Thevenin and Norton Theorem

In this tutorial, you will explore the fundamentals of Thevenin and Norton equivalents in AC circuits. You will learn how to simplify complex circuits using Thevenin’s and Norton’s theorems, including steps for circuits with independent and dependent sources. Additionally, you will understand the process of calculating equivalent voltages and currents, and discover practical applications of these concepts in circuit analysis.

Contents:

1. Thevenin Equivalent of an AC Circuit
2. Norton Equivalent of an AC Circuit
3. AC Circuits with Independent Sources Using Thevenin’s Theorem
4. AC Circuits with Independent Sources Using Norton’s Theorem
5. AC Circuits with Independent and Dependent Sources Using Thevenin’s Theorem
6. AC Circuits with Independent and Dependent Sources Using Norton’s Theorem

Thevenin Equivalent of an AC Circuit

Thevenin’s theorem states that any linear AC circuit with complex impedances can be replaced by an equivalent circuit consisting of a single voltage source (Thevenin voltage) in series with a Thevenin impedance. The load connected to the original circuit can then be reconnected to this equivalent circuit for analysis.

The given diagram represents the Thevenin equivalent circuit of an AC circuit.

• Thevenin’s theorem helps in simplifying a circuit. An AC circuit with complex impedance can be converted to a Thevenin equivalent circuit by obtaining the equivalent voltage and equivalent impedance.
• The equivalent voltage is the terminal voltage when they are open-circuited while equivalent impedance is the impedance seen from the output when the independent sources are turned off.
• The equivalent voltage or the Thevenin voltage is connected in series to the equivalent impedance. The load is reconnected to the terminals of the Thevenin equivalent circuit.
• The load current I_L can be obtained by the following expression:
  \(I_L = \frac{V_{\text{source}}}{Z_{\text{total}}}\)
  \(I_L = \frac{V_{\text{Th}}}{Z_{\text{Th}} + Z_L} = \frac{V_{\text{Th}}}{(R_{\text{Th}} + jX_{\text{Th}}) + (R_L + jX_L)}\)
• The load voltage V_L can be obtained by voltage division rule.
  \(V_L = \frac{V_{\text{source}}}{Z_{\text{total}}} \cdot Z_L\)
  \(V_L = \frac{V_{\text{Th}}}{Z_{\text{Th}} + Z_L} \cdot Z_L = \frac{V_{\text{Th}} \cdot Z_L}{(R_{\text{Th}} + jX_{\text{Th}}) + (R_L + jX_L)}\)
  \(V_L = I_L \cdot Z_L\)

Norton Equivalent of an AC Circuit

Norton’s theorem simplifies an AC circuit by replacing it with a current source (Norton current) in parallel with a Norton impedance (ZN).

The given diagram represents the Norton equivalent of an AC circuit.

• Norton’s theorem simplifies the circuit for analysis. An AC circuit with complex impedance can be converted to a Norton equivalent circuit by obtaining the equivalent current and equivalent impedance.
• The equivalent current or the Norton current is the current flowing in the terminals when they are short-circuited while Norton impedance is the equivalent impedance when the independent sources are turned off.
• The Norton current source is connected in parallel to the equivalent impedance.
• The load is reconnected to the terminals of the Norton equivalent circuit and the load voltage and current are calculated.
• The load current (I_L) can be obtained by applying the current division rule.
  \(I_L = \frac{I_{\text{source}}}{Z_{\text{total}}} \cdot Z_N\)
  \(I_L = \frac{I_N}{Z_N + Z_L} \cdot Z_N = \frac{I_N \cdot Z_N}{(R_N + jX_N) + (R_L + jX_L)}\)
• The load voltage V_L can be obtained by the following expression:
  \(V_L=I_L Z_L\)

Solving AC Circuits with Independent Sources Using Thevenin’s Theorem

The following steps to solve an AC circuit with all independent sources using Thevenin’s theorem.

• The load impedance (Z_L) in the circuit is removed.
• The open-circuit voltage across the load terminals is the Thevenin voltage of the circuit (V_Th). This voltage is calculated.
  \(V_{Th}=V_{oc}\)
• The sources are replaced by their internal resistance. In the case of ideal sources, the voltage source is short-circuited and the current source is open-circuited.
• The equivalent impedance as seen from the load terminals is calculated. This impedance is known as the Thevenin Impedance Z_Th.
  \(Z_{Th}=Z_{in}\)
• The Thevenin equivalent circuit is drawn by connecting the Thevenin voltage and Thevenin impedance in series.
• Reconnect the load to the equivalent circuit and calculate the load parameters.

Solving AC Circuits with Independent Sources Using Norton’s Theorem

The following are the steps to solve an AC circuit with all independent sources using Norton’s theorem.

• The load impedance (Z_L) in the circuit has to be removed as it is not required in the analysis.
• The load terminals are short-circuited. The short circuit current flowing in the terminals is calculated and this current is the Norton current (I_N).
  \(I_N=I_{sc}\)
• The sources present in the circuit must be replaced by their internal resistance. If the sources are ideal, the voltage source must be short-circuited and the current source must be open-circuited.
• The equivalent impedance seen from the load terminals is calculated. This Norton impedance (Z_N) is the same as the Thevenin impedance (Z_Th).
• The Norton’s equivalent circuit is drawn by connecting the current source with Norton current in parallel with Norton impedance.
• Reconnect the load across the load terminals. The load parameters can be calculated.

Solving AC Circuits with Independent and Dependent Sources Using Thevenin’s Theorem

For circuits with independent as well as dependent sources, the steps are:

• The load impedance (Z_L) in the circuit must be removed to form an open circuit terminal. The open-circuit voltage across the open-circuited terminals is calculated. This is the Thevenin voltage (V_Th).
  \(V_{Th}=V_{oc}\)
• The sources present in the circuit must be replaced by their internal resistance.
• The dependent sources cannot be replaced by their internal resistance directly as few circuit variables control them.
• A voltage source giving voltage (v_o) is placed at the load terminals. v_o can be any value but is taken as 1 V for simplicity. The resulting current (i_o) from the voltage source is calculated. The ratio of voltage to current will give the Thevenin impedance (Z_o).
  \(Z_{\text{Th}} = \frac{v_o}{i_o} = \frac{1}{i_o} \)
• A current source giving any current (i_o) may also be placed at the load terminals. i_o can be chosen as 1 A. The resulting voltage (v_o) across the current source is calculated. The ratio of voltage to current will give the Thevenin impedance (Z_Th).
  \(Z_{\text{Th}} = \frac{v_o}{i_o} = \frac{v_o}{1} \)
• The Thevenin equivalent circuit is obtained by connecting the Thevenin voltage and Thevenin impedance in series. The load impedance is reconnected to the equivalent circuit.

Solving AC Circuits with Independent and Dependent Sources Using Norton’s Theorem

The following points give the steps to solve an AC circuit with independent as well as dependent sources using Norton’s theorem.

• The load impedance (Z_L) in the circuit is removed and the load terminals are short-circuited. Calculate the short-circuit current flowing through the terminal to find the Norton current (I_N).
  \(I_N=I_{sc}\)
• For ideal sources, the voltage source must be short-circuited and the current source must be open-circuited.
• The dependent sources cannot be replaced by their internal resistance, thus it is kept as it is.
• A voltage (v_o) is applied at the load terminals, where v_o is chosen as 1 V. The current (i_o) from the voltage source in the terminal is calculated. The ratio of voltage to current gives the Norton’s impedance (Z_N).
  \(Z_N = \frac{v_o}{i_o} = \frac{1}{i_o} \)

A current source (i_o) at the load terminals may be placed instead of a voltage source. The value of i_o can be chosen as 1 A for simplicity. The voltage (v_o) across the current source is calculated. The ratio of voltage to current gives the Norton’s impedance (Z_N).
\(Z_N = \frac{v_o}{i_o} = \frac{v_o}{1} \)

• The Norton’s equivalent circuit is obtained by connecting the current source with Norton current in parallel with Norton impedance. The load impedance is connected across the load terminals to obtain the final circuit.