3.1 Introduction

This chapter presents a concise treatment of S-parameters, meant primarily as an introduction to the more general formalism of large-signal approaches of the next chapter. The concepts of time invariance and spectral maps are introduced at this stage to enable an easier generalization in the ensuing chapter. The interpretations of S-parameters as calibrated measurements, intrinsic properties of the device-under-test (DUT), intellectual property (IP)-secure component behavioral models, and composition rules for linear system design are presented. The cascade of two linear S-parameter components is considered as an example to be generalized to the nonlinear case later. The calculation of S-parameters for a transistor from a simple nonlinear device model is used as an example to introduce the concepts of (static) operating point and small-signal conditions, both of which must be generalized for the large-signal treatment.

3.2 S-parameters

Since the 1950s, S-parameters, or scattering parameters, have been among the most important of all the foundations of microwave theory and techniques.

S-parameters are easy to measure at high frequencies with a vector network analyzer (VNA). Well-calibrated S-parameter measurements represent properties of the DUT, independent of the VNA system used to characterize it. Calibration procedures [2] remove systematic measurement errors and enable a separation of the overall values into numbers attributable to the device, independent of the measurement system used to characterize it. In this context, we call such properties intrinsic.¹ These DUT properties (gain, loss, reflection coefficient, etc.) are familiar, intuitive, and important [3]. Another key property of S-parameters is that the S-parameters of a composite system are completely determined from knowledge of the S-parameters of the constituent components and their connectivity. S-parameters provide the complete specification of how a linear component responds to an arbitrary signal. Therefore, simulations of linear systems that are designed by cascading S-parameter blocks are predictable with certainty. S-parameters define a complete behavioral description of the linear component at the external terminals, independent of the detailed physics or specifics of the realization of the component. S-parameters can be shared between component vendors and system integrators freely, without the possibility that the component implementation can be reverse engineered, protecting IP and promoting sharing and reuse. Indeed, one may ask the question, “are S-parameters measurements, or do they constitute a model?” The answer is really “both.” They are numbers that can be measured accurately at RF and microwave frequencies, and these numbers are the coefficients of a behavioral model that expresses the scattered waves as linear contributions of the incident waves.

S-parameters need not come only from measurements. They can be calculated from physics by solving Maxwell’s equations, by linearizing the semiconductor equations, or computed from matrix analysis of linear equivalent circuits. In this way, the many benefits of S-parameters can be realized starting from a more detailed representation of the component from first principles or from a complicated linear circuit model.

Graphical methods based around the Smith Chart were invented to visualize and interpret S-parameters and graphical design methodologies soon followed for circuit design [3], [4]. These days, electronic design automation (EDA) tools provide simulation components – S-parameter blocks – and design capabilities using the familiar S-parameter analysis mode.

One of the great utilities of S-parameters is the interoperability among the measurement, modeling, and design capabilities they provide. One can characterize the component with measured S-parameters, use them as a high-fidelity behavioral model of the component with complete IP protection, and design systems with them in the EDA environment.

3.3 Wave Variables

The term scattering refers to the relationship between incident and scattered (reflected and transmitted) travelling waves.

By convention, in this text the circuit behavior is described using generalized power waves [5], although there are alternative wave-definitions used in the industry.

The wave variables, AA and BB, corresponding to a specific port of a network, are defined as simple linear combinations of the voltage and current, VV and II, at the same port, according to Figure 3.1 and equations (3.1).

A=V+Z0I2Z0B=V−Z0I2Z0(3.1)

The reference impedance for the port, Z₀Z0, is, in general, a complex value. For the purpose of simplifying the concepts presented, the reference impedance is restricted to real values in this text.

Figure 3.1 Wave definitions.

The currents and voltages can be recovered from the wave variables, according to equations (3.2).

