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):

Ap,kBq,l(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 maps² defines the behavior of the DUT under all steady-state conditions consistent with our previous assumptions.

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

Bp,k=Fp,k(B)(V1,A1,1,A1,2,…,I2,A2,1,A2,2,…)I1=F1(I)(V1,A1,1,A1,2,…,I2,A2,1,A2,2,…)V2=F2(V)(V1,A1,1,A1,2,…,I2,A2,1,A2,2,…)(4.4)

In (4.4) there are many independent variables. These include the DC applied voltage at port one, V₁; the incident waves at port one at each positive harmonic frequency, A_1,1 – A_1,N; the DC applied current at port 2, I₂; 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, V₁ and I₂, and also on each complex-valued phasor of the incident RF signals at each port. Likewise, the DC responses of the DUT, current I₁I1 and voltage V₂V2, depend on all of the RF incident signal complex amplitudes, as well as applied biases V₁ and I₂, 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), F₁^(I), and F₂^(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).

4.3.1 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, ττ), the resulting output signal must be b(t − τ)bt−τ (the same response as before but delayed by precisely the same time, ττ).

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, ττ, produces a phase-shift, e^jkω₀τejkω0τ, 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).

Bp,k=Fp,k(B)(V1,A1,1,A1,2,…,I2,A2,1,A2,2,…)⇒Bp,ke−jkω0τ=Fp,k(B)(V1,A1,1e−jω0τ,A1,2e−2jω0τ,…,I2,A2,1e−jω0τ,A2,2e−2jω0τ,…)(4.5)

Since (4.5) must hold for all real values of ττ, we choose the particular value given in (4.6). The selection of ττ 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).

τ=ϕA1,1ω0(4.6)

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

Bp,kP−k=Fp,k(B)(V1,A1,1P−1,A1,2P−2,…,I2,A2,1P−1,A2,2P−2,…)⇒Bp,k=Fp,k(B)(V1,A1,1P−1,A1,2P−2,…,I2,A2,1P−1,A2,2P−2,…)Pk≡Xp,k(V1,|A1,1|,A1,2P−2,…,I2,A2,1P−1,A2,2P−2,…)Pk(4.8)

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 kth 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(A_{1, 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, ττ, 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(A_{1, 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, ττ, 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.

4.6.1 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_kBk, 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^jkω₀tejkω0t for positive values of the harmonic index k, for ω₀ω0 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 f₁ and f₂f1andf2, we will get scattered signal components at frequencies f₁ ± f₂f1±f2. This is fine for f₁>f₂f1>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 f₁ < f₂f1<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},  …)

Bp,k=Fp,k(A1,1,A1,−1,A1,2,A1,−2,…,A2,1,A2,−1,A2,2,A2,−2,…)(4.13)

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:

Bp,k=Fp,k(A1,1,A1,−1,A1,2,A1,−2,…,A2,1,A2,−1,A2,2,A2,−2,…)≈Fp,kLSOPS+∑l=−NN∑q=1P∂Bp,k∂Aq,lLSOPS⋅Aq,l≡Xp,kFRefLSOPSPk+∑l=−NN∑q=1PXp,k;q,ldRefLSOPS⋅Aq,lPk−l(4.14)

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,

Ak=A−k∗andBk=B−k∗fork<0(4.15)

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.

Bp,k=Xp,kFRefLSOPSPk+∑q=1,..,Pl=+1not11NXp,k;q,lSRefLSOPS⋅Aq,lPk−l+∑q=1,..,Pl=+1not11NXp,k;q,lTRefLSOPS⋅Aq,l∗Pk+l(4.16)

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).

Bp,k1=Xp,kFPkBp,k2=Xp,kFPk+Xp,k;q,lSPk−lAq,l1+Xp,k;q,lTPk+lAq,l1∗Bp,k3=Xp,kFPk+Xp,k;q,lSPk−lAq,l2+Xp,k;q,lTPk+lAq,l2∗(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).

B1,1(A1,1,A2,1)=X1,1FA1,1P+X1,1;2,1SA1,1A2,1+X1,1;2,1TA1,1P2A2,1∗B2,1(A1,1,A2,1)=X2,1FA1,1P+X2,1;2,1SA1,1A2,1+X2,1;2,1TA1,1P2A2,1∗(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:

Xp,k;q,ld=∂Bp,kP−k∂Aq,lP−l=∂Bp,kn∂Aq,ln=Xp,k;q,lSXp,k;q,−ld=∂Bp,kP−k∂Aq,−lPl=∂Bp,kP−k∂Aq,l∗Pl=∂Bp,kn∂Aq,l∗n=Xp,k;q,lT(4.19)

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

Xp,−k;q,ld=∂Bp,−kn∂Aq,ln=∂Bp,kn∗∂Aq,ln=∂Bp,kn∂Aq,ln∗∗=Xp,k;q,lT∗Xp,−k;q,−ld=∂Bp,−kn∂Aq,−ln=∂Bp,kn∗∂Aq,−ln=∂Bp,kn∂Aq,ln∗=Xp,k;q,lS∗(4.20)

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 AA 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 S₂₂”) 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_kBk, 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^kPk 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_lAl, 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_lAl 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 Nonlinearity⁷

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 + γx³

fx=αx+γx3(4.21)

x(t) = x₀(t) + Δx(t)

xt=x0t+Δxt(4.22)

Case 1

x0t=AΔxt=δejωt+δ∗e−jωt2(4.23)

In this case, shown in (4.23), the main signal, x₀(t)x0t, is just a constant value (a DC term) while the perturbation is a small tone at angular frequency ωω. The constant A is real and δδ 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(x₀(t) + Δx(t)) − f(x₀(t))≈f^′(x₀(t))Δx(t)

Δyt=fx0t+Δxt−fx0t≈f′x0tΔxt(4.24)

For this case, the conductance nonlinearity, f^′(x₀)f′x0, is evaluated at the fixed value x₀ = Ax0=A, where, using (4.21), we obtain (4.25), where G(A) is defined as being,

f^′(A) ≡ G(A) = α + 3γA²

f′A≡GA=α+3γA2(4.25)

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

Δyt=α+3γA2⋅δejωt+δ∗e−jωt2(4.26)

If we look at the complex coefficient of the term proportional to e^jωtejωt we find it is given simply in (4.27):

α+3γA22δ(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)

Yω=GAXω(4.28)

ΔYωΔXω=GA(4.29)

Case 2

x0t=AcosωtΔxt=δejωt+δ∗e−jωt2(4.30)

This time we take x₀(t)x0t to be a pure sinusoid given in (4.30). There is no loss of generality by taking AA to be real, which is equivalent to taking the phase of the main signal at time zero to be zero. We can interpret δδ 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.

f′Acosωt=α+3γA2cos2ωt=α+3γA22+3γA22cos2ωt(4.31)

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

Δyt=α+3γA22+3γA22e2jωt+e−2jωt2⋅δejωt+δ∗e−jωt2(4.32)

This time we get contributions proportional to e^jωtejωt and e^3jωte3jωt, and their complex conjugates, four terms in all. If we restrict our attention, as in case 1, to output terms proportional to e^jωtejωt, we obtain the coefficient given in (4.33):⁸

α2+3γA24δ+3γA28δ∗(4.33)

We observe that the output phasor at frequency ωω is not proportional to the input phasor, δδ, at frequency ωω but instead has distinct contributions proportional to both δδ 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, δδ, and not on both δδ 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:

ΔYωΔXω=α+3γA22+3γA24e−2jϕδ(4.34)

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