each positive harmonic frequency, *A*_{1,1} *– A*_{1,N}; the DC applied current at port 2, *I*_{2}; and the incident waves at port two, *A*_{2,1} *– A*_{2,N}.

The scattered RF waves at each port depend nonlinearly on the applied port biases, *V*_{1} and *I*_{2}, and also on each complex-valued phasor of the incident RF signals at each port. Likewise, the DC responses of the DUT, current *I*_{1}I1 and voltage *V*_{2}V2, depend on all of the RF incident signal complex amplitudes, as well as applied biases *V*_{1} and *I*_{2}, respectively. Note that this is different from the typical S-parameter situation where the DC conditions are taken to be independent of the applied RF signals since the RF signals are assumed to be arbitrarily small.

The system defined by equations (4.4) is really not much more than book-keeping at this point. For this two-port example there are *2N + 2* nonlinear functions of *2N + 2* variables. The model definition is not complete until all the functions *F*^{(B)}*, F*_{1}^{(I)}, and *F*_{2}^{(V)} are specified mathematically or otherwise identified from data associated with the component. Even so, the structure of (4.4) makes clear that *cross-frequency stimulus-response interactions* are taken into account. For example, the third harmonic of the scattered wave at port two depends on the complex value of the second harmonic wave incident at port one. Thus harmonic time-domain load-pull under steady-state large-signal conditions is covered by the system of equations (4.4).

### 4.3 Time-Invariant Large-Signal Spectral Maps

A fundamental property of components such as transistors, power amplifiers, nonlinear resistors, etc.^{4} is that of time invariance, discussed in earlier chapters. We saw in Chapter 3 that for linear components the S-parameter mathematical relationship between incident and scattered waves is automatically consistent with the time-invariance principle. This is not true in the nonlinear case, as we will see in section 4.3.1. Since a model inconsistent with the time-invariance property of the physical component *must* give wrong answers, it behooves us to properly ensure this property is correctly implemented.

Referring to the previous example of (4.4), we will show not all possible functions *F*^{(B)}, F1I, and F2V are consistent with the mathematical representation of the time-invariance property. In fact, most are not. The restriction of such functions to those spectral maps obeying the time-invariance property results in X-parameters. This could be taken as the fundamental definition of X-parameters. We will derive these restrictions in Section 4.3.1

Other frequency-domain large-signal behavioral models also obey the time-invariance property. One of these, the Cardiff Model, will be described in section 4.9.2.

#### 4.3.1 Derivation of Time-Invariant Spectral Maps

We don’t need to restrict ourselves to periodic signals for the statement of time invariance, and we drop the port indices for simplicity. We assume the DUT produces an output signal, *b*(*t*)bt, when stimulated by an incident signal, *a*(*t*)at. A time-invariant DUT means that for an incident signal, *a*(*t* − *τ*)at−τ (the same signal as before but now delayed by an arbitrary time, *τ*τ), the resulting output signal *must* be *b*(*t* − *τ*)bt−τ (the same response as before but delayed by precisely the same time, *τ*τ).

Returning to our case of periodic stimulus and response, we can write the time-invariance condition in terms of the Fourier coefficients of the incident and scattered periodic signals. We recall that a time-shift, *τ*τ, produces a phase-shift, *e*^{jkω0τ}ejkω0τ, for the *k*^{th} harmonic term of the Fourier series of a periodic signal. This results in the following frequency-domain expression of time invariance for one of the equations of (4.4).

Since (4.5) must hold for all real values of *τ*τ, we choose the particular value given in (4.6). The selection of *τ*τ according to (4.6) can be interpreted as selecting the phase reference to be one of the system’s signals, and that signal is *A*_{1, 1}A1,1. This has the effect of *phase-normalizing* each harmonic coefficient of the incident and scattered waves to the phase of *A*_{1, 1}A1,1, and in particular results in (4.7).

*A*

_{1, 1}

*e*

^{−jω0τ}= |

*A*

_{1, 1}|

*e*

^{jϕ(A1, 1)}

*e*

^{−jω0τ}= |

*A*

_{1, 1}|

Equation (4.5) can now be recast as (4.8), using the notation *P* = *e*^{jϕ(A1, 1)}P=ejϕA1,1.