V=Z0⋅A+BI=1Z0⋅A−B(3.2)

Here AA and BB represent the incident, scattered waves VV and II are the port voltage and current, respectively, and Z₀Z0 is the reference impedance for the port. Z₀Z0 can be different for each port, but we do not consider that further here. A typical value of Z₀Z0 is 50 Ω by convention, but other choices may be more practical for some applications. A value for Z₀Z0 closer to 1 Ω is more appropriate for S-parameter measurements of power transistors, for example, given that power transistors typically have very small output impedances.

The variables in equations (3.1) and (3.2) are complex numbers representing the RMS-phasor description of sinusoidal signals in the frequency domain. Later we will generalize to the envelope domain by letting these complex numbers vary in time.

A, B,  V,A,B,V, and II can be considered RMS-vectors, the components of which indicate the values associated with sinusoidal signals at particular ports labeled by positive integers. Thus A_jAj is the incident wave RMS-phasor at port j and I_kIk is the current RMS-phasor at port k. For now, Z₀Z0 is taken to be a fixed real constant, in particular, 50 Ω.

A graphical representation of the wave description is given in Figure 3.2 for the case of a system described by a two-port device. For definiteness, we assume a port description with a common reference pin, as indicated in the figure.

Figure 3.2 Incident and scattered waves of a two-port device. All ports should be referenced to the same pin for modeling purposes.

To retrieve the time-dependent sinusoidal voltage signal at the ith port, the complex value of the phasor and also the angular frequency, ωω, to which the phasor corresponds, must be known. The voltage is then given by (3.3), and, similarly, for the other variables, where Vipk are peak values.

vit=ReVipkejωt(3.3)

It is convenient to keep track of the frequency associated with a particular set of phasors by rewriting (3.1) according to (3.4), and (3.2) according to (3.5), where the port indexing notation is made explicit.

Aiω=12Z0⋅Viω+Z0IiωBiω=12Z0⋅Viω−Z0Iiω(3.4)

Viω=Z0⋅Aiω+BiωIiω=1Z0Aiω−Biω(3.5)

For each angular frequency, ωω, Eq. (3.4) is a set of two equations defined at each port.

The assumption behind the S-parameter formalism is that the system being described is linear and therefore there must be a linear relationship, implying superposition holds, between the phasor representation of incident and scattered waves. This is expressed in (3.6) for an N-port network:

Biω=∑j=1NSijωAjω,∀i∈12…N(3.6)

The set of complex coefficients, S_ij(ω)Sijω, in (3.6) define the S-parameter matrix, or simply, the S-parameters, at that frequency. Equation (3.6), for the fixed set of complex S-parameters, determines the output phasors for any set of input phasors. The summation is over all port indices, so that incident waves at each port, j, contribute in general to the overall scattered wave at each output port, i. For now we consider all frequencies to be positive (ω>0ω>0). Note that contributions to a scattered wave at frequency ωω come only from incident waves at the same frequency. This is not the case for the more general X-parameters, where a stimulus at one frequency can lead to scattered waves at different frequencies.

For definiteness and later reference, the S-parameter equations for a linear two-port are written explicitly in (3.7).

B1=S11A1+S12A2B2=S21A1+S22A2(3.7)

The set of equations (3.6) represent a model of the network under study. For the purpose of creating a model for the network, all ports should be referenced to the same pin, as shown in Figure 3.2.

Such connectivity is the natural option for the measurement and modeling process of a 3-pin network (like a transistor) but it has to be extended in the general case of an arbitrary network and it is necessary for all networks, linear and/or nonlinear. This connectivity convention is considered by default (unless otherwise specified) for the remaining of this text.

Using the topological connection in Figure 3.2, the set of equations (3.6) represent a complete model of the network under test.

From (3.6) we note a stimulus (incident wave) at a particular port j will produce a response (scattered wave) at all ports, including the port at which the stimulus is applied.

3.4 Geometrical Interpretation of S-parameters

For linear two-port components described by (3.7), the scattering of A₂ at port 2 is given simply by S₂₂ A₂, where S₂₂ is just a fixed complex number. This means we can write for ΔB₂ = B₂ − S₂₁A₁ΔB2=B2−S21A1

For fixed |A₂|A2, S₂₂A₂ = S₂₂|A₂|e^jϕ(A₂)S22A2=S22A2ejϕA2, so by varying ϕ(A₂)ϕA2 from 0 to 2π2π radians, equation (3.8) traces out a circle with radius |A₂|S22A2. This is shown in Figure 3.3.

