Fundamentals of Electrical Engineering

Chapter 4
Network Theorems and Engineering Decisions

Bachelors Program in Electrical Engineering
Fundamentals of Electrical Engineering • Bachelor

Network Theorems and Engineering Decisions

Introduction

Electrical networks encountered in power supplies, instrumentation, communication systems and embedded hardware may contain many sources and components. Direct application of Kirchhoff’s laws is always possible for a lumped network, but it is not always the most efficient engineering approach. Network theorems reduce a complex linear circuit to simpler forms while preserving the terminal behavior relevant to a design decision.

This chapter develops linearity, superposition, source transformation, Thévenin and Norton equivalents, and maximum-power-transfer principles. Particular attention is given to theorem conditions, dependent sources, stored energy, simulation, and the distinction between a mathematically correct result and an appropriate engineering decision.

Learning Objectives

After studying this chapter, a student should be able to:

determine whether a circuit satisfies the conditions for linear analysis;
apply superposition to voltages and currents;
transform practical voltage and current sources without changing terminal behavior;
construct Thévenin and Norton equivalents, including networks with dependent sources;
correctly use test sources and incremental resistance;
determine loads for maximum power transfer in DC and AC networks;
compare hand analysis with circuit simulation; and
select an efficient and defensible solution method for a practical network.

1. Linearity and Network Response

A network is linear when its output satisfies additivity and homogeneity. If excitation x₁ produces response y₁ and excitation x₂ produces response y₂, then

y(a x₁ + b x₂)=a y(x₁)+b y(x₂)

where a and b are constants. The excitation may be a voltage or current source, while the response may be any branch voltage or current.

Ideal resistors with constant resistance are linear because v=Ri. Ideal inductors and capacitors are described by

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

where L is inductance in henries, C is capacitance in farads, v_L is in volts, and i_C is in amperes. These elements support linear analysis when their parameters are constant.

For circuits containing inductors or capacitors, linear response with respect to the applied independent sources normally assumes zero initial conditions. If an inductor has initial current or a capacitor has initial voltage, the stored energy must be represented as an additional excitation. Holding a nonzero initial condition fixed while varying only the external sources generally produces an affine response—an offset plus a linear response—not a homogeneous one.

Linear dependent sources remain active during linear analysis, provided their controlling relationship is linear. Diodes, transistors over large signal ranges, saturating inductors and temperature-dependent resistors are generally nonlinear. They may nevertheless be replaced by linearized small-signal models around a specified operating point.

2. Superposition Theorem

2.1 Statement and conditions

In a linear network containing multiple independent sources, any branch voltage or current equals the algebraic sum of the responses caused by each independent source acting separately.

When one independent source is considered:

replace every other ideal voltage source by a short circuit;
replace every other ideal current source by an open circuit;
retain all dependent sources;
retain passive impedances.

A nonideal source must not be removed together with its internal resistance. For example, deactivating a practical voltage source sets its ideal source voltage to zero but leaves its series resistance in the circuit.

Superposition applies directly to voltages and currents, not to power, because power is nonlinear in these quantities:

P=VI=I²R=\frac{V²}{R}

Power must therefore be calculated from the total voltage or current after all component responses have been summed.

2.2 Worked example: node voltage by superposition

A node is connected to a 12 V source through 4 kΩ, to a 6 V source through 6 kΩ, and to ground through 3 kΩ. Both source voltages are referenced to ground. Determine the node voltage V.

With only the 12 V source active, the 6 V source is short-circuited:

V₁₂=\frac{12/4}{1/4+1/6+1/3}=4.00\ \text{V}

The resistance values may be entered in kilohms because the common scale cancels.

With only the 6 V source active:

V₆=\frac{6/6}{1/4+1/6+1/3}=1.333\ \text{V}

Therefore,

V=V₁₂+V₆=5.333\ \text{V}

A direct nodal check gives

\frac{V-12}{4\ \text{kΩ}}+\frac{V-6}{6\ \text{kΩ}}+\frac{V}{3\ \text{kΩ}}=0

which produces the same result.

2.3 When superposition is useful

Superposition is especially useful when an engineer needs to identify the contribution of noise, bias, interference or several supplies. It becomes inefficient when a circuit contains many independent sources because one subproblem is required for each source. In such cases, nodal analysis or simulation may be faster.

3. Source Transformation

A practical voltage source 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, where

I_s=\frac{V_s}{R_s}

Conversely,

V_s=I_sR_s

Here V_s is in volts, I_s is in amperes, and R_s is in ohms. The current-source arrow must be oriented so that the open-circuit voltage has the same polarity as the original voltage source.

Equivalent practical voltage and current sources

Figure: Equivalent practical voltage and current sources

Figure: Terminal equivalence of a practical voltage source and a practical current source.

For example, a 10 V source in series with 5 Ω is equivalent to a 2 A source in parallel with 5 Ω. Source transformation preserves external terminal voltage and current for every connected load. It does not necessarily preserve internal branch currents or the power assigned to individual ideal elements.

The transformation extends to sinusoidal steady-state circuits by replacing resistance with impedance:

\underline{I}_s=\frac{\underline{V}_s}{Z_s}

where phasors are normally RMS quantities and Z_s is measured in ohms.

4. Thévenin and Norton Theorems

4.1 Thévenin equivalent

Any linear two-terminal network can be represented at its port by an ideal voltage source V_th in series with an equivalent resistance or impedance R_th or Z_th. For a resistive network,

v=V_th-R_thi

where i is defined as current delivered by the equivalent network to the load. The open-circuit terminal voltage is

V_th=V_oc

4.2 Norton equivalent

The same network can be represented by a current source I_N in parallel with R_N. For ordinary finite equivalents,

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

If I_sc is defined as the short-circuit current delivered by the network, then

I_N=I_sc, \qquad R_th=\frac{V_oc}{I_sc}

The last expression requires a linear one-port, consistent reference directions, and finite nonzero quantities. Degenerate ideal-source cases must be handled separately.

4.3 Derivation from the terminal relation

The voltage–current characteristic of a linear resistive one-port containing independent sources is affine. Using delivered current,

v=V_oc-R_thi

At open circuit, i=0, so v=V_oc. At short circuit, v=0, giving I_sc=V_oc/R_th. Thus the Thévenin and Norton models describe the same straight-line terminal characteristic.

4.4 Methods for finding equivalent resistance

#### Method 1: deactivate independent sources

Set independent voltage sources to zero by replacing them with short circuits and set independent current sources to zero by replacing them with open circuits. Keep dependent sources active. Determine the resistance viewed from the port.

This method is convenient when the remaining resistor network can be simplified directly.

#### Method 2: apply a test source

After deactivating independent sources, apply a test voltage V_t and calculate the entering current I_t, or apply a test current and calculate the voltage. Then

R_th=\frac{V_t}{I_t}

This form applies because the deactivated network passes through the origin of its port characteristic. Dependent sources must remain active.

#### Method 3: incremental test with independent sources active

If independent sources remain active, the port relation generally contains an offset. Therefore, V_t/I_t is not generally equal to R_th. Instead, use two operating points:

R_th=\frac{\Delta V_t}{\Delta I_t}

with current defined as entering the network. In differential form,

R_th=\frac{dV_t}{dI_t}

This is the slope of the port characteristic. For a linear network the slope is constant; for a nonlinear network it is a local small-signal resistance at the operating point.

As an illustration, suppose a port contains a 10 Ω resistor and a voltage-controlled current source of transconductance 0.2 S, both drawing current from the port node to ground, plus a 1 A independent source directed from ground to the port. With port current entering the network,

I_t=\frac{V_t}{10}+0.2V_t-1=0.3V_t-1

Thus,

R_th=\frac{\Delta V_t}{\Delta I_t}=\frac{1}{0.3}=3.333\ \text{Ω}

The ratio V_t/I_t varies with the test level because of the 1 A offset and is therefore not the equivalent resistance. If the independent source is first deactivated, the relation becomes I_t=0.3V_t, and the ordinary ratio gives 3.333 Ω.

4.5 Worked example: Thévenin and Norton models

A 24 V ideal source supplies a 6 Ω resistor in series with a node. A 12 Ω resistor connects that node to ground. Find the equivalent seen between the node and ground.

The open-circuit voltage is the divider voltage:

V_th=24\left(\frac{12}{6+12}\right)=16\ \text{V}

After deactivating the 24 V source, the 6 Ω and 12 Ω resistors appear in parallel:

R_th=6\parallel12=\frac{6\times12}{6+12}=4\ \text{Ω}

The Norton current is

I_N=\frac{16}{4}=4\ \text{A}

For an 8 Ω load,

I_L=\frac{16}{4+8}=1.333\ \text{A}

V_L=I_LR_L=10.667\ \text{V}

P_L=I_L²R_L=14.22\ \text{W}

The Norton model gives identical terminal results.

