2.40d are the reference circuits for these conversions. They are useful when analyzing the series element models, such as coupling capacitors, and the models for series resistors and inductors. The underlying computation for the transmission conversions is
(2.26)
The Z‐transmission conversion would be well suited to view the series resistance of a coupling capacitor. The Y‐transmission would show the resistive value of a series‐mounted surface‐mount technology (SMT) resistor with a shunt capacitance as a constant conductance with a reactance increasing as 2πf, forming a straight reactance line.
These conversions are often confused with conversion to Y‐ or Z‐parameters, but they are not, in general, related. These provide simple modeling functions based on a single S‐parameter, whereas the Y‐, Z‐, and related parameters provide a matrix result and require knowledge of all four S‐parameters as well as the reference impedance. These other matrix parameters are described in the next section.
2.6 Other Linear Parameters
Even though a VNA measures S‐parameters as its fundamental information, many other figures of merit may be computed directly from these measurements, through the use of transformations, found in several references (Hong and Lancaster 2001; Keysight Technologies n.d.-a). Most of these common parameters relate the voltage and current at the ports, rather than the a and b waves. Many of these transformations arise out of different definitions of terminal conditions as applied to Figure 1.2. These definitions arise out of DC or low‐frequency measurements, where it is an easy matter to short a terminal, meaning ZL = 0, or open a terminal, meaning ZL → ∞. An often confusing point is that it is not necessary that the terminals actually be opened or shorted, but most commonly the parameter is described in those terms. Just as it is most common to terminate the a 2‐port network in Z0 to define S21, making a2 = 0, it is not necessary to do so, and S21 can be determined with any terminal impedance as long as sufficient changes in a1 and a2 are made to solve Eq. (1.17), as shown in Eq. (1.21). Since the voltage and current relationships on the terminals of a DUT are easily determined from the S‐parameters, many other linear parameters can be determined as well. Unless otherwise noted, these transformations apply to the simple case where the S‐parameters are defined with a single, real‐valued reference impedance.
2.6.1 Z‐Parameters, or Open‐Circuit Impedance Parameters
Z‐parameters are one of the more commonly defined parameters and often the first characterization parameter introduced in engineering courses on electrical circuit fundamentals.
The Z‐parameters are defined in terms of voltages and currents on the terminals as
(2.27)
where the V's and I's are defined in Figure 1.2. If we apply the condition of driving a voltage source into the first input terminal and opening the first output terminal, which forces I2 to zero, and measure the input and output voltages, we can determine two of the parameters; similarly, the other two parameters are determined by driving the output terminal and opening the input terminal. Mathematically this can be stated as
(2.28)
But, the conditions for measurement of these parameters directly cannot be realized in RF and microwave systems for several key reasons, listed here:
1 When the ports of an RF circuit are left open, fringing capacitance from the center pin of the RF terminal to ground reduces the impedance at high frequency, and phase shift from the DUT reference plane makes the practical value of an open circuit deviate from the ideal.
2 The measuring equipment to sense V1, I1, V2, and I2 has a parasitic impedance to ground, which also shunts some of the terminal current. At the driving port, this means that the measured current does not match the actual current into the DUT, and at the open port it means that while being measured, the output impedance does not match that of an open circuit
3 For many active devices, the DUT is only conditionally stable and may oscillate if larger reflections are presented at the ports. Phase shift of the open circuit from the terminal port to the DUT active device can cause the reflection to take on almost any phase, and it ensures that at some frequency the reflection at the port will be such that the device will oscillate. This is perhaps the primary reason for S‐parameters being used on active devices; they provide a consistent low‐reflection load, which in general prevents oscillations of the DUT.
Of course, Z‐parameters are not restricted to just two ports, and the Z‐parameters can be put in a matrix form of
(2.29)
where [Z] is called the Z‐matrix.
The Z‐matrix and S‐matrix can be computed from each other, given the reference impedance is known for the S‐parameters, and the same on each port; then
(2.30)
where ΔS = (1 − S11)(1 − S22) − S21S12
(2.31)
where ΔZ = (Z11 + Z0)(Z22 + Z0) − Z21Z12
An attribute of the Z‐matrix is that if a DUT is lossless, the Z‐matrix will contain only pure imaginary numbers; this is commonly found in filter design applications. If Z21 = Z12, then the DUT is reciprocal, and if also Z11 = Z22, the network is symmetrical. Note that in general ZIn ≠ Z11 except for a 1‐port network. ZIn represents the ratio of V1 and I1 for the DUT as it is terminated, normally in the system reference impedance Z0, where Z11 is the ratio of V1 and I1 when all the other ports are open circuited, a not very useful case in practice. Another important attribute of the Z‐matrix is that its values do not depend upon the measurement system, unlike S‐parameters whose values depend upon the reference impedance for each port and whose values can change for the same network, if these reference impedances change. Put another way, the S11 of a 50 Ω load will be quite different when measured in a 75 Ω reference impedance, but the Z‐parameters will not change.
2.6.2 Y‐Parameters, or Short‐Circuit Admittance Parameters
Y‐parameters are essentially an inverse, of Z‐parameters, and in fact the Y‐matrix is the inverse of the Z‐matrix. The definition of Y‐parameters is derived from
(2.32)
From this, the common description of Y‐parameters are
(2.33)
The Y‐parameters can also be defined for more than 2‐port devices, and the matrix form is
(2.34)