Equation (4.8) shows that the principle of time invariance *restricts the functional form* of the spectral mappings to those that change only by a multiplicative phase factor when the arguments to the function are appropriately transformed (phase-shifted) according to their harmonic indices. Exercise 4.1 explores cases where (4.8) is not satisfied. Given (4.7), we can therefore define X-parameter functions as in the bottom of (4.8). Notice that the X-parameter functions, denoted for now by *X*_{p, k}Xp,k, are defined on a set of independent variables of *dimension one fewer* than the original mappings Fp,kB. This follows because only the real-valued quantity |*A*_{1, 1}|A1,1, rather than the general complex quantity *A*_{1, 1}A1,1, enters the domain of the *X*_{p, k}Xp,k functions. The final complex coefficient of the scattered wave is obtained by multiplying the value of the X-parameter function by the appropriate phase factor depending on the harmonic index of the scattered wave.

#### 4.3.2 Graphical Representation of Time-Invariant Spectral Maps

The evaluation of an X-parameter model makes explicit use of the time-invariant spectral mapping condition given by (4.8). For example, consider the case of a two-port DUT with signals incident at port one at the fundamental frequency, and simultaneously at port two with components at the fundamental frequency, second, and third harmonics. These four phasors are shown in the top row of Figure 4.3. The length of a vector indicates its magnitude, and the angle represents the phase.

Figure 4.3 Phase normalization of incident waves to the reference stimulus.

Since *ϕ*(*A*_{1, 1})ϕA1,1, the phase of *A*_{1, 1}A1,1, is not zero, the X-parameter functions that appear in the bottom of (4.8) can’t be evaluated. The prescription of (4.8) requires that we first phase-shift the *k*th spectral components of the incident waves by *k* times the phase of *A*_{1, 1}A1,1 in the clockwise direction. This is accomplished by multiplying each variable *A*_{p, k}Ap,k by *P*^{−k} = *e*^{−jkϕ(A1, 1)}P−k=e−jkϕA1,1. The second harmonic terms will therefore be phase-shifted twice as much as the fundamental component, and third harmonic terms three times as much, etc. DC terms (not shown in the figure) are not modified – consistent with DC being the “zeroth” harmonic, or *k = 0*. The resulting phasors, denoted by Ap,kn=Ap,kP−k, define the *reference stimulus.*^{5} These are shown in the bottom of Figure 4.3, emphasizing the relative phase rotations of the various harmonic components. In the time domain, the reference stimulus is simply the original set of periodic incident waveforms *translated in time* with respect to the original waveforms by a time, *τ*τ, given by (4.6) [7].

The X-parameter functions of (4.8) can be evaluated at the reference stimulus because *A*_{1, 1(n)}A1,1n is a real number. We denote by *reference response* the values of the scattered waves returned by the X-parameter functions evaluated at the reference stimulus. The corresponding phasors of the reference response are denoted Bp,kn. Since the DUT is nonlinear, there will generally be scattered waves at each port with nonvanishing components at multiple harmonics beyond those of the incident signals. Values for four of these scattered phasors of the reference response are shown in the top part of Figure 4.4.

Figure 4.4 Phase denormalization from the reference response to the actual response.

Finally, the desired *B*_{p, k}Bp,k values (corresponding to the actual *A*_{p, k}Ap,k incident waves) are obtained by a *phase denormalization* process applied to the reference response. This is accomplished by a phase-rotation in the counterclockwise sense (complex multiply by *P*^{k} = *e*^{jkϕ(A1, 1)}Pk=ejkϕA1,1) of the values returned by the X-parameter functions according to (4.8). In the time-domain, this corresponds to the appropriate delay of the periodic reference response by the precise amount, *τ*τ, by which the incident signals were delayed compared to the reference stimulus.

In summary, we have taken an arbitrary periodic stimulus, time-translated it to a reference condition where the phase of *A*_{1, 1}A1,1 can be taken to be zero, evaluated the X-parameter functions, and time-translated the result back to obtain the final periodic response. Since this is a manifestly time-invariant process, we are guaranteed the DUT’s time-invariant behavior is built in to the mathematical description through the X-parameter formalism.

### 4.4 Large-Signal Behavioral Modeling Framework in Wave-Space

The steady-state nonlinear behavior of RF and microwave components are defined in terms of time-invariant spectral maps defined in wave-space – with DC conditions specified in terms of real-valued currents and voltages that also depend on the RF incident waves. Schematically, this is represented by Figure 4.5, for an example of a component that we label, simply, as a (frequency) converter.

The example of Figure 4.5 has three RF ports, labeled 1 through 3, as well as a DC port, labeled 4. The incident RF waves are indicated by *A*_{p}, where the port index *p* ranges from 1 to 3. The waves have multiple harmonic components, but these second indices are suppressed in the figure for simplicity. The incident waves and the DC applied voltage, *V*_{4}, are taken as the independent variables. The scattered waves, *B*_{p}, and the DC current, *I*_{4}, are given by X-parameter relations such as (4.8). For example, the RF outputs at port 2 are specified by the X-parameter expression (4.9), for Ap,kn=Ap,kP−k

For port 4, the expression for the bias current response is given in (4.10). In the general case, there are both DC and RF input and outputs at each port.

