Electrical Circuit Analysis

Chapter 4
Linearity and Network Theorems

Bachelors Program in Electrical Engineering
Electrical Circuit Analysis • Bachelor

Linearity and Network Theorems

Introduction

Electrical networks encountered in power systems, instrumentation, communications and electronic design can contain many interconnected sources and components. Direct application of Kirchhoff’s laws always remains possible, but repeated solution of a large network—especially for several source or load conditions—is inefficient. Network theorems provide mathematically justified ways to decompose, transform and replace portions of a circuit without losing the terminal behavior relevant to the engineering problem.

This chapter develops superposition, source transformation, Thévenin, Norton, substitution, reciprocity and maximum power-transfer theorems. Particular attention is given to their applicability conditions. These results are powerful only when their assumptions—such as linearity, bilateral behavior and correct source deactivation—are respected.

Learning Objectives

After studying this chapter, students should be able to:

distinguish linear circuit elements from networks whose responses are linear in their source excitations;
apply superposition to calculate voltages and currents in multisource circuits;
perform valid voltage-source and current-source transformations;
determine Thévenin and Norton equivalents using open-circuit, short-circuit and test-source methods;
apply substitution and reciprocity theorems to appropriate networks;
derive and use maximum power-transfer conditions for DC and AC loads;
evaluate the limitations, safety implications and design trade-offs associated with network equivalents.

1. Linearity in Circuit Analysis

1.1 Mathematical meaning of linearity

An operator or system F is linear if it satisfies both additivity and homogeneity:

F(x₁+x₂)=F(x₁)+F(x₂)

F(αx)=αF(x)

Together, these properties imply

F(αx₁+βx₂)=αF(x₁)+βF(x₂)

where α and β are constants.

A resistor with constant resistance is a linear element because v=Ri. An ideal inductor and capacitor are also linear when their parameters are constant:

v_L=L\frac{di_L}{dt}, \qquad i_C=C\frac{dv_C}{dt}

Here, L is inductance in henries and C is capacitance in farads. Linear dependent sources also preserve linearity when their controlling relationships have constant coefficients, such as i=g_mv_x.

An ideal independent source should not itself be described as a linear circuit element in the homogeneous constitutive sense. For example, the relation v=12\text{ V} does not pass through the origin and does not satisfy homogeneity. Instead, a network constructed from linear passive elements and linear controlled sources can have responses that are linear functions of its independent source excitations. This source-to-response linearity is the basis of superposition.

For dynamic circuits, superposition applies when initial conditions are zero or when initial-condition responses are represented as separate excitations. Nonlinear devices—including diodes, transistors over large signal ranges, saturating inductors and temperature-dependent resistors—do not generally satisfy superposition.

1.2 Linear and nonlinear quantities

Voltage and current are linear response variables in a linear network. Power is not, because

p=vi=i²R=\frac{v²}{R}

Consequently, powers produced by individual source contributions cannot normally be added to obtain total power. The voltage or current contributions must first be added algebraically, after which power is calculated from the total response.

2. Superposition Theorem

2.1 Statement and procedure

In a linear network containing two or more independent sources, any branch voltage or current equals the algebraic sum of the responses produced by each independent source acting alone. Linear dependent sources remain active because they are part of the network relationships.

To apply superposition:

Select one independent source to remain active.
Set every other independent source to zero.
Replace a zero-valued ideal voltage source by a short circuit.
Replace a zero-valued ideal current source by an open circuit.
Retain all resistances, impedances and dependent sources.
Calculate the selected source’s contribution with the chosen reference polarity or direction.
Repeat for every independent source and add the contributions algebraically.

Source deactivation does not mean physically removing every source branch. Any internal or series resistance associated with a practical source remains in the network.

2.2 Worked Example 1: Superposition in a two-source network

The following network has a 12 V ideal source whose upper terminal is positive, a 3 Ω resistor, a 6 Ω resistor and a 2 A source directed upward into node v. Determine the node voltage relative to ground.

