Calculations and Simulations with Pads and Feedback Amplifiers
The result is converted to a function of the load, Z. We then consider four common cases for 50 Ohm Pads: 1 dB (R1=910, R2=5.6), 3 dB (R1=300, R2=18), 6 dB (R1=150, R2=36), and 10 dB (R1=100, R2=68). In all cases R3=R1. These resistor values are standard 5% values that are used when building practical circuits. The design equations are found in EMRFD, Ch. 7.
We allowed the output load resistance, Z, to vary from 5 to 500 Ohms and examined the input resistance of the four pads. The results are plotted here:
The terminated 1 dB pad has an input resistance that varies over a range nearly as large as the termination. In contrast, the 10 dB pad has an input R that is close to 50 Ohms, no matter what output termination is used on the output. Owing to symmetry, the usually pi pad will have an output resistance similarly related to the input termination, or source impedance.
A similar behavior occurs when complex terminations with series reactance are used. The analysis is not, however, as straight forward. Rather than go through these esoteric and probably less than illuminating calculations, we will take a more direct approach based on the Smith Chart. A Smith Chart program that we wrote several years ago for DOS, MicroSmith, is used. (MicroSmith is still available, now free of charge. It is part of the collection of DOS software distributed with the ARRL edition of Introduction to RF Design. The software is found on the ARRL web site by entering IRFD in the search box.) The analysis uses a little known, although not completely hidden feature that allows random component values to be programmed into the program. (The feature is accessed by pressing Alt-Z while viewing the Smith Chart.)
This circuit starts with a short circuit termination of 0+j0. This is then followed by a resistor that randomly varies from 5 to 500 Ohms. The next element is a random reactance that randomly assumes values from -300 to 300 Ohms. The next three elements are then the resistors that compromise the pi-pad. (The random variations are set up with Ctrl-F10 after activated with Alt-Z.) This is illustrated with a figure:
For example, assume MicroSmith generates a random resistance of 80 Ohms and a reactance of -100 Ohms. The resulting impedance is 80-j100. This Z is placed in parallel with R3 to generate a new complex impedance. R2 is added to this and the result is paralleled with R1 to produce a final input complex impedance, Z-in. The final result is plotted as a point (a small rectangle) on the Smith Chart. This process is repeated a large number of times to generate a family of points filling region, suggesting the possible impedances that are possible.
Remember that the Smith Chart is merely a way of plotting impedances. All possible impedances with positive resistive parts, but all possible reactive parts (positive and negative) appear within the circle. The impedance at the center of the circle is the characteristic resistance of the chart, often 50 Ohms.
The results follow: