Chapter 33: Thevenin and Norton Theorems

Date: 1/8/2019

33.1 Introduction

Many of the networks analysed by Kirchhoff’s, mesh current, and nodal analysis can be analysed more quickly and evenly by using Thevenin and Norton theorems.

Thevenin’s Theorem

This states: “The current which flows in any branch of a network is the same as that which would flow in the branch if it were connected across a source of electrical energy, the emf of which is equal to the potential difference which would appear across the branch if it were open-circuited, and the internal impedance of which is equal to the impedance which appears across the open-circuited branch terminals when all sources are replaced by their internal impedances.”

The theorem applies to any linear active network. The term “active” means it contains a source or sources of emf. The term “linear” means the measured values of circuit components are independent of the direction and magnitude of the current flowing through them.

This theorem simply means that a complicated network with terminals AB can be replaced by a source E in series with an impedance Z. E is the open circuit voltage measured at terminals AB, and Z is the equivalent impedance of the network at terminals AB when all internal sources of emf are made zero. The polarity of E is made to reflect the polarity of AB such that the current for the network and the Thevenin transform for an impedance Z_L connected across AB is the same.

Procedure for Thevenin’s Theorem

The following procedure can be used when determining the current flowing in a branch:

1. Remove the load/target impedance Z_L.
2. Determine the open-circuit voltage E across the break.
3. Remove each source of emf and replace it by its internal impedance (if it has zero internal impedance, replace it by a short circuit), and then determine the internal impedance looking in at the break.
4. Determine the current from the Thevenin equivalent circuit using the formula:

Where Z is the internal impedance and Z_L is the resistance of the load.

33.4 Norton’s Theorem

In a Thevenin circuit, the source of electrical energy is represented by an EMF in series with an impedance. It can also be represented by a current source in parallel with an impedance; these two circuits are electrically equivalent.

Norton’s Theorem states: “The current that flows in any branch of a network is the same as that which would flow in the branch if it were connected across a source of electrical energy, the short-circuit current of which is equal to the current that would flow in a short-circuit across the branch, and the internal impedance of which is equal to the impedance which appears across the open-circuited branch terminals.”

Meaning that a network containing voltage sources and impedances can be replaced by a current source in parallel with an impedance.

Procedure for Norton’s Theorem

The method of finding the Norton equivalent circuit consists of 4 steps:

1. Short circuit AB.
2. Determine the short-circuit current I_sc.
3. Remove each source of emf and replace it with a short circuit or with an internal resistance. (If current sources exist, replace them with an open circuit), then determine the impedance Z “looking in” at a break between A and B.
4. Determine the value of current flowing in the target impedance from the equation of the Norton equivalent network:

33.5 Thevenin and Norton Equivalent Networks

A Thevenin equivalent circuit (voltage source E in series with impedance Z) can be converted to a Norton equivalent circuit (current source I_sc in parallel with impedance Z) and vice versa. The relationship is:

To convert from Thevenin to Norton, divide the voltage by the impedance to get I_sc, and keep the impedance the same but place it in parallel. To convert from Norton to Thevenin, multiply the current by the impedance to get E, and keep the impedance the same but place it in series.