5. Maximum Power Transfer

5.1 Resistive DC networks

For a Thévenin source supplying a resistive load R_L,

P_L=\left(\frac{V_th}{R_th+R_L}\right)²R_L

where V_th is in volts, resistances are in ohms, and power is in watts. This formula assumes R_th>0 and a purely resistive load.

Differentiating with respect to R_L gives

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

The stationary point is a maximum when

R_L=R_th

and the maximum load power is

P_L,max=\frac{V_th²}{4R_th}

At this operating point, equal power is dissipated in R_th and R_L, so the source-to-load efficiency of this simple model is only 50%. Maximum power transfer is therefore important in communication interfaces, sensing and impedance matching, but it is usually undesirable in bulk power distribution, where high efficiency and voltage regulation dominate.

5.2 AC networks

Let the Thévenin impedance be

Z_th=R_th+jX_th

and the load impedance be Z_L=R_L+jX_L. For RMS phasors, average load power is

P_L=\frac{|\underline{V}_th|²R_L}{|Z_th+Z_L|²}

Maximum average power occurs under conjugate matching:

Z_L=Z_th^*=R_th-jX_th

This requires an adjustable complex load and R_th>0. If the load must be purely resistive, the optimum is instead

R_L=|Z_th|=\sqrt{R_th²+X_th²}

These two conditions must not be confused.

6. Choosing an Efficient Solution Method

Network feature	Usually efficient method	Reason
One dominant essential node	Nodal analysis	Few unknown voltages
Several series loops	Mesh analysis	Few loop-current equations
Need each source’s contribution	Superposition	Separates source effects
Repeated changes of one load	Thévenin or Norton	Network solved once
Mixed series voltage and parallel current sources	Source transformation	Reveals simpler topology
Dependent sources at a port	Test-source method	Preserves controlled-source action
Large or tolerance-sensitive circuit	Simulation plus analytical checks	Rapid sweeps and verification
Maximum load power required	Thévenin impedance and matching	Direct design criterion

An efficient engineer defines the required output before selecting a method. Finding every branch variable is unnecessary when only load voltage is required. Conversely, a terminal equivalent cannot answer questions about internal component stress unless the original network is revisited.

7. Analytical Methods and Circuit Simulation

Circuit simulators based on SPICE can perform operating-point, transient, AC, noise, sensitivity and Monte Carlo analyses. They are particularly valuable for networks containing many components, frequency dependence, tolerances or realistic nonlinear device models.

Analytical methods remain essential because they:

reveal parameter dependence and limiting behavior;
expose sign, topology and unit errors;
provide independent checks on simulation;
clarify assumptions hidden inside device models; and
support design decisions without relying on a single software implementation.

A robust workflow is:

define the port, reference directions and required quantities;
identify whether the network is linear and whether initial energy is present;
estimate expected magnitudes and limiting cases;
select a theorem or systematic equation method;
calculate analytically where practical;
construct the simulation with explicit ground and source definitions;
compare results, power balance and limiting cases; and
investigate discrepancies rather than selecting the preferred answer.

Simulation results are not automatically correct. Frequent errors include omitted grounds, wrong source polarity, incorrect AC magnitude, peak-versus-RMS confusion, unrealistic ideal models and unintended initial conditions. AI-assisted schematic generation may accelerate model creation, but generated netlists and component assumptions require expert verification.

8. Illustrative Case: Emergency Beacon Supply Interface

Background

An illustrative engineering team is designing an emergency beacon supplied by a battery and cable assembly. Measurements at the beacon connector produce a 12 V open-circuit voltage and an 8 A short-circuit current.

Problem

The beacon requires at least 9 V and 15 W during activation. The team initially proposes matching the load for maximum power transfer.

Analysis

The source equivalent is

V_th=12\ \text{V}, \qquad R_th=\frac{12}{8}=1.5\ \text{Ω}

Maximum power transfer requires R_L=1.5\ \text{Ω}. The resulting load voltage and power are

V_L=12\frac{1.5}{1.5+1.5}=6\ \text{V}

P_L=\frac{12²}{4(1.5)}=24\ \text{W}

Although power is maximized, the 6 V terminal voltage violates the 9 V requirement.

At nominal conditions, a 4.5 Ω load gives

V_L=12\frac{4.5}{1.5+4.5}=9\ \text{V}

P_L=\frac{9²}{4.5}=18\ \text{W}

This appears acceptable, but battery and cable tolerances must be considered. Suppose V_th may fall to 11.4 V and R_th may rise to 1.8 Ω. The worst-case voltage becomes

V_L=11.4\frac{4.5}{1.8+4.5}=8.14\ \text{V}

The requirement is no longer met.

For a passive resistor to maintain at least 9 V under this worst case,

R_L\geq\frac{9(1.8)}{11.4-9}=6.75\ \text{Ω}

Yet delivering 15 W at 9 V would require an effective resistance no greater than

R_L=\frac{9²}{15}=5.4\ \text{Ω}

The two conditions are incompatible for a fixed resistive load.

Approach and outcome

The team selects a regulated DC–DC converter rather than a fixed resistor interface. At the worst-case source condition, the maximum available source power is

P_max=\frac{11.4²}{4(1.8)}=18.05\ \text{W}

A converter delivering 15 W would therefore require efficiency above approximately 15/18.05=83.1\%, together with suitable input-voltage and current ratings. The final design must also include startup-current analysis, thermal margins, cable protection and battery-discharge testing.

Learning points

Maximum power is not synonymous with correct design. Voltage regulation, efficiency, tolerances, thermal limits, reliability and safety can dominate the decision. A Thévenin equivalent efficiently establishes feasibility, while simulation and testing validate converter dynamics and component stresses.

9. Limitations, Safety and Professional Practice

Network theorems are only as valid as their models. Important limitations include:

nonlinear or time-varying components;
temperature-dependent parameters;
magnetic saturation and semiconductor limits;
distributed effects at high frequency;
unknown initial energy in reactive components;
negative incremental resistance and potentially unstable active networks; and
measurement loading that alters the circuit being characterized.

Maximum-power-transfer calculations do not override current, voltage, thermal or insulation ratings. Short-circuit testing may be unsafe on batteries, power converters and mains-connected equipment; I_sc should often be inferred from controlled tests rather than measured directly.

Professional practice requires traceable assumptions, appropriate uncertainty margins, validated simulation models, safe laboratory procedures and compliance with relevant electrical, electromagnetic-compatibility and product-safety standards. Engineers should report whether values are nominal, worst-case, measured or simulated.

Chapter Summary

Linearity permits complex responses to be decomposed and recombined. For reactive circuits, zero initial conditions are required for homogeneous response with respect to external sources; otherwise stored energy must be included as an excitation. Superposition applies to voltages and currents but not directly to power.

Source transformations preserve terminal behavior and lead naturally to Thévenin and Norton equivalents. Thévenin voltage is the open-circuit voltage, while Norton current is the delivered short-circuit current. Independent sources are deactivated when resistance is determined by direct inspection or a simple test-source ratio, while dependent sources remain active. If independent sources remain active, resistance must be obtained from \Delta V/\Delta I or the differential slope—not generally from V/I.

Maximum power occurs at resistance matching in resistive circuits and conjugate impedance matching in AC circuits. This condition may conflict with efficiency, voltage regulation and safety. Analytical methods provide insight and verification; simulation handles complexity and variation. Sound engineering combines both approaches with testing, tolerance analysis and documented assumptions.

Key Terms

Affine response: A response consisting of a constant offset plus a linear function of the excitation.

Conjugate matching: Selection of a load impedance equal to the complex conjugate of the source impedance.

Dependent source: A source controlled by another voltage or current in the circuit.

Equivalent resistance: The resistance representing a network’s terminal voltage–current slope.

Independent source: A voltage or current source whose specified value does not depend on another circuit variable.

Incremental resistance: The ratio \Delta V/\Delta I or derivative dV/dI at a port.

Linearity: Satisfaction of additivity and homogeneity.

Maximum power transfer: Operation at the load condition that maximizes delivered power.

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

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

Port: A pair of terminals through which a network interacts with another system.

Short-circuit current: Current delivered when terminal voltage is zero.

Small-signal model: A linearized model valid for perturbations around an operating point.

Source transformation: Conversion between terminally equivalent practical voltage and current sources.

Superposition: Addition of individual source responses in a linear network.

Test source: An imposed port voltage or current used to determine terminal characteristics.

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

Zero initial conditions: Absence of initial capacitor voltage and inductor current before an excitation is applied.