Superposition circuit with a positive 12 volt source feeding a node through 3 ohms, a 6 ohm resistor to ground, and a 2 ampere source directed upward from ground

Figure: Superposition circuit with a positive 12 volt source feeding a node through 3 ohms, a 6 ohm resistor to ground, and a 2 ampere source directed upward from ground

Figure: Unambiguous superposition circuit with the 12 V source positive at its upper terminal and the 2 A source injecting current into node v.

Contribution of the 12 V source. Open-circuit the 2 A source. The 3 Ω and 6 Ω resistors form a voltage divider:

v₁₂=12\left(\frac{6}{3+6}\right)=8\text{ V}

Contribution of the 2 A source. Short-circuit the 12 V source. The resistors are then in parallel:

R_eq=3\parallel6=\frac{3(6)}{3+6}=2\ Ω

The node-voltage contribution is

v_2A=(2\text{ A})(2\ Ω)=4\text{ V}

Therefore,

v=v₁₂+v_2A=8+4=12\text{ V}

A direct nodal check confirms the result:

\frac{v-12}{3}+\frac{v}{6}-2=0 \quad\Rightarrow\quad v=12\text{ V}

Power must now be calculated from the total voltage. For example, the 6 Ω resistor dissipates P₆=v²/6=24\text{ W}, not the sum of separately calculated contribution powers.

3. Source Transformation

A practical voltage-source representation consisting of an ideal voltage source V_s in series with resistance R_s is terminally equivalent to an ideal current source I_s in parallel with the same resistance, provided

I_s=\frac{V_s}{R_s}

The current-source arrow points from the negative toward the positive terminal of the corresponding voltage source. Conversely,

V_s=I_sR_s

For a terminal voltage v and load current i delivered by the source, the voltage-source form gives

v=V_s-iR_s

The current-source form gives i=I_s-v/R_s, or v=R_s(I_s-i). Substitution of I_s=V_s/R_s produces the same terminal equation.

Source transformation preserves external terminal behavior, not internal branch currents or powers. It is invalid to transform an ideal voltage source with zero series resistance directly because the equivalent current would be unbounded. Transformations may also be used with linear impedances in sinusoidal steady state and with dependent sources when the control relationship is retained correctly.

Example. A 12 V source in series with 3 Ω transforms into a 4 A source in parallel with 3 Ω. Either representation has an open-circuit voltage of 12 V and a short-circuit current of 4 A.

4. Thévenin and Norton Theorems

4.1 Terminal equivalents

Any linear two-terminal network can be represented at its terminals by either:

a Thévenin voltage source V_th in series with resistance or impedance R_th or Z_th; or
a Norton current source I_N in parallel with the same resistance or impedance.

The equivalent parameters satisfy

V_th=V_oc

I_N=I_sc

R_th=R_N=\frac{V_th}{I_N}

Here, V_oc is the open-circuit terminal voltage and I_sc is the short-circuit current under consistent polarity and direction conventions. The ratio is usable when the quantities are finite and I_N≠0.

These equivalents reproduce terminal voltage and current for every connected load within the linear model. They do not necessarily reproduce the internal voltages, currents or component powers of the original network.

4.2 Determining equivalent resistance

Three standard methods are available.

Method 1: Source deactivation. Deactivate all independent sources, retain dependent sources, and calculate the resistance seen at the terminals. This is convenient for resistive networks containing only independent sources and resistors.

Method 2: Test source. First deactivate all independent sources. Keep all dependent sources active. Apply a test voltage V_test and determine the resulting input current I_test, or apply a test current and determine the terminal voltage. Then

R_th=\frac{V_test}{I_test}

The current direction must be defined as entering the positive terminal of the test voltage. Deactivating independent sources before forming this ratio is essential; otherwise the measured voltage and current include an offset caused by the original excitation and their simple ratio need not equal the Thévenin resistance.

Method 3: Open- and short-circuit quantities. Calculate V_oc and I_sc, then use R_th=V_oc/I_sc. This method can be effective when dependent sources are present.