To model a particular device, the abstract constitutive relations (4.9) and (4.10) – as well as those for every X-parameter function corresponding to Figure 4.5 – must be identified from measured (or simulated) input–output characteristics of the component. In principle, this can be obtained by an exhaustive process of applying DC and RF sources at each port and varying the magnitudes and phases of all RF components and the value of the applied bias voltage, *V*_{4}, and acquiring the resulting responses. The outputs can be fitted with smooth functions of all the inputs or tabulated and interpolated by the simulator during simulation. This process is limited by “the curse of dimensionality,” however. That is, it becomes exponentially complex in the number of independent variables. To deal with this, we present systematic approximations that can be used to more conveniently apply the X-parameter principles to many important cases of practical interest, starting in Section 4.6.

### 4.5 Cascading Nonlinear Behavioral Model Blocks

How is it possible to combine, in analogy with linear S-parameters, multiple large-signal behavioral functional blocks to simulate a more complicated nonlinear circuit or system? We illustrate the concept by connecting port 2 of the converter block of Figure 4.5 to the input of a simple power amplifier (PA) block in Figure 4.6. The input port of the PA is labeled port 5, and the PA output is labeled port 6. The scattered waves at port 5 of the PA are defined by X-parameter constitutive relations that depend only on the incident waves (and biases, not shown) at ports 5 and 6.

Figure 4.6 Cascading large-signal functional blocks in wave-space.

To solve the circuit of Figure 4.6, Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL), must be applied at the node connecting the two blocks. These conditions impose constraints on the sets of incident and scattered waves, given by the simple relationships (4.11). Specifically, at the solution of the circuit, the incident waves at port 2 must be equal to the scattered waves from port 5, and the scattered waves from port 2 must be equal to the incident waves at port 5. These equalities must hold separately for each harmonic index, *k, k = 1,…,N*. That is, (4.11) is a set of *2N* equations, or if including DC terms, *2(N + 1)* equations. It is a simple matter to prove that (4.11) is equivalent to the circuit laws, KVL and KCL, when expressed in terms of wave variables using equations 3.1:

The solution for the entire circuit is given by eliminating the scattered wave variables and solving the set of self-consistent implicit nonlinear equations for the incident waves *A*_{2, k} and *A*_{5, k}A2,kandA5,k for *k = 1,…,N*, given by (4.12). Contrary to the case of cascaded S-parameters considered in Chapter 3, because the present problem is nonlinear, no closed form solution can be found in general for the X-parameters of the cascade as a function of the X-parameters of each block. The solution to (4.12) is therefore obtained numerically by the nonlinear circuit simulator:

Of course most available circuit simulators first convert the wave variables in (4.12) to currents and voltages, using equations (3.2), and then solve the resulting implicit nonlinear equations in current-voltage space.

In fact, the simple relations (3.1) and (3.2) between wave variables and current-voltage variables enable functional blocks to be cascaded with nonlinear (and linear) functional blocks expressed in wave-space and/or conventional current-voltage space. That is, large-signal frequency-domain behavioral model functional blocks are completely compatible with distributed passive elements, lumped nonlinear elements defined in the time domain, and “compact” nonlinear transistor models. However, the present formulations of X-parameters and similar frequency-domain nonlinear behavioral models (e.g., the Cardiff Model – see Section 4.9.2) are presently limited to harmonic balance and circuit envelope analyses or their subsets (S-parameters, small-signal mixer analysis, etc.).

It should be stressed that the cascadability of these functional blocks depends only on the accuracy of the sampled constitutive relations of each block. The results for the cascade follow directly by applying only the circuit laws. No additional approximations are required. The cascadability becomes arbitrarily accurate by taking into account enough harmonic components of the incident and scattered waves at sufficiently many complex values of the independent incident wave variables.

### 4.6 Spectral Linearization

#### 4.6.1 Simplification for Small Mismatch

The considerations used for the definition and application of X-parameters and related approaches described thus far apply to the steady-state behavior of time-invariant nonlinear components with incident and scattered waves on a harmonic frequency grid. The generality of the formalism comes at the cost of considerable complexity. Each spectral map is a nonlinear function of every applied DC bias condition and all the magnitudes and phases of each spectral component of every signal at every port of the component. Sampling such behavior in all variables for many ports and harmonics would be prohibitive in terms of data acquisition time, data file size, and model simulation time.

