Notes › Fundamentals of Electric Circuits (Sadiku) Lecture 4

Circuit Theorems

calendar_today June 23, 2025 article 757 words access_time 8 min replay Modified June 26, 2025

Table of Contents

Linearity

Combination of two properties: homogeneity and additivity
Homogeneity requires that if the excitation is multiplied by a constant, the output is multiplied by the same constant
Additivity requires that the response to a sum of excitations is the sum of the responses if each excitation was applied separately

Definition 04.1 (Linear Circuit).

A circuit whose output is linearly related (directly proportional) to its input. Satisfies the homogeneity and additivity properties.

Example 04.2.

(Practice Problem 4.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 154])

Example 04.3.

(Practice Problem 4.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 154])

Superposition

For circuits with more than one independent source, you can determine voltage/current of an element by summing the contribution of each source to the variable

Definition 04.4 (Superposition Principle).

The voltage across (or current through) an element of a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone. Employ the following:

Voltage sources are 0V with a short circuit
Current sources are 0A with an open circuit
Dependent sources are left intact, since they are controlled by circuit variables

Example 04.5.

(Example 4.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 156])

Example 04.6.

(Practice Problem 4.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 156])

Example 04.7.

(Example 4.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 156])

Example 04.8.

(Practice Problem 4.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 158])

Example 04.9.

(Practice Problem 4.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 159])

Source Transformation

Replacing a voltage source in series with a resistor by a current source in parallel with a resistor (or vice versa)

$$V_{s} = R i_{s} \ \text{or} \ i_{s} = \frac{V_{s}}{R}$$

Can be applied to dependent sources if the equation is kept satisfied

Example 04.10.

(Example 4.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 161])

Example 04.11.

(Practice Problem 4.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 161])

Example 04.12.

(Example 4.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 162])

Example 04.13.

(Practice Problem 4.7 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 162])

Thevenin’s Theorem

Definition 04.14.

Any linear two-terminal circuit can be replaced by a voltage source $V_{\text{th}}$ in series with a resistor $R_{\text{th}}$.

$V_{\text{th}} = V_{\text{oc}}$
$R_{\text{th}} = R_{\text{eq}}$ with independent sources off

Finding $R_{\text{th}}$

Two cases:
1. The network has no dependent sources. Then, turn off all independent sources, and $R_{\text{th}} = R_{\text{eq}}$
2. The network has dependent sources. Then, turn off all independent sources, and apply a voltage source $v_o$ at the terminals. Determine $i_o$, and $R_{\text{th}} = \frac{v_o}{i_o}$
  - Alternatively, apply a current source $i_o$ at the terminals and find the terminal voltage $v_o$
If $R_{\text{th}} < 0$, the circuit is supplying power

Example 04.15.

(Example 4.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 164])

Example 04.16.

(Practice Problem 4.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 166])

Example 04.17.

(Example 4.9 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 166])

Example 04.18.

(Practice Problem 4.9 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 167])

Example 04.19.

(Practice Problem 4.10 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 169])

Norton’s Theorem

Definition 04.20.

Any linear two-terminal circuit can be replaced by a current source $i_{N}$ in parallel with a resistor $R_{N}$.

$i_{N} = i_{\text{sc}}$
$R_{\text{N}} = R_{\text{eq}}$ with independent sources off

An implication from this is $R_{N} = R_{\text{th}}$; this is true when we looked at source transformations previously
Another implication is that Thevenin and Norton’s Theorems are related by:

$$V_{\text{th}} = I_{N} R_{\text{th}} \ \text{or} \ V_{\text{oc}} = i_{\text{sc}}R_{N}$$

Example 04.21.

(Example 4.11 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 170])

Example 04.22.

(Practice Problem 4.11 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 171])

Example 04.23.

(Example 4.12 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 172])

Example 04.24.

(Practice Problem 4.12 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 172])

Maximum Power Transfer

Definition 04.25 (Maximum Power Theorem).

Maximum power is transferred to $R_{L}$ iff ${R_{L} = R_{\text{th}}}$. In terms of $V_{\text{th}}$:

$$p_{\text{max}} = \frac{V_{\text{th}}^2}{4R_{\text{th}}}$$

In terms of $I_{N}$:

$$p_{\text{max}} = \frac{I_{N}^2 R_{N}}{4}$$

Example 04.26.

(Example 4.13 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 175])

Example 04.27.

(Practice Problem 4.13 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 176])

Sources

Alexander & Sadiku, Fundamentals of Electric Circuits

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