Figure 3.3 Locus of points produced from S-parameter equation (3.8) evaluated for all phases of A₂A2. The radius of the circle is |S₂₂ A₂|S22A2.

This fact is trivial in the case of S-parameters. We will see in the next chapter that the scattering of even a small-signal A₂ has a different (non-circular) geometry when a nonlinear component is driven with a large A₁ signal.

3.5 Nonlinear Dependence on Frequency

Equation (3.6) shows that the scattered waves are linear functions of the complex amplitudes (the phasors) of the incident waves, with the S-parameters being the coefficients of the linear relationship. These coefficients, however, have a frequency dependence that is usually not linear. That is, we generally have S_ij(λω)≠λS_ij(ω)Sijλω≠λSijω for λλ a positive real number. For example, an ideal band-pass filter response is linear in the incident wave phasor, but the filter response is a nonlinear function of the frequency of the incident wave. This is shown in Figure 3.4.

Figure 3.4 A linear network has a linear behavior versus input power level, but the dependence on frequency is usually not linear.

3.6 S-parameter Measurement

3.6.1 Conventional Identification

By setting all incident waves to zero in (3.6), except for A_jAj, one can deduce the simple relationship between a given S-parameter (S-parameter matrix element) and a particular ratio of scattered to incident waves according to (3.9). Sometimes (3.9) is taken as the definition of the S-parameters for a linear system, instead of our starting point (3.6). However, since we will show an alternative identification method for S-parameters in Section 3.6.2, we prefer to interpret (3.9) as a simple consequence of the more fundamental linearity principle, (3.6).

Sijω=BiωAjωAk=0∀k≠j(3.9)

Equation (3.9) corresponds to a simple graphical representation shown in Figure 3.5 for the simple case of a two-port component. In Figure 3.5 a, the stimulus is a wave incident at port 1. The fact that A₂A2 is not present (A₂ = 0A2=0) is interpreted to mean that the B₂B2 wave scattered and travelling away from port 2 is not reflected back into the device at port 2. Under this condition, the device is said to be perfectly matched at port 2. Two of the four complex S-parameters, specifically S₁₁ and S₂₁S11andS21, can be identified using (3.9) for this case of exciting the device with only A₁A1. Figure 3.5b shows the case where the device is stimulated with a signal, A₂A2, at port 2, and assumed to be perfectly matched at port 1 (A₁ = 0A1=0). The remaining S-parameters, S₁₂ and S₂₂S12andS22, can be identified from this ideal experiment.

Figure 3.5 S-parameter experiment design.

3.6.2 Alternative Identification

The S-parameters may also be determined by a direct solution of (3.6) from multiple measurements of the B_iBi in response to excitations in which the A_jAj have multiple (relative) phases.

The S-parameters at a particular frequency may also be determined by a direct solution of (3.6) by measuring the scattered waves in response to known excitations by incident waves at multiple ports simultaneously. Exciting the device by multiple signals can be useful, even for linear components, because, for example, there is no need to switch a signal from one port to the others or to perfectly match the other ports as required in the ideal conventional case. For the nonlinear case of the next chapter, we will have to use simultaneous excitations. Let’s use a simple two-port [N = 2 in (3.6)] example to illustrate the concept.

In this case there are four complex S-parameters to be determined from the measured responses at both ports, B_iBi, to the known excitations simultaneously at each port, A_jAj. Therefore, we need at least two sets of independent excitations, Ajn, and the corresponding responses, Bin, where the superscript, n, labels the experiment number. In matrix notation we have

B11B12B21B22=S11S12S21S22⋅A11A12A21A22(3.10)

Equation (3.10) is a set of four complex equations for the four unknown S-parameters, S_ijSij given the four measured Bin responses to the four stimuli Ajn. The formal solution is given in matrix notation in (3.11).