4.3 Worked Example 2: Thévenin and Norton load analysis

An 18 V source supplies a 6 Ω series resistor leading to output terminal a. A 3 Ω resistor connects terminal a to ground terminal b. Find the equivalent seen at a-b, and determine the current in a 4 Ω load.

With the load removed, voltage division gives

V_th=V_oc=18\left(\frac{3}{6+3}\right)=6\text{ V}

Deactivate the 18 V ideal source by replacing it with a short circuit. The 6 Ω and 3 Ω resistors are then in parallel:

R_th=6\parallel3=2\ Ω

The Norton current is

I_N=\frac{V_th}{R_th}=\frac{6}{2}=3\text{ A}

With R_L=4\ Ω, the load current from the Thévenin equivalent is

I_L=\frac{V_th}{R_th+R_L}=\frac{6}{2+4}=1\text{ A}

Thus V_L=I_LR_L=4\text{ V} and P_L=I_L²R_L=4\text{ W}.

5. Substitution Theorem

The substitution theorem states that if the voltage across and current through a branch are known, that branch may be replaced by any other element or source arrangement that maintains the same branch voltage and current. Under these conditions, all voltages and currents in the remainder of the network remain unchanged.

The theorem is an operating-point statement, not a claim that the replacement has the same behavior for every possible load or excitation. It is therefore different from a Thévenin equivalent, which preserves the complete linear terminal relationship.

Worked Example 3: Branch substitution

A 10 V ideal source drives a 2 Ω resistor and a 3 Ω resistor in series. The current is

I=\frac{10}{2+3}=2\text{ A}

The 3 Ω branch has a voltage drop

V₃=(2)(3)=6\text{ V}

Replace the 3 Ω resistor by an ideal 6 V source having a 6 V drop in the direction of current. The remaining loop current becomes

I=\frac{10-6}{2}=2\text{ A}

The voltage and current at the substituted branch are unchanged, so the current through the 2 Ω resistor and all other network variables remain unchanged. However, if the 10 V excitation changes, the fixed 6 V replacement will not generally reproduce the original resistor behavior. Substitution is useful for verifying solutions, constructing equivalent operating-point models and replacing a complicated branch whose voltage and current are already known.

6. Reciprocity Theorem

Reciprocity concerns the interchange of excitation and response locations. In a linear, bilateral, time-invariant network composed of reciprocal elements such as ideal resistors, inductors and capacitors, the transfer response remains unchanged when the excitation and response branches are interchanged, provided their orientations and units are defined consistently.

In two-port notation, impedance reciprocity is expressed as

Z₂₁=Z₁₂

where Z₂₁=V₂/I₁ with port 2 open except for measurement, and Z₁₂=V₁/I₂ under the reciprocal experiment.

The theorem is not generally applicable to networks containing unilateral devices, independent simultaneous excitations, nonreciprocal controlled-source arrangements, gyrators or magnetically biased nonreciprocal components. Linearity alone is insufficient; reciprocity is an additional structural property.

Worked Example 4: Reciprocal transfer resistance

Consider a two-node resistor network. Node 1 is connected to ground through 6 Ω, node 2 is connected to ground through 3 Ω, and a 2 Ω resistor joins the nodes. Inject 1 A into node 1 while node 2 has no external current injection. The nodal equations are

\begin{bmatrix}2/3&-1/2\\-1/2&5/6\end{bmatrix}
\begin{bmatrix}v₁\\v₂\end{bmatrix}
=
\begin{bmatrix}1\\0\end{bmatrix}

The conductance-matrix determinant is

\left(\frac{2}{3}\right)\left(\frac{5}{6}\right)-\left(\frac{1}{2}\right)²=\frac{11}{36}

Therefore,

v₂=\frac{1/2}{11/36}(1\text{ A})=\frac{18}{11}\text{ V}\approx1.636\text{ V}

The transfer resistance is Z₂₁=18/11\ Ω. If the 1 A source is moved to node 2 and the response is measured at node 1, the symmetry of the conductance matrix gives

v₁=\frac{18}{11}\text{ V}

Thus Z₁₂=Z₂₁=18/11\ Ω, demonstrating reciprocity numerically.