Review Questions

Short Answer Questions

State the superposition theorem and explain how ideal independent sources are deactivated.
Why must dependent sources remain active when equivalent resistance is calculated?
What is the relationship between Thévenin and Norton equivalents?
Why cannot powers caused by individual sources generally be superposed?
Distinguish resistance matching from conjugate impedance matching.

Long Answer Questions

Derive the maximum-power-transfer condition for a resistive Thévenin source.
Explain three methods for determining Thévenin resistance and state the conditions for each.
Discuss how nonzero capacitor voltage or inductor current affects the use of linearity and superposition.
Compare analytical circuit methods with SPICE simulation, including their strengths and failure modes.
Explain why maximum power transfer may be inappropriate for an electrical power distribution system.

Analytical / Application Questions

A 30 V source with a 5 Ω series resistor supplies a variable load. Determine the Thévenin and Norton models, the load for maximum power, the maximum load power and the efficiency at that point.
A linear one-port obeys I_t=0.4V_t-2 with current entering the port. Determine its incremental resistance and explain why a single ratio V_t/I_t is generally unsuitable.
A source has V_th=20\ \text{V} and Z_th=4+j3\ \text{Ω}. Determine the load for maximum average power and compare it with the optimum when the load is restricted to a pure resistance.

Multiple Choice Questions with Answers

Question 1

When applying superposition, how is an ideal independent voltage source deactivated?

A. It is replaced by an open circuit.

B. It is replaced by a short circuit.

C. Its voltage is doubled.

D. It is replaced by its load resistance.

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

Question 2

A circuit contains a capacitor with a fixed nonzero initial voltage. Which statement is correct?

A. The response to external sources is necessarily homogeneous.

B. The capacitor must always be replaced by a short circuit.

C. The initial stored energy should be treated as an additional excitation; otherwise the response may be affine.

D. Superposition becomes valid only for power.

Correct Answer: C. The initial stored energy should be treated as an additional excitation; otherwise the response may be affine.

Question 3

A 10 V source in series with 5 Ω is transformed into which Norton source?

A. 0.5 A in parallel with 5 Ω

B. 2 A in parallel with 5 Ω

C. 2 A in series with 5 Ω

D. 50 A in parallel with 2 Ω

Correct Answer: B. 2 A in parallel with 5 Ω

Question 4

A network has V_oc=18\ \text{V} and delivers I_sc=3\ \text{A}. What is its Thévenin resistance?

A. 6 Ω

B. 15 Ω

C. 21 Ω

D. 54 Ω

Correct Answer: A. 6 Ω

Question 5

When a test source is applied while independent sources remain active, equivalent resistance is generally found from which quantity?

A. V_tI_t

B. V_t/I_t at any single operating point

C. \Delta V_t/\Delta I_t

D. I_t/\Delta V_t

Correct Answer: C. \Delta V_t/\Delta I_t

Question 6

A 16 V Thévenin source with R_th=4\ \text{Ω} supplies an 8 Ω load. What is the load current?

A. 0.50 A

B. 1.00 A

C. 1.33 A

D. 4.00 A

Correct Answer: C. 1.33 A

Question 7

For a positive resistive Thévenin source, maximum load power occurs when:

A. R_L=0

B. R_L=R_th

C. R_L=2R_th

D. R_L\to\infty

Correct Answer: B. R_L=R_th

Question 8

A source has Z_th=5+j2\ \text{Ω}. Which adjustable complex load receives maximum average power?

A. 5+j2\ \text{Ω}

B. 2-j5\ \text{Ω}

C. 5-j2\ \text{Ω}

D. 7\ \text{Ω}

Correct Answer: C. 5-j2\ \text{Ω}

Question 9

At maximum power transfer in a simple resistive Thévenin circuit, what is the source-to-load efficiency?

A. 25%

B. 50%

C. 75%

D. 100%

Correct Answer: B. 50%

Question 10

Which task is most directly simplified by constructing a Thévenin equivalent?

A. Calculating responses for many different loads connected to the same port

B. Determining every internal branch current after each component is replaced

C. Superposing power values from independent sources

D. Modeling a strongly nonlinear diode over its entire operating range

Correct Answer: A. Calculating responses for many different loads connected to the same port

Fundamentals of Electrical Engineering
Chapter 4 of 15

Chapter 4 Network Theorems and Engineering Decisions