S = BA⁻¹

S=BA−1(3.11)

More generally, for more robust results, there can be more data [columns for the first and third matrices in (3.10)] than unknowns and the formal solution is given in terms of pseudo-inverses according to (3.12). According to the discussion in Chapter 1, this provides the best S-matrix in the least squares sense:

S = BA^T(AA^T)⁻¹

S=BATAAT−1(3.12)

3.6.3 S-parameter Measurement for Nonlinear Components

It is important to note that the ratio on the right-hand side of (3.9) can be computed from independent measurements of incident and scattered waves for actual components corresponding to any nonzero value for the incident wave, A_jAj. The value of this ratio, however, will generally vary with the magnitude of the incident wave. Therefore, the identification of this ratio with the S-parameters of the component is valid for any particular value of incident A_jAj only if the component behaves linearly, namely according to (3.6). In other words, the values of the incident waves, A_jAj, need to be in the linear region of operation for this identification to be valid. For nonlinear components, such as transistors biased at a fixed voltage, the scattered waves don’t continue indefinitely to increase as the incident waves get larger in magnitude (this is compression). Therefore, different values of (3.9) result from different values of incident waves. A better definition of S-parameters for a nonlinear component is a modification of (3.9) given by (3.13). That is, for a general component, biased at a constant DC-stimulus, the S-parameters are related to ratios of output responses to input stimuli in the limit of small input signals. This emphasizes that S-parameters properly apply to nonlinear components only in the small-signal limit.

Sijω≡lim∣Aj∣→0BiωAjωAk=0∀k≠j(3.13)

3.7 S-parameters as a Spectral Map

If there are multiple frequencies present in the input spectrum, one can represent the output spectrum in terms of a matrix giving the contributions to each output frequency from each input frequency.

An example in the case of three input frequencies is given by Eq. (3.14). Here we assume a single port, for simplicity, and therefore drop the port indices. It is clear from (3.14) that S-parameters are a diagonal map in frequency space. This means each output frequency contains contributions only from inputs at that same frequency. Or, in other words, each input frequency never contributes to outputs at any different frequency.

Bω1Bω2Bω3=Sω1000Sω2000Sω3Aω1Aω2Aω3(3.14)

A graphical representation is given in Figure 3.6 for the case of forward transmission through a two-port network with matched terminations at both ports.

Figure 3.6 Linear spectral map through S-parameters matrix.

The interpretation of Figure 3.6, mathematically represented by (3.14), is that S-parameters define a particularly simple linear spectral map relating incident to scattered waves. S-parameters are diagonal in the frequency part of the map, namely they predict a response only at the particular frequencies of the corresponding input stimuli. It will be demonstrated in later chapters that the large-signal approaches provide for richer behavior.

For signals with a continuous spectrum, the diagonal nature of the S-parameter spectral map can be written as

Biω=∑j=1N∫Sijωδω−ω′Ajω′dω′,∀i∈12…N.(3.15)

Performing the integration over input frequencies in (3.15) results in Equation (3.6), the form usually given for S-parameters.

3.8 Superposition

Any linear theory, like S-parameters, enables the general response to an arbitrary input signal to be computed by superposition of the responses to unit stimuli (see the discussion of the impulse response in Chapter 1, Section 2). Superposition enables great simplifications in analysis and measurement. Superposition is the reason S-parameters can be measured by independent experiments with one sinusoidal stimulus at a time, one stimulus per port per frequency using Eq. (3.9). The general response to any set of input signals can be obtained by superposition using Eq. (3.6).

An example of superposition is shown in Figure 3.7, with all signals represented both in time and frequency domains.

Figure 3.7 Superposition example.

This example uses two signals, each containing two frequency components, as stimuli incident at each port, independently, with the other port perfectly matched. The example shows that the response to a linear combination of the stimuli is the same linear combination of the individual responses.

As always, the caveat is that the component actually behaves linearly over the range of signal levels used to stimulate the device. There is no a priori way to know whether a component will behave linearly without precise knowledge about its composition or physical or measured characteristics.

3.9 Time Invariance of Components Described by S-parameters

A DUT description in terms of S-parameters defined by (3.6) naturally embodies an important principle known as time invariance. Time invariance states that if y(t)yt is the DUT response to an excitation, x(t)xt, then the DUT response to the time-shifted excitation, x(t − τ)xt−τ, must be y(t − τ)yt−τ. This must be true for all time-shifts, ττ. That is, if the input is shifted in time, the output is shifted by the corresponding amount but is otherwise identical with the DUT response to the nonshifted input. This is stated mathematically in (3.16), where O is the operator taking input to output:

Time invariance is a property of common linear and nonlinear components, such as passive inductors, capacitors, resistors, and diodes, and active devices such as transistors. Examples of components not time-invariant (in the usual sense) are oscillators and other autonomous systems.

The proof follows from elementary properties of the Fourier Transform, where a phase shift by e^jωτejωτ in the frequency domain corresponds to a time shift of ττ in the time domain. The time-domain waves incident at the ports (the stimuli) are a_j(t)ajt and their Fourier transforms are Ajpkω, as in equation (3.17). In (3.17), the symbol F denotes the Fourier transform. The superscript, (pk), in (3.17) refers to peak value (as opposed to the power-wave value).

Fajt=Ajpkω(3.17)

The time-domain waves scattered from the ports (the response) are b_i(t)bit and their Fourier transforms are Bipkω, as in equation (3.18).

Fbit=Bipkω=∑j=1NSijωAjpkω=∑j=1NSijωFajt(3.18)

If all stimuli are delayed with the same time-delay, ττ, the response becomes

F−1∑j=1NSijωFajt−τ=F−1∑j=1NSijωFajteiωτ=F−1Bipkωeiωτ=bit−τ(3.19)

Equation (3.19) proves that S-parameters are automatically consistent with the principle of time invariance. Therefore, any set of S-parameters describes a time-invariant system.

Unlike the case for S-parameters, a more general (e.g., nonlinear) relationship between incident A waves and scattered B waves is not automatically consistent with the property (3.16) of time invariance. This will be demonstrated in Chapter 4. Therefore, in order to have a consistent representation of a nonlinear time-invariant DUT, the time-invariance property must be manifestly incorporated into the mathematical formulation of the behavioral model relating input to output waves. A representation of a time-invariant DUT by equations not consistent with (3.16) means the model is fundamentally wrong and can yield very inaccurate results for some signals, even if the model “fitting” (or identification) appears good at time t.

3.10 Cascadability

Another key property of S-parameters is that the response of a linear circuit or system can be computed with perfect certainty knowing only the S-parameters of the constituent components and their interconnections. The overall S-parameters of the composite design can be calculated by using (3.6) in conjunction with Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) applied at the internal nodes created by connections between two or more components.

This is illustrated for a cascade of two-ports in Figure 3.8a. Each component is characterized by its own S-parameter matrix, S⁽¹⁾S1 and S⁽²⁾S2, respectively. The output port (subscript number 2) of the first component is connected to the input port (subscript number 1) of the second component, creating an internal node.

Figure 3.8 The cascade of 2 two-ports.

It is a straightforward exercise to write the two equations that follow from applying KVL and KCL at the internal node. These are given by equations (3.20) and (3.21)

B21=A12(3.20)

A21=B12(3.21)

Each two-port relates two input and two output variables. When cascaded, equations (3.20) and (3.21) can be used to compute the overall S-parameter matrix of the composite. The result is given in equation (3.22).

Scomposite=S111+S112S121S2111−S221S112S121S1221−S221S112S211S2121−S221S112S222+S221S122S2121−S221S112(3.22)

A network characterized by the S^compositeScomposite is equivalent to the cascade of the two DUTs, yielding the same response under the same stimuli. This is shown in Figure 3.8b.

3.11 DC Operating Point: Nonlinear Dependence on Bias

For devices such as diodes and transistors, the S-parameter values depend very strongly on the DC bias conditions defining the component operating conditions. For example, a FET transistor biased in the saturation region of operation will have S-parameters appropriate for an active amplifier (where |S₂₁|>>1>>|S₁₂|S21≫1≫S12) whereas when biased “off” (say V_DS = 0VDS=0) the S-parameters will represent the characteristics of a passive switch (S₂₁ = S₁₂S21=S12 with |S_ij| < 1Sij<1).

3.12 S-parameters of a Nonlinear Device

It is possible to derive the S-parameters of a transistor starting from a simple nonlinear model of the intrinsic device. The process illustrates the concept of linearizing a nonlinear mapping about an operating point. This concept, suitably generalized, is a foundation for X-parameters and similar approaches in the next chapter.

The equivalent circuit of a simple, unilateral FET model is shown in Figure 3.9. More complete model topologies and equations are discussed in Chapter 5 and [7].

Figure 3.9 Simple nonlinear equivalent circuit model of a FET.

A current source, i_DSiDS, represents the nonlinear bias-dependent channel current from drain to source, and a simple nonlinear two-terminal capacitor, q_GSqGS, represents the nonlinear charge storage between gate and source terminals.

The large-signal model equations corresponding to this equivalent circuit are given by (3.23) and (3.24), where the stimulus applied to the device is the set of port voltages, v_GSvGS and v_DSvDS, and the device response is the set of port currents, i_GSiGS and i_DSiDS:

iGt=dqGSvGStdt(3.23)

Equations (3.23) and (3.24) are evaluated for time-varying voltages assumed to have a fixed DC component and a small sinusoidal component at a single RF or microwave angular frequency, ω = 2πfω=2πf. Port 1 is associated with the gate and port 2 with the drain terminals, referenced to the source.

For a stimulus comprising a combination of DC and one sinusoidal signal per port, the voltages are formally expressed as shown in (3.25). The symbol δδ is used to emphasize that such terms are considered sufficiently small that simple perturbation theory can be applied to simplify the ensuing analysis.

vit=viDC+δVi⋅cosωt+ϕi(3.25)

Expressions in (3.26) and (3.27) are obtained by substituting (3.25) for i = 1,2, into (3.23) and (3.24), and evaluating the result to first order in the real quantities δV_iδVi.

i1t=i1DC+δi1t=ddtqGSv1DC+dqGSdv1v1DC,v2DCδV1⋅cosωt+ϕ1=cGSv1DCddt(δV1⋅cosωt+ϕ1)(3.26)

i2t=i2DC+δi2t=iDSv1DCv2DC+gmv1DCv2DC⋅δV1⋅cosωt+ϕ1+gDSv1DCv2DC⋅δV2⋅cosωt+ϕ2(3.27)

Here the following definitions of the linearized equivalent circuit elements have been used:

cGSv1DC=dqGSdv1v1DCgmv1DCv2DC=∂iDS∂v1v1DC,v2DCgDSv1DCv2DC=∂iDS∂v2v1DC,v2DC(3.28)

By equating the 0th order terms in δV_iδVi, the operating point conditions are obtained:

i1DC=0Ai2DC=iDSv1DCv2DC(3.29)

Equating the first order terms in δV_iδVi leads to

δi1t=cGSv1DCddt(δV1⋅cosωt+ϕ1)δi2t=gmv1DCv2DCδV1⋅cosωt+ϕ1+gDSv1DCv2DCδV2⋅cosωt+ϕ2(3.30)

Equation (3.30) is expressed in the frequency domain by defining complex phasors δI(ω)δIω and δV(ω)δVω according to (3.31):

δiit=ReδIiωejωtδvit=ReδViωejωt=ReδViejϕiejωt(3.31)

Equations (3.30) can be rewritten for the complex phasors in matrix notation as

δI1ωδI2ω=Yv1DCv2DCω⋅δV1ωδV2ω(3.32)

where

Yv1DCv2DCω=jωcGSv1DC0gmv1DCv2DCgDSv1DCv2DC(3.33)

Equation (3.33) defines the (common source) admittance matrix of the model. The matrix elements are evidently functions of the DC operating point (bias conditions) of the transistor and also the (angular) frequency of the excitation.

Since the phasors representing the port voltage and currents in (3.32) can be reexpressed as linear combinations of incident and scattered waves using (3.5), it is possible to derive the expression for the S-parameters in terms of the Y-parameters (admittance matrix elements). This results in the well-known conversion formula (3.34) [3]:²

Here II is the two-by-two unit matrix, Z₀Z0 is the reference impedance used in the wave definitions in (3.4), YY is the two-port admittance matrix, and SS is the corresponding S-parameter matrix. Substituting (3.33) into (3.34) results in an explicit expression, (3.35), for the S-parameters corresponding to this simple model in terms of the linear equivalent circuit element values given in (3.28):

Sv1DCv2DCω=1−jωcGSv1DCZ01+jωcGSv1DCZ00−2gmv1DCv2DCZ01+gDSv1DCv2DCZ01+jωcGSv1DCZ01−gDSv1DCv2DCZ01+gDSv1DCv2DCZ0(3.35)

In summary, the S-parameters of a nonlinear component can be derived or computed by linearizing the full nonlinear characteristics of the circuit equations around a static (DC) operating point defined by the voltage or current bias conditions. The S-parameters define a linear relationship between the incident and scattered waves at a fixed DC operating point of the device and fixed frequency for the incident waves. The S-parameters are an accurate description of how the device responds to signals, provided the signal amplitude is sufficiently small that the DC operating point is not significantly affected by the signal. This will almost always be the case for signals of sufficiently small amplitude.

3.13 Additional Benefits of S-parameters

3.14 Limitations of S-parameters

Interestingly, S-parameters are still commonly used for nonlinear devices such as transistors and amplifiers. The problem, often forgotten or taken for granted, is that S-parameters only describe properly the behavior of a nonlinear component in response to small signal stimuli for which the device behavior can be approximated as linear around a fixed DC, or static, operating point. That is, only when the nonlinear device is assumed to depend linearly on all RF components of the incident signals is the S-parameter paradigm valid. S-parameters contain no information about how a nonlinear component generates distortion, such as manifested by energy generated at harmonics and intermodulation frequencies by the component in response to excitation by one or more tones (sinusoidal signals). S-parameters are inadequate to describe nonlinear devices excited by large signals despite many ad hoc attempts. Attempts to generalize S-parameters to the nonlinear realm have led to a wide variety of problems, including measurements that appear nonrepeatable, and the inability to make predictable design inferences from such measurements or simulations. Techniques such “hot S-parameter” measurements and modeling nonlinear components using “large-signal S-parameter analysis” are examples of incomplete, insufficient, and ultimately incorrect approaches. Under large-signal conditions, totally new phenomena, for which there are no analogues in S-parameter theory, appear and must be taken into account.

For example, when a nonlinear component such as a transistor or power amplifier is stimulated simultaneously by two sinusoidal signals at different frequencies, the output spectrum is not consistent with the superposition principle discussed in Section 3.8. Rather, the response of a nonlinear system to two or more excitations generally contains more nonzero frequency components than were present in the input signal (see Figure 3.10). These terms are generally referred to as intermodulation distortion.³ It is clear that this phenomenon cannot be described by S-parameters. A more general framework is required to address this type of behavior, which is the subject of the next chapter.

Figure 3.10 Intermodulation spectrum of typical transistor output in response to a simultaneous excitation by two sinusoidal signals at different frequencies. Superposition is not obeyed in general.

3.15 Summary

S-parameters are linear time-invariant spectral maps defined in the frequency domain. They represent intrinsic properties of the DUT, enabling the hierarchical design of linear circuits and systems given only the S-parameters of the constituent functional blocks and their topological arrangement in the design. S-parameters can be defined, measured, and calculated for nonlinear components, but they are valid descriptions only under small-signal conditions. Phenomena such as harmonic and intermodulation distortion, generated by nonlinear components, require a more comprehensive framework for their consistent description and application.

3.16 Exercises