7. Maximum Power-Transfer Theorem

7.1 Resistive DC networks

Let a linear network be replaced by a Thévenin source V_th and positive resistance R_th, connected to a variable positive load resistance R_L. The load current and power are

I_L=\frac{V_th}{R_th+R_L}

P_L=I_L²R_L=\frac{V_th²R_L}{(R_th+R_L)²}

Differentiating with respect to R_L gives

\frac{dP_L}{dR_L}=\frac{V_th²(R_th-R_L)}{(R_th+R_L)³}

For R_th>0 and R_L>0, the stationary point is a maximum at

R_L=R_th

The maximum load power is

P_L,max=\frac{V_th²}{4R_th}

At this operating point, the load and Thévenin resistance dissipate equal power, so the efficiency of the Thévenin source model is 50%. Consequently, maximum power transfer is often appropriate in communications and impedance-matching applications but undesirable in high-efficiency power distribution.

7.2 AC conjugate matching

For sinusoidal steady-state operation, let

Z_th=R_th+jX_th, \qquad Z_L=R_L+jX_L

Maximum average load power occurs under conjugate matching:

Z_L=Z_th^*=R_th-jX_th

If V_th is an RMS phasor and R_th>0, the maximum average power is

P_L,max=\frac{|V_th|²}{4R_th}

The formula is not valid without modification if peak rather than RMS phasors are used. Practical matching may also be constrained by component losses, bandwidth, stability and allowed voltage or current ratings.

8. Illustrative Design Case: Loading a Sensor Interface

Background

This fictional case represents an industrial sensor output connected to a monitoring module. Measurements show that the sensor and its conditioning network have a Thévenin equivalent of 9 V in series with 1.5 kΩ.

Problem

The monitoring input is represented by a selectable load resistance R_L. It must receive at least 6 V. The designer also wishes to understand the maximum available power and the consequences of selecting the power-matched load.

Analysis

The Norton equivalent is

I_N=\frac{9\text{ V}}{1.5\text{ kΩ}}=6\text{ mA}

Maximum power transfer occurs at R_L=1.5\text{ kΩ}. At this value,

V_L=9\left(\frac{1.5}{1.5+1.5}\right)=4.5\text{ V}

P_L,max=\frac{9²}{4(1500)}=0.0135\text{ W}=13.5\text{ mW}

Although power is maximized, the 4.5 V output violates the 6 V minimum requirement. The voltage requirement gives

9\frac{R_L}{1500+R_L}\ge6

Solving yields

R_L\ge3000\ Ω=3\text{ kΩ}

At the limiting value of 3 kΩ, the load voltage is 6 V, current is 2 mA and load power is

P_L=\frac{6²}{3000}=12\text{ mW}

Approach and outcome

The designer should select at least 3 kΩ; in a voltage-measurement application, a substantially higher input resistance is normally preferable because it reduces loading. Maximum power matching is rejected because the governing requirement is voltage accuracy rather than power extraction.

Learning points

A Thévenin equivalent makes repeated load evaluation straightforward.
Maximum power transfer is not automatically the optimum design objective.
Input resistance, measurement accuracy, current demand and component ratings must be considered together.
Equivalent circuits support decisions but do not replace verification against the physical sensor’s permitted operating range.

9. Selection of Theorems

Engineering task	Preferred theorem or method	Important caution
Isolate effects of several sources	Superposition	Do not superpose power
Convert source form	Source transformation	Preserve resistance and terminal polarity
Evaluate many possible loads	Thévenin or Norton	Equivalent is terminal-specific
Find resistance with dependent sources	Test-source method	Deactivate independent sources first
Preserve one known operating condition	Substitution	Not a general terminal equivalent
Exchange source and response locations	Reciprocity	Network must be reciprocal and bilateral
Select a load for maximum power	Maximum power transfer	May conflict with efficiency or voltage requirements

10. Practical Limitations and Professional Practice

Circuit theorems are exact only within the adopted model. Real sources have output limits, resistors vary with temperature, inductors may saturate, capacitors have parasitic resistance and semiconductor devices are nonlinear. A Thévenin equivalent obtained at one frequency may be invalid at another because impedances change with frequency.