Fortunately, in most cases of practical interest, only a few large-signal components need to be considered with complete generality, while most spectral components can be considered small and can be dealt with effectively by methods of perturbation theory. In fact, as stated previously, S-parameters are the limiting case where *all* RF signals are considered small. Specifically, we can apply the following methodology, depicted in Figure 4.7.

– Identify the few large tones that drive the nonlinear behavior of the system.

– Identify the nonlinear spectral maps defined when only these few large tones are incident on the DUT. This nonlinear spectral map represents the specific steady-state of the system, designated the

*large-signal operating point (LSOP).*

– Linearize the spectral maps around this

*LSOP*defined by the few important large tones.

– Treat the remaining small signals as small perturbations, using superposition appropriately, based on the above linearized maps around the

*LSOP.*

This methodology approximates the complicated general multivariate nonlinear maps, *B*_{k}Bk, with much simpler nonlinear maps, XkF, defined on the few preselected large tones only, and *simple linear maps*, Xk,k’SandXk,k’T, which account for the contributions of the many small amplitude tones. These approximations result in a dramatic reduction of complexity while providing an excellent description for many important practical applications. The details of the equations appearing in Figure 4.7 will be derived and discussed further in Section 4.6.2.

Figure 4.7 Illustration of spectral linearization approximation. The equations will be discussed in more detail in Section 4.6.2

The simplest such approximation is to assume there are *no large RF signals* that need to be treated fully nonlinearly. The LSOP is therefore just the DC operating point. All the RF components can then be treated with perturbation theory, by linearizing around the DC operating point. This is just the S-parameter approximation of the previous chapter.

#### 4.6.2 Simplest Nontrivial X-Parameter Case

Beyond S-parameters, the next simplest case of X-parameters involves one large RF signal that is treated generally while the remaining spectral components are treated as small. This is often the case for power amplifiers, at least those “nearly matched” to 50 Ω, where the main stimulus signal is a narrow-band modulation around a fixed carrier incident at the input port. The DUT response generally includes several harmonics. If the DUT is not perfectly matched at the output port, there will be small reflections at the fundamental and the harmonic frequencies back into the output port of the DUT. These small reflected signals will be treated as perturbations to simplify the description. Similarly, scattered waves at the harmonic frequencies at the input could be reflected back into the DUT if it is not perfectly matched at the source. These incident signals can also be treated as small in many applications. The validity of the approximation depends on whether these signals are small enough so that their contributions do not affect the LSOP. This is quite analogous to the condition on an RF signal to be small enough such that it does not change the DC operating point of a device for a valid S-parameter measurement, as discussed in Chapter 3. In Section 4.9 we will consider multiple large signals, in particular to deal with arbitrary load-dependence for large mismatch at the output port (load-pull) and two large incommensurate tones for mixer analysis.

##### 4.6.2.1 Digression: Double-Sided Spectra and Spectral Linearization

So far we have simply assumed our phasor description corresponds to single-sided spectra. That is, the complex value, *A*_{p, k} or *B*_{p, k}Ap,korBp,k, is associated with *e*^{jkω0t}ejkω0t for positive values of the harmonic index *k*, for *ω*_{0}ω0 is a positive angular frequency. The spectral maps take complex amplitudes for positive frequencies into scattered wave amplitudes at positive frequencies (and DC). The simultaneous presence of signals incident at multiple frequencies on a nonlinear device, however, creates *self-mixing*, with signals scattered at frequencies corresponding to the sum and difference frequencies of the incident signals. For example, if we have signals simultaneously incident at positive frequencies *f*_{1} and *f*_{2}f1andf2, we will get scattered signal components at frequencies *f*_{1} ± *f*_{2}f1±f2. This is fine for *f*_{1}>*f*_{2}f1>f2, and we can map incident phasors associated with positive frequencies into scattered phasors also associated with positive frequencies. But we must consider also the case where *f*_{1} < *f*_{2}f1<f2. If we restrict ourselves, as we have so far, to positive frequencies – single-sided spectra – we will have to be careful about how we consistently deal with some of the resulting mixing terms.

For this and other reasons, it is convenient to expand the domain of incident and scattered complex amplitudes to *double-sided* spectra. That is, we consider signals with positive and negative frequencies incident and scattered by the DUT.

We now have the situation depicted in Figure 4.8.

Figure 4.8 Double-sided spectra on a harmonic grid.

Formally, this means we have to consider the arguments to all spectral maps to include negative frequency components as well as positive. Thus we must generalize equations (4.3) to a form given by (4.13). Similar conditions apply to (4.4). We note (4.13) also depends on the DC bias but this is suppressed for simplicity.

