# Network Theorems

In this chapter, we will discuss the following network theorems:

- Substitution Theorem
- Superposition Theorem
- Reciprocity Theorem
- Theveninâ€™s Theorem
- Nortonâ€™s Theorem
- Maximum Power Transfer Theorem
- Millmanâ€™s Theorem

# Superposition Theorem

*Statement*This theorem states that in a linear bilateral network, the current at any point (or voltage between any two points) due to the simultaneous action of a number of independent sources in the network is equal to the summation of the component currents (or voltages). A component current (or voltage) is defined as that due to one source acting alone in the network with all the remaining sources removed.

# Theveninâ€™s Theorem

**A linear active bilateral network can be replaced at any two of its terminals, by an equivalent voltage source (Theveninâ€™s Voltage source),**

*Statement**V*

_{oc}, in series with an equivalent Impedance (Theveninâ€™s impedance),

*Z*

_{th}.

Here,

*V*

_{oc}is the open circuit voltage between the two terminals under the action of all sources and initial conditions, and

*Z*

_{th}is the impedance obtained across the terminals with all sources removed by their internal impedance and initial conditions reduced to zero.

# Nortonâ€™s Theorem

*Statement*A linear active bilateral network can be replaced at any two of its terminals, by an equivalent current source (Nortonâ€™s current source),

*I*

_{sc}, in parallel with an equivalent admittance (Nortonâ€™s admittance),

*Y*.

_{N}**Illustration of Nortonâ€™s Theorem**

Here,

*I*_{sc}is the short circuit current flowing from one terminal to the other under the action of all sources and initial conditions, and*Y*is the admittance obtained across the terminals with all sources removed by their internal impedance and initial conditions reduced to zero._{N}# Maximum Power Transfer Theorem

*Statement*Maximum power is absorbed by one network from another connected to it at two terminals, when the impedance of one is the complex conjugate of the other.

This means that for maximum active power to be delivered to the load, load impedance must correspond to the conjugate of the source impedance (or in the case of direct quantities, be equal to the source impedance).

# Millmanâ€™s Theorem

Consider a number of admittances*Y*

_{1},

*Y*

_{2},

*Y*

_{3}â€¦

*Y*â€¦

_{p}*Y*,â€¦

_{q}*Y*are connected together at a common point

_{n}*S*. If the voltages of the free ends of the admittances with respect to a common reference

*N*are known to be

*V*

_{1N},

*V*

_{2N},

*V*

_{3N}â€¦

*V*â€¦

_{pN}*V*,â€¦

_{qN}*V*, then Millmanâ€™s theorem gives the voltage of the common point

_{nN}*S*with respect to the reference

*N*, as follows.

Applying Kirchhoffâ€™s Current law at node

*S*,

**Illustration of millmanâ€™s theorem**

â€‹
or
or
â‡’

An extension of the Millmanâ€™s theorem is the

An extension of the Millmanâ€™s theorem is the

*equivalent generator theorem*