Computer-aided circuit tools can automate operating-point, AC and parameter-sweep calculations. They should be used to verify—not replace—the engineer’s understanding of polarity, reference direction and theorem assumptions. Automated or AI-assisted circuit reduction may silently apply a theorem outside its valid conditions; results should therefore be checked using Kirchhoff’s laws, dimensional consistency and limiting cases.

Safe professional practice also requires attention to isolation, grounding, source short-circuit capability and component power ratings. Measuring a Norton current by physically shorting a high-energy source may be unsafe even when the mathematical operation is valid. In such cases, calculate the current from a trusted model or use properly rated test equipment. Designs must comply with applicable electrical safety, electromagnetic-compatibility and product standards.

Chapter Summary

Linearity permits source-response relationships to be decomposed through additivity and homogeneity. Independent sources are excitations rather than homogeneous linear elements, while constant-parameter resistors, inductors, capacitors and linear controlled sources provide linear network relationships. Superposition calculates voltage or current contributions one source at a time but cannot be applied directly to power.

Source transformations interchange equivalent voltage-source and current-source representations. Thévenin and Norton theorems replace a linear two-terminal network with simple terminal equivalents, greatly simplifying repeated load analysis. When a test source is used to find equivalent resistance, every independent source must first be deactivated while dependent sources remain active. Substitution preserves a known branch operating condition, whereas reciprocity relates transfer responses after interchanging excitation and response in a reciprocal bilateral network. Maximum power is delivered to a resistive load when R_L=R_th and to an AC load when Z_L=Z_th^*, although maximum power may conflict with efficiency, voltage regulation and measurement accuracy.

Key Terms

Linearity: Satisfaction of additivity and homogeneity.

Homogeneity: Scaling an excitation scales the response by the same factor.

Additivity: The response to summed excitations equals the sum of individual responses.

Bilateral element: An element whose voltage-current behavior is unchanged when direction is reversed consistently.

Superposition theorem: A method that sums individual source contributions in a linear network.

Source deactivation: Setting an independent voltage source to zero by shorting it or an independent current source to zero by opening it.

Source transformation: Conversion between terminally equivalent voltage-source and current-source forms.

Thévenin equivalent: A voltage source in series with an equivalent resistance or impedance.

Norton equivalent: A current source in parallel with an equivalent resistance or impedance.

Open-circuit voltage: Terminal voltage measured when external load current is zero.

Short-circuit current: Terminal current when the terminal voltage is zero.

Test source: A known voltage or current applied to determine terminal resistance or impedance.

Substitution theorem: Replacement of a branch while preserving its known voltage and current.

Reciprocity theorem: Equality of interchanged transfer responses in a reciprocal network.

Transfer impedance: Ratio of a response voltage at one port to an excitation current at another.

Maximum power transfer: The load condition that maximizes received power.

Conjugate matching: AC load selection satisfying Z_L=Z_th^*.

Loading effect: Change in a source or measurement voltage caused by a connected load.

Review Questions

Short Answer Questions

State the additivity and homogeneity conditions for linearity.
Explain why powers cannot be added using superposition.
How are ideal independent voltage and current sources deactivated?
Distinguish a Thévenin equivalent from a substitution-theorem replacement.
State the maximum power-transfer condition for an AC load.

Long Answer Questions

Derive the equivalence between a practical voltage source and a practical current source using their terminal equations.
Explain three methods for determining Thévenin resistance, including the treatment of independent and dependent sources.
Develop a complete superposition procedure and discuss its applicability to circuits with initial energy or nonlinear components.
Explain reciprocity in two-port terms and identify networks for which the theorem may fail.
Derive the maximum power-transfer condition for a positive resistive load and discuss the associated efficiency trade-off.