*B*

_{p, k}=

*F*

_{p, k}(

*A*

_{1, 1},

*A*

_{1, − 1},

*A*

_{1, 2},

*A*

_{1, − 2}, …,

*A*

_{2, 1},

*A*

_{2, − 1},

*A*

_{2, 2},

*A*

_{2, − 2}, …)

Not only do the arguments of (4.13) include phasors corresponding to negative frequencies on a double-sided harmonic grid, but the index *k* for the scattered wave harmonic is now permitted to take negative integer values as well as positive integer values. While this appears to increase the complexity, we will be able to use this approach as a simple starting point from which to derive the formalism of spectrally linearized X-parameters for small mismatch.

We now define spectral linearization of double-sided spectral maps. We assume all spectral components except *A*_{1, 1} and *A*_{1, − 1}A1,1andA1,−1 are small so we can expand (4.13) in a complex Taylor series. We write, noting the summations omit terms with joint values of (*q* = 1, *l* = ± 1)q=1l=±1:

Here we have introduced the notation *large-signal operating point stimulus*, or *LSOPS*, to refer to the set of large-signal incident waves (and DC applied biases) in the absence of the small signals. Note the summation in (4.14) excludes the phasors *A*_{1, ± 1}A1,±1 that are arguments of the *LSOPS*. The *reference large-signal operating point stimulus*, *RefLSOPS*, is obtained from the *LSOPS* by phase-normalizing the arguments. This transformation brings out the phase factors *P*^{k} and *P*^{k − k‘}PkandPk−k’ in the respective terms using the time-invariance principle discussed in Section 4.3. The derivation of the last equality of (4.14) is left as an exercise.

We have introduced the notation Xp,kF to denote the X-parameter function evaluated at the *RefLSOPS*. The symbols *X*^{(d)}Xd denote the partial derivatives of the double-sided spectral maps evaluated at the *RefLSOPS.*

Now we can get back to real signals by imposing conjugate symmetry. That is,

We now return to the single-sided spectral representation. We restrict ourselves to positive indices but invoke (4.15) to define the contributions from negative frequencies. We can therefore re-express (4.14) as (4.16) where *X*^{(S)}XS and *X*^{(T)}XTare defined for positive indices as the corresponding terms of *X*^{(d)}Xd for only positive indices. For notational simplicity, in this and what follows, we omit the double summation sign even though we are summing over two indices, *q* and *l*. In (4.16) the summation does not include the term for which both *q = 1* and *l = 1* together.

Equation (4.16) is the classical form of the spectrally linearized maps around a single large tone at the fundamental frequency at the input port of a nonlinear DUT [6]. This is defined in terms of single-sided spectra for positive frequencies only, since this is what is measured. Unlike S-parameters, there are independent contributions linear in *A* and also linear in *A**, with separate coefficients *X*^{(S)}XS and *X*^{(T)}XT, respectively. The terms *X*^{(F)}XF, *X*^{(S)}XS, and *X*^{(T)}XT are nonlinear functions of their arguments, namely the applied DC biases and (phase-normalized) complex amplitudes of the RF signals. We can now properly identify the terms of (4.16) with the corresponding labels in Figure 4.7.

Expressions similar in form to (4.16) hold when the *RefLSOPS* is more complicated than the case considered in this section, as we will see when considering two large tones for “load-dependent” X-parameters and for mixers in Section 4.9.

#### 4.6.3 Identification of X-Parameter Terms from Measured Data

##### 4.6.3.1 Ideal Experiment Design and Identification by Offset Phase Method

The objective is to identify (extract), for each specific *RefLSOPS* of the DUT, all of the X-parameter values, specifically Xp,kF,Xp,k;q,lS,andXp,k;q,lT, that enter equation (4.16). We will treat this algebraically in this section and provide a geometrical interpretation in Section 4.6.7.

It is assumed for now (this restriction will be removed in Section 4.6.3.2) that the source and the measurement instrument are ideal. Specifically, we assume an ideal incident tone at port one and that no signals generated by the DUT in response to the large excitation signal are reflected back into the DUT.

For each port number, *p*, and harmonic index *k*, the ideal experiment design for each *RefLSOPS* consists of a sequence of three distinct RF excitations, labeled by integer *m* and measurements of the corresponding scattered wave, Bp,km for each excitation.

The first excitation is a single large tone of complex amplitude *A*_{1, 1}A1,1, applied at port one, with measurement result denoted by Bp,k1. The second excitation involves application of the same large-amplitude tone, *A*_{1, 1}A1,1, and, simultaneously, the application at port *q* and harmonic index *l* of a small perturbation tone with complex amplitude Aq,l1. The measurement result for this case is Bp,k2. Finally, the third excitation is of the same type as the second experiment, but with a distinct phase for the small tone at port *q* and harmonic index *l*, which we label as Aq,l2. The measurement result is Bp,k3.

