|Pi Network||T Network|
The impedances of the pi network (Za, Zb, and Zc) can be found from the impedances of the T network with the following equations:
Za = ( (Z1*Z2)+(Z1*Z3)+(Z2*Z3) ) / Z2
Zb = ( (Z1*Z2)+(Z1*Z3)+(Z2*Z3) ) / Z1
Zc = ( (Z1*Z2)+(Z1*Z3)+(Z2*Z3) ) / Z3
Note the common numerator in all these expressions which can prove useful in reducing the amount of computation necessary.
The impedances of the T network (Z1, Z2, Z3) can be found from the impedances of the equivalent pi network with the following equations:
Z1 = (Za * Zc) / (Za + Zb + Zc)
Z2 = (Zb * Zc) / (Za + Zb + Zc)
Z3 = (Za * Zb) / (Za + Zb + Zc)
Note the common denominator in these expressions.
In the case where all the impedances are reactive (i.e. they are all
in the form jX), it is handy to note that the -1 factors from squaring
j*j on the top cancels the -1 from bringing the j in the denominator up
Z1 = -j * (R1 *cos(beta) - sqrt( R1 * R2))/ sin (beta)
Z2 = -j * (R2 *cos(beta) - sqrt(R1 * R2))/sin(beta)
Z3 = -j * sqrt(R1*R2) / sin(beta)
Za = j *R1*R2 * sin(beta) / (R2 * cos(beta) - sqrt(R1 * R2))
Zb = j * R1*R2 * sin(beta) / (R1 * cos(beta) - sqrt(R1 * R2))
Zc = j * sqrt( R1 * R2) * sin (beta)
Beta is the phase lag passing through the network from either port 1 to port 2 or vice versa. Note that if beta is 0 or pi, these expressions break down, except if R1=R2. If you need to transform resistive impedances and you don't want any phase shift, you have to use a transformer.
In many practical applications, the load or generator impedances may be reactive (i.e. Z (port 1) and Z (port 2) are some general R+jX). This can be accomodated by absorbing the external reactive impedance into the network, reducing or increasing the series or shunt impedance as requred. For instance, if a T network is required to connect between two impedances: 50+j0 and 100-j20 with 45 degrees of phase shift:
First, calculate the Z's assuming resistive impedances: R1=50, R2= 100
Z1 = -j * (50 * .707 - sqrt(50*100))/.707 = +j 50 ohms
Z2 = -j * (100 * .707 - sqrt(50*100))/.707 = 0 ohms
Z3 = -j * sqrt( 50 * 100) / .707 = -j 100 ohms
(the example is somewhat contrived, and it winds up creating an L network for the resistive case).
Now, a reactive component is added to Z2 to exactly cancel the external reactive component. This changes Z2 from 0 ohms to +j20 ohms. The final network is then:
Z1 = +j50, Z2= +j20, Z3 = -j100 ohms
If you are working with a pi network, you would want to transform the external impedances into their corresponding shunt forms first, so that the reactive component is a shunt value, which would be absorbed (or combined) with the corresponding shunt component of the pi network.