Analytical / Application Questions

A 24 V source with a 4 Ω series resistor feeds a terminal having an 8 Ω resistor to ground. Determine the Thévenin and Norton equivalents and evaluate a 6 Ω load.
A network has V_th=15\text{ V} and R_th=5\ Ω. Find the load required for maximum power, the maximum load power and the source-model efficiency at that condition.
In a reciprocal two-port, a 2 A injection at port 1 produces 5 V at open-circuit port 2. Predict the response at port 1 when the same 2 A injection is moved to port 2, stating all required assumptions.

Multiple Choice Questions with Answers

Question 1

Which statement most accurately describes ideal independent sources in linear circuit analysis?

A. They are homogeneous linear elements because their values are constant.

B. They are excitations for which a linear network may exhibit a linear source-to-response relationship.

C. They must be removed before any linear analysis is performed.

D. They make superposition valid for nonlinear elements.

Correct Answer: B. They are excitations for which a linear network may exhibit a linear source-to-response relationship.

Question 2

In Worked Example 1, the 12 V source contributes 8 V and the 2 A source contributes 4 V at node v. What is the total node voltage?

A. 4 V

B. 8 V

C. 12 V

D. 32 V

Correct Answer: C. 12 V

Question 3

How is an ideal independent voltage source deactivated during superposition?

A. It is replaced by an open circuit.

B. It is replaced by a short circuit.

C. Its polarity is reversed.

D. Its voltage is doubled.

Correct Answer: B. It is replaced by a short circuit.

Question 4

A 12 V source in series with 3 Ω is equivalent to which Norton source?

A. 4 A in parallel with 3 Ω

B. 4 A in series with 3 Ω

C. 36 A in parallel with 3 Ω

D. 0.25 A in parallel with 12 Ω

Correct Answer: A. 4 A in parallel with 3 Ω

Question 5

An 18 V source feeds a 6 Ω series resistor and a 3 Ω shunt resistor. What is the Thévenin resistance seen at the shunt resistor's terminals after removing the load and deactivating the source?

A. 1 Ω

B. 2 Ω

C. 3 Ω

D. 9 Ω

Correct Answer: B. 2 Ω

Question 6

Before using R_th=V_test/I_test in the test-source method, what must be done?

A. Deactivate all independent sources while retaining dependent sources.

B. Deactivate all dependent sources while retaining independent sources.

C. Short every resistor in the network.

D. Open every controlled-source branch.

Correct Answer: A. Deactivate all independent sources while retaining dependent sources.

Question 7

What does the substitution theorem guarantee?

A. The replacement behaves identically for every possible excitation.

B. Maximum power is transferred to the replacement.

C. The rest of the network is unchanged when the replacement preserves the known branch voltage and current.

D. The original network must be reciprocal.

Correct Answer: C. The rest of the network is unchanged when the replacement preserves the known branch voltage and current.

Question 8

A reciprocal resistor network has a transfer resistance of 18/11\ Ω from port 1 to port 2. What is its reverse transfer resistance?

A. 11/18\ Ω

B. 18/11\ Ω

C. 1\ Ω

D. 36/11\ Ω

Correct Answer: B. 18/11\ Ω

Question 9

A network has V_th=10\text{ V} and R_th=5\ Ω. Which resistive load receives maximum power?

A. 2 Ω

B. 5 Ω

C. 10 Ω

D. 20 Ω

Correct Answer: B. 5 Ω

Question 10

A 9 V Thévenin source with R_th=1.5\text{ kΩ} must maintain at least 6 V across a resistive load. What is the minimum load resistance?

A. 0.75 kΩ

B. 1.5 kΩ

C. 3 kΩ

D. 6 kΩ

Correct Answer: C. 3 kΩ

Electrical Circuit Analysis
Chapter 4 of 15

Chapter 4 Linearity and Network Theorems