Handbook of Microwave Component Measurements. Joel P. Dunsmore

      where

      This is a generalized formula so that an impedance, [Z_n], may be defined for any port of the original [S] matrix and any other impedance may defined for any other port for the new [S^′] matrix. However, the two most common cases are where the transformation occurs for all impedances at the ports being equal, so that each element in the X matrix and Γ matrix are identical, and where in the 2‐port case only one impedance is transformed, as when the S‐parameters of a network are defined in two different impedances.

      If the measurement system impedance is pure‐real, an alternative method for obtaining S‐parameters at a different real impedance than the measurement system is to de‐embed an ideal transformer at each port, with the turns ratio set to the square root of the impedance change. De‐embedding methods are discussed in Chapter 9.

      

       2.4.3 Concatenating Circuits and T‐Parameters

      In many instances, it is convenient to concatenate devices, and signal‐flow charts provide a useful tool for understanding the interactions and determining the resulting S‐parameter matrix. With appropriate transformations, the concatenation of S‐parameter devices can be greatly simplified. One such transformation is from S‐parameters to T‐parameters, which also depend upon the wave functions but in a different relationship.

Figure 2.39 shows a concatenation of two devices, with a and b waves for each independently identified. Using normal signal‐flow properties, the combined S‐parameters of two devices is

      (2.18)

Figure 2.39 Concatenation of two devices.

      However, signal‐flow‐graph techniques get tedious for concatenating a long series of devices, and other transformations make this work easier and more programmatic.

The T‐parameters (Keysight Application Note 154 n.d.-b) create a new functional relationship between input and output waves, with the independent variables being waves on the right and the dependent variables being waves on the left.

      (2.19)

      Or in the matrix form

      (2.20)

      From this, the T‐matrix describing the first and second devices are

      (2.21)

      From inspection, one can recognize that the waves a_2A = b_1B and b_2A = a_1B so that the concatenation becomes this simple result

      (2.22)

      Or

      (2.23)

      Using this definition of T‐parameters, the following conversions can be defined

      (2.24)

Note that in this conversion, S21 always appears in the denominator. This can cause numerical difficulties in devices with transmission zeros and can sometimes cause de‐embedding functions to fail. More robust de‐embedding algorithms check for this condition and modify the method of concatenation in such a case.

      Other definitions of T‐parameter type relationships have been described, which exchange the position variables a₁ and b₁ on the dependent variable side and exchange the position of a₂ and b₂ on the independent variable side (Mavaddat 1996). This version has similar properties, but care must be taken not to confuse the two methods as, of course, the resulting T‐parameters are different. Another definition, which might seem more intuitive, would set the input terms a₁ and b₁ as the independent variable. Unfortunately, this has the undesirable effect of setting S12 in the denominator of the transformation parameter and thus gives difficulties when applied to unilateral gain devices such as amplifiers.

2.5 Modeling Circuits Using Y and Z Conversion

One common desire in evaluating the performance of a component is to model that component as an impedance comprised of a resistive element with a single series or shunt reactive element, as demonstrated in Section 2.4.1.1. This desire was furthered by some built‐in transformation functions on VNAs, first introduced with the HP8753A but common now on many models. The goal was to model a device in such a way that the S‐parameters mapped to a single resistive and reactive element in the so‐called Z‐transform case (not to be confused with the discrete time z‐transform) or a single conductance and susceptance in the Y‐transform case. These are quite simple models and represented in Figure 2.40.

Figure 2.40 Y and Z conversion circuits.

       2.5.1 Reflection Conversion

      Reflection conversions are computed from the S11 trace and are essentially the same values as presented by impedance or admittance readouts of the Smith chart markers. Thus, Z‐reflection conversion would be used with the circuit description from Figure 2.40a and display the impedance in the real part of the result and the reactance in the imaginary part of the result. Y‐reflection would be used with the circuit of Figure 2.40b and display the conductance in the real part and the susceptance in the imaginary part of the result. The computations for these conversions are

      (2.25)

      Typically, these conversions would be used on one‐port devices and measurements. If it is used on a 2‐port device, one must remember that the load impedance will affect the measured value of the Z‐ or Y‐reflected conversion.

       2.5.2 Transmission Conversion

These reflection conversions are already well known as the models represented by the Smith chart, but a similar conversion can be performed for a simple transmission measurement. In this case, the circuits of Figure