Using (4.16), we obtain three linear equations for the unknown X-parameter values Xp,kF,Xp,k;q,lS,andXp,k;q,lT, in terms of known excitations A1,1,Aq,l1,andAq,l2 and the measured values Bp,k1,Bp,k2,andBp,k3 shown in equation (4.17).

This procedure is then applied, separately, for small perturbation tones at other ports and harmonic indices (different values for the indices *q* and *l*), and the entire process repeated for each distinct *RefLSOPS* for which the X-parameter values are desired. The phases of the small tones, *A*_{q, l}Aq,l, in (4.17) can be arbitrary,^{6} but the system of equations is better conditioned when the phase difference between Aq,l1andAq,l2 is close to 90 degrees. More than two phases for the applied small tones, Aq,lm, can be used as stimuli and the resulting augmented system (4.17) solved for the X-parameter values in a least squares sense in order to reduce the effect of measurement noise.

##### 4.6.3.2 X-Parameter Identification in an Imperfect Environment

The method of Section 4.6.3.1 describes X-parameter identification in a perfect world. However, an actual microwave source, when used at high power conditions, may generate its own complex-valued harmonic distortion components, so the intended ideal excitations of the previous section may not be realized precisely. The output of the DUT in response to such a “dirty source” is therefore a combination of its response to a pure tone and that of the amplified spurious signals that contaminated the input. In addition, the measurement HW may not present a perfect 50 Ω environment to all signals generated by the DUT in response to the large-signal excitation. Both of these nonidealities, if not properly accounted for, create errors in the values of the X-parameters that we wish to attribute to the DUT.

An extension of the method described in Section 4.6.3.1 was developed in [8] for measuring and identifying X-parameters using nonideal HW. The method can be used to characterize the imperfect source and then “predistort” it so that it produces a pure, single-tone input, or, equivalently, correct the DUT response to a dirty source and imperfect match, in order to predict the DUT response to an ideal source or sources. The method is a practical way to calibrate out the imperfections of the source and measurement HW so as to properly return X-parameters that can be regarded as *intrinsic properties of the DUT on its own*, rather than the combined source-DUT-instrument system. A schematic representation of the idea is given in Figure 4.9.

Figure 4.9 Method to identify X-parameters in an imperfect environment

Despite the imperfect environment, the complete set of *X*^{(S)} and *X*^{(T)} parameters can be identified from a global analysis of the measured data from the entire set of experiments of the type described in Section 4.6.3.1 for all values of small tone indices *q* and *l*. The desired stimulus (a large pure tone at the input) is subtracted from the actual measured stimulus that consists of one large tone and several complex harmonic components. This results in an “imperfection” signal with small components at each harmonic of the fundamental. This imperfection signal is then used with the previously identified *X*^{(S)} and *X*^{(T)} parameters to calculate corrections to the measured response so as to return the actual DUT response to the ideal large stimulus.

This method is particularly effective in X-parameter applications where (far from 50 Ω) loads are controlled and presented to the DUT by the HW system (e.g. with a tuner or an active source at the output port), but where harmonic loads may be uncontrolled and vary as the fundamental load is changed. This will be described in the context of an example in section 4.9.1.5. The method, so applied, is used to calibrate out the harmonic terminations to help present X-parameters referenced to ideal excitations under known loads with perfectly matched harmonic terminations.

#### 4.6.4 Example: *X*-Parameters of a Two-Port Amplifier with Small Mismatch

A simple and illustrative example of the basic X-parameter formulation (4.16) is a nearly matched two-port power amplifier driven by a large signal at port 1. Neglecting any harmonics, for simplicity, (4.16) reduces to (4.18).

Thus a two-port, in this simple “fundamental only” approximation, has six X-parameter functions, whereas a linear two-port is described by only four S-parameters. The six X-parameter functions in (4.18), and therefore the scattered waves, *B*_{p, 1}Bp,1, depend on |*A*_{1, 1}|A1,1 (shown) and also depend on the DC bias conditions, but we suppress the DC arguments for simplicity.

In Figure 4.10, we plot the values of the three X-parameter functions, X2,1F,X2,1;2,1S,andX2,1;2,1T in (4.18), that contribute to *B*_{2, 1}B2,1, versus the input port drive, |*A*_{1, 1}|A1,1. The function X2,1F is divided by the magnitude of the incident wave to make the result unitless, like the other two functions shown.

Figure 4.10 Some X-parameter functions of a power amplifier.

