Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 129

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.