Pi and T networks

This page has a variety of useful network synthesis equations to do the following things:

Transforming from Pi to T and vice versa

Any pi network can be transformed to an equivalent T network. This is also known as the Wye-Delta transformation, which is the terminology used in power distribution and electrical engineering. The pi is equivalent to the Delta and the T is equivalent to the Wye (or Star) form.
 
pi network T network
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 top.
 

Synthesis of pi and T networks to transform resistances and create phase shifts

Assuming that the desired port impedances are purely resistive (i.e. real), you can design a T or pi network with purely reactive components both to produce a desired phase shift (beta) and transform the impedances with the following equations. Note that beta can be any value, except for zero or pi.

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.



radio/math/wyedelta.htm - revised 31 Dec 2002, Jim Lux
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