We observe two things about the various X-parameter functions. As the power of the incident signal diminishes, so does the value of the *X*^{(T)}XT term. In fact, in the limiting case of vanishing power, the *X*^{(T)}XT term goes all the way to zero, as we’ll see in Section 4.8. This should be no surprise. After all, we know that in the small-signal limit, the S-parameter formalism of Chapter 3 is a complete description. Since the *X*^{(T)}XT terms in (4.18) contribute to the scattered waves terms proportional to the conjugates of the incident phasors, and these conjugate terms don’t appear in the S-parameter description, we logically conclude all *X*^{(T)}XT terms must vanish in the small-signal limit.

On the other hand, at large incident power levels at port 1, the magnitudes of the *X*^{(T)}XT terms can increase to the point where they can become comparable to, or even exceed, the magnitudes of the *X*^{(S)}XS terms. The *X*^{(T)}XT terms are therefore just as important as the *X*^{(S)}XS terms at very high incident drive levels. This is true no matter how small the incident signals are at port two when the DUT is driven hard at port one.

##### 4.6.4.1 *X*^{(S)} and *X*^{(T)} Functions in Terms of Double-Sided Sensitivities

If we want, we can relate *X*^{(S)}XS and *X*^{(T)}XT defined on positive frequencies, to the double-sided derivatives of the double-sided spectral maps as follows [9]:

Comparing (4.16) with (4.14) we have, for *k, l >0*:

In (4.19) we again use the superscript “(*n*)” to denote phase-normalized waves. Using (4.15), we also find

The double-sided expression (4.14) is an analytic function (in the sense of complex variable theory) of each complex phasor *A*_{q, l}Aq,l. For single-sided expressions, however, the spectral map (4.16) is nonanalytic. That is, we must differentiate with respect to *A*_{p, k}Ap,k and *A*_{p, k∗}Ap,k∗ independently in this expression. The coefficients, *X*^{(S)}XS and *X*^{(T)}XT, correspond to *partial derivatives of the single-sided spectral map with respect to* *A*_{q, l}Aq,l and *A*_{q, l∗}Aq,l∗, respectively.

#### 4.6.5 Origins of the Conjugate Terms

The appearance in (4.16) of terms linear in both *A*A and *A*^{∗}A∗ is perhaps surprising. The preceding analysis shows that the origin of the *A*^{∗}A∗ terms is the mixing of the small amplitude negative frequency components back into the positive domain. Of course there is no mixing when the DUT is perfectly linear, so these terms have no analogue in the simple S-parameter world. Simply making S-parameter measurements at high input power levels (e.g. so-called “hot S_{22}”) cannot account for such terms and their measurable consequences [10].

Expression (4.16) allows us a more detailed interpretation of Figure 4.7. We suppress the port indices on the diagram for simplicity. The top line of Figure 4.7 shows a single-sided spectral map from four RF signal components (plus DC) into an output spectrum. The nonlinear function depends on the complex values of all independent variables. The approximation made now is to designate one of the incident RF signals as large and the remaining ones as small. Of the four incident signal components (apart from DC), the one at the fundamental frequency is supposed large and the others small. This is indicated by the groupings in the top left of Figure 4.7.

The approximate decomposition of the scattered waves, *B*_{k}Bk, using (4.16), is indicated in Figure 4.7. The simple nonlinear map, designated by XkF, depends only on the DC and the single real number corresponding to the magnitude of the large tone. This is much simpler than the full nonlinear map that depends on all variables. The phase factor *P*^{k}Pk multiplies the X-parameter function to ensure the map is properly time invariant. Note how close the output spectrum on the right column second row is to the full result (top row). The minor differences are now nearly completely accounted for by the contributions from the multiple linear maps, Xk;lSandXk;lT. Each small tone, *A*_{l}Al, contributes linearly to the response around all harmonics due to contributions weighted by *X*^{(S)}XS terms and also linearly in Al∗ with contributions weighted by independent *X*^{(T)}XT terms. Equivalently, there is a contribution at each harmonic, *k*, linear in each of the *A*_{l}Al and each of the Al∗ terms. The “wiggly” nature of the lines indicates the phase-dependent contributions of the complex numbers, added as vectors. The length of the sum of all the vectors at each frequency depends on each of their phases and magnitudes; they add vectorially.

##### 4.6.5.1 More on the Origin of Conjugate Terms: An Example with a Cubic Nonlinearity^{7}

A concrete example is now provided as another illustration of the origins of the conjugate terms. We consider the simple case of a static algebraic nonlinearity (e.g., a polynomial) in the time domain and compute the mapping in the spectral domain. We start by considering a system described by a simple instantaneous nonlinearity containing both a linear and cubic term. We look at the following three cases, for which the analysis can be computed exactly. The first case is the linear response of this nonlinear system around a static (DC) operating point. This is the familiar condition for which linear S-parameters apply. The second case is the linearization of the system around a time-varying large-signal operating state, with the time variation and perturbation having the same fundamental period. The third case is a simple generalization of the second where the perturbation is applied at a distinct frequency compared to the fundamental frequency of the periodically driven nonlinear system.

