# Pi-Network and "Dipping the Final"

-July 15, 2013

An LC circuit often called a "pi-network" is commonly used to match the impedance of a transmitter's load (an antenna) to something more suitable for the transmitter's output amplifier. That output amplifier is very often a beam power vacuum tube whose output impedance, the plate resistance of that tube, is high and therefore well approximated by a current source.

The basic circuit looks like this sketch. The two capacitors, C1 and C2, are shown as variable because we will see how adjusting their respective values affects the pi-network's behavior.

We look at three things. They are 1) the transfer function of the output voltage, e2, in response to the current excitation, i1, 2) the impedance, Zin, that is presented to the current source and 3) the phase angle associated with the algebraic expression for Zin.

To serve as an example, let L = 2 µHy and F = 10 MHz. When C2 is made large, say 2000 pF, the maximum value of the presented impedance Zin tends to be high. As we adjust C1, we get this kind of result. The peak in Hdb aligns with the peak in Zin and also aligns with a 180° value of the phase function. We see a presented Zin of 11255 Ohms for that value of C1. Let's call this our Light Load.

We now reduce the value of capacitor C2 to 1000 pF and again examine the effects of varying C1. As before, the peak in Hdb aligns with the peak in Zin and also aligns with a 180° value of the phase function, but now we see a presented Zin of only 2693 Ohms for that value of C1. We may call this our Moderate Load.

We now again reduce the value of capacitor C2 to 200 pF and again examine the effects of varying C1. As before, the peak in Hdb aligns with the peak in Zin and also aligns with a 180° value of the phase function, but now we see a presented Zin of only 333 Ohms for that value of C1. We may call this our Full Load.

Taking a set of idealized characteristic curves for an assumed vacuum tube amplifier operating in class-C (which now replaces the excitation current source), we see this:

Okay then, all you ham radio operators out there, you've seen this before. C2 is the loading control of your Heathkit DX-20, DX-35, DX-40, DX-60, DX-100, Apache or your Globe Scout 680 or whatever while C1 is the knob that you use to tune the plate current for a "dip" as you go for the highest value of Zin as shown in this analysis.

Derivation of transfer function:

Derivation of impedance equation:

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