If a second network can be replaced for a first network without affecting the currents and voltages that appear at the network terminals, the two networks are said to be equivalent.

Reducing a complicated network can be achieved by using two methods: T network and π network.

T network: A T network is a type of electrical network that is commonly used to match impedances in RF and microwave circuits. The "T" in T network stands for "transformation." In this network, two impedances are transformed into a third impedance by means of two reactive components, typically capacitors or inductors.

The T network can be used to reduce a complicated network by transforming it into a T network and then simplifying it by combining reactive components that are in parallel or series. The process is repeated until the desired level of simplicity is achieved.

π network: A π network is a type of electrical network that is used to match impedances in RF and microwave circuits. The "π" in π network refers to the shape of the network, which resembles the Greek letter pi. In this network, two reactive components are used to transform two impedances into a third impedance.

The π network can be used to reduce a complicated network by transforming it into a π network and then simplifying it by combining reactive components that are in parallel or series. The process is repeated until the desired level of simplicity is achieved.

Both T and π networks are useful for reducing the complexity of electrical networks and for impedance matching in RF and microwave circuits. However, the choice between a T network and a π network depends on the specific requirements of the circuit and the desired level of performance.

In fig 1, if there is a linear passive electric network, no matter how complicated in internal connection, it can be made equivalent to T network as in fig 2 then

With these values for a T network which will be equivalent in performance to the original network.

Now, the original network if made equivalent with π network fig 3 then

Multiplying equation 9 and 10, we get

Again, subtracting equation 9 from 8, we get

Again multiplying equation 13 by equation 10, we get

Dividing equation 12 by equation 14, we get

Similarly, from equations 14 and 8 we get

These three equations 16, 17 and 18 permits designing a π - network, equivalent to any complicated network.

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