The objective is to look at the linearized response of the system in the (single-sided) frequency domain and demonstrate that the relationship between the perturbation phasor and the response phasor is not an analytic function in cases 2 and 3, namely when the nonlinear system is driven. That is, these examples illustrate the simultaneous presence of the perturbation phasors and their complex conjugate phasors, with distinct coefficients, in the response of a nonlinear driven system to additional injected signals.

The nonlinearity is described by (4.21). The signal, *x*(*t*)xt, is written in (4.22) as the sum of a main signal and a perturbation term, assumed to be small. We will calculate, to first order in the perturbation, the response of the system (4.21) to the excitation (4.22). We will consider three cases.

*f*(

*x*) =

*αx*+

*γx*

^{3}

*x*(

*t*) =

*x*

_{0}(

*t*) +

*Δx*(

*t*)

**Case 1**

In this case, shown in (4.23), the main signal, *x*_{0}(*t*)x0t, is just a constant value (a DC term) while the perturbation is a small tone at angular frequency *ω*ω. The constant *A* is real and *δ*δ is a complex number of small magnitude that allows for the phase of the perturbation to be arbitrary. The total signal is obviously real.

The first order response is computed in (4.24). The approximation becomes exact as *Δx*(*t*)→0Δxt→0

*Δy*(

*t*) =

*f*(

*x*

_{0}(

*t*) +

*Δx*(

*t*)) −

*f*(

*x*

_{0}(

*t*))≈

*f*

^{′}(

*x*

_{0}(

*t*))

*Δx*(

*t*)

For this case, the conductance nonlinearity, *f*^{′}(*x*_{0})f′x0, is evaluated at the fixed value *x*_{0} = *A*x0=A, where, using (4.21), we obtain (4.25), where *G(A)* is defined as being,

*f*

^{′}(

*A*) ≡

*G*(

*A*) =

*α*+ 3

*γA*

^{2}

Substituting (4.25) into (4.24) and using (4.23) we obtain (4.26):

If we look at the complex coefficient of the term proportional to *e*^{jωt}ejωt we find it is given simply in (4.27):

Since (4.26) is a linear input–output relationship with constant coefficients, the complex Fourier component at the output frequency is linearly proportional to the complex Fourier coefficient of the input small-signal phasor. In fact we can read off the frequency-domain phasor relationship (4.28), or equivalently (4.29), where the gain, *G*(*A*)GA, is given by (4.25) and *X*(*ω*)Xω is the complex input phasor corresponding to the positive frequency component of the small perturbation tone. We note in this case the gain, *G*(*A*)GA, depends nonlinearly on the DC operating point, through (4.25), but is constant in time.

*Y*(

*ω*) =

*G*(

*A*)

*X*(

*ω*)

**Case 2**

This time we take *x*_{0}(*t*)x0t to be a pure sinusoid given in (4.30). There is no loss of generality by taking *A*A to be real, which is equivalent to taking the phase of the main signal at time zero to be zero. We can interpret *δ*δ as the relative phase between the main signal and perturbation signal at the same frequency.

We go through the same procedure as in Case 1 to derive the conductance, but this time it is evaluated at the periodically time-varying operating condition (essentially the *LSOPS*) according to (4.31). The second form follows from the simple trigonometric identity cos2ωt=12+cos2ωt2.

Using (4.31) to evaluate (4.24) for this case, we obtain (4.32):

This time we get contributions proportional to *e*^{jωt}ejωt and *e*^{3jωt}e3jωt, and their complex conjugates, four terms in all. If we restrict our attention, as in case 1, to output terms proportional to *e*^{jωt}ejωt, we obtain the coefficient given in (4.33):^{8}

We observe that the output phasor at frequency *ω*ω is not proportional to the input phasor, *δ*δ, at frequency *ω*ω but instead has distinct contributions proportional to *both* *δ*δ and *δ*^{∗}δ∗. That is, the linearization of the nonlinear system around the periodically time-varying operating point determined by the large tone, is not analytic in the sense of complex variable theory. If it were analytic, (4.33) would depend only on the complex variable, *δ*δ, and not on both *δ*δ and *δ*^{∗}δ∗.

Taking the ratio of the complex output Fourier Coefficient to the input Fourier coefficient, we obtain the result (4.34), where *ϕ*(*δ*)ϕδ is the relative phase of the main and perturbation signal: