### 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, *A*A and *B*B, corresponding to a specific port of a network, are defined as simple linear combinations of the voltage and current, *V*V and *I*I, at the same port, according to Figure 3.1 and equations (3.1).

The reference impedance for the port, *Z*_{0}Z0, 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).

Here *A*A and *B*B represent the incident, scattered waves *V*V and *I*I are the port voltage and current, respectively, and *Z*_{0}Z0 is the reference impedance for the port. *Z*_{0}Z0 can be different for each port, but we do not consider that further here. A typical value of *Z*_{0}Z0 is 50 Ω by convention, but other choices may be more practical for some applications. A value for *Z*_{0}Z0 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 *I*I 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*_{j}Aj is the incident wave RMS-phasor at port *j* and *I*_{k}Ik is the current RMS-phasor at port k. For now, *Z*_{0}Z0 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 *i*th port, the complex value of the phasor and also the angular frequency, *ω*ω, 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.

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.

For each angular frequency, *ω*ω, 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:

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 *ω*ω 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).

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 A_{2} at port 2 is given simply by *S*_{22} *A*_{2}, where *S*_{22} is just a fixed complex number. This means we can write for *ΔB*_{2} = *B*_{2} − *S*_{21}*A*_{1}ΔB2=B2−S21A1

*ΔB*

_{2}=

*S*

_{22}

*A*

_{2}

For fixed |*A*_{2}|A2, *S*_{22}*A*_{2} = *S*_{22}|*A*_{2}|*e*^{jϕ(A2)}S22A2=S22A2ejϕA2, so by varying *ϕ*(*A*_{2})ϕA2 from 0 to 2*π*2π radians, equation (3.8) traces out a circle with radius |*A*_{2}|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 *A*_{2}A2. The radius of the circle is |*S*_{22} *A*_{2}|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 *A*_{2} has a different (non-circular) geometry when a nonlinear component is driven with a large *A*_{1} 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 *λ*λ 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*_{j}Aj, 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).

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 *A*_{2}A2 is not present (*A*_{2} = 0A2=0) is interpreted to mean that the *B*_{2}B2 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 *S*_{11} and *S*_{21}S11andS21, can be identified using (3.9) for this case of exciting the device with only *A*_{1}A1. Figure 3.5b shows the case where the device is stimulated with a signal, *A*_{2}A2, at port 2, and assumed to be perfectly matched at port 1 (*A*_{1} = 0A1=0). The remaining S-parameters, *S*_{12} and *S*_{22}S12andS22, 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*_{i}Bi in response to excitations in which the *A*_{j}Aj 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*_{i}Bi, to the known excitations simultaneously at each port, *A*_{j}Aj. 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

Equation (3.10) is a set of four complex equations for the four unknown S-parameters, *S*_{ij}Sij 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}

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}

#### 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*_{j}Aj. 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*_{j}Aj only if the component behaves linearly, namely according to (3.6). In other words, the values of the incident waves, *A*_{j}Aj, 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.