3. Network Theorems and Equivalent Circuits

Introduction and Context

Network theorems provide structured ways of simplifying and interpreting linear electrical circuits. Instead of solving every branch current and node voltage simultaneously, an engineer may select a theorem that exposes the particular feature of interest: the contribution of an individual source, the effect of a component change, the behavior seen by a load, or the condition for maximum power delivery. These theorems are therefore not merely algebraic shortcuts. They are analytical models that support design decisions, sensitivity studies, fault analysis, measurement, and system interfacing.

The theorems considered in this chapter apply primarily to lumped, linear, bilateral, time-invariant networks, although several have useful extensions. Resistors, inductors, capacitors, dependent sources, and ideal independent sources can be treated within the appropriate linear-domain formulation. In sinusoidal steady state, resistance is replaced by impedance and current or voltage by phasors. In the Laplace domain, impedances become functions of the complex variable s, allowing the same ideas to be applied to transient and frequency-domain problems.

The central theme is equivalence. A complicated network may be replaced, at selected terminals, by a simpler network that produces the same terminal voltage-current relationship. Such equivalence is always conditional: it is defined with respect to specified terminals, operating assumptions, and a stated class of allowable loads. A Thévenin equivalent that is correct at one pair of terminals does not necessarily preserve internal branch currents or power dissipation elsewhere in the original network.

Learning Objectives

After completing this chapter, the learner should be able to:

1. State the assumptions and scope of the principal network theorems.

2. Apply superposition to voltage, current, power-related, and dependent-source problems with appropriate caution.

3. Derive and use substitution, reciprocity, compensation, and Tellegen theorems.

4. Determine Thévenin and Norton equivalents using open-circuit, short-circuit, and impedance methods.

5. Distinguish source resistance, driving-point impedance, transfer impedance, and load impedance.

6. Apply the maximum-power-transfer theorem in resistive and AC networks.

7. Interpret equivalent circuits for sensitivity, design, measurement, and system integration.

8. Identify common errors, limitations, and professional constraints in theorem-based analysis.

The Linear-Network View

A network is linear when its response satisfies both homogeneity and additivity. If an excitation x produces response y, then excitation ax produces response ay. If excitations x1 and x2 produce responses y1 and y2, then x1 + x2 produces y1 + y2. Resistors with constant resistance, inductors with constant inductance, capacitors with constant capacitance, and ideal independent or dependent sources are linear elements under the usual circuit assumptions.

Linearity makes two kinds of reasoning possible. First, a response can be decomposed into contributions from separate sources. Second, a network at its terminals can be represented by a simpler linear relation. For a one-port network, the terminal relation may be written as

v = Voc − iZth,

where Voc is the open-circuit voltage and Zth is the Thévenin impedance under the selected sign convention. The equivalent Norton form is

i = Isc − vYn,

where Isc is the short-circuit current and Yn = 1/Zth is the Norton admittance.

These equations are terminal descriptions. They do not claim that the internal topology of the replacement network is identical to that of the original network.

Superposition: Separating the Contributions of Sources

The superposition theorem states that in a linear network containing several independent sources, the voltage or current in any element equals the algebraic sum of the voltages or currents produced by each independent source acting alone. When one independent source is retained, all other independent voltage sources are replaced by their internal impedances. Thus an ideal voltage source is short-circuited and an ideal current source is open-circuited. Dependent sources are not suppressed because their values depend on circuit variables and they remain active.

Suppose a response r is caused by independent sources s1, s2, ..., sn. Linearity gives

r = r1 + r2 + ... + rn.

The theorem is most useful when the required response is associated with one branch or node and the circuit becomes substantially simpler after source deactivation. It is also useful in systems containing sources of different types, such as a DC bias source and a small AC signal source, provided the underlying model remains linear.

A significant qualification concerns power. Power is proportional to v2/R or i2R and is therefore nonlinear in voltage and current. The power delivered when all sources act together is not generally equal to the sum of the powers calculated with each source acting alone. Superposition may be applied to voltage and current first; the resulting total voltage or current must then be used to calculate power.

For dependent sources, source deactivation does not mean deleting the dependent source. If the circuit contains a dependent source, the response due to an independent source may be found by superposition while retaining all dependent-source relationships. This frequently requires nodal or mesh analysis for each excitation.

Illustrative example: source contribution

Consider a 12 V source in series with R1 = 4 Ω feeding a node connected to R2 = 8 Ω and a load RL = 8 Ω in parallel. The parallel combination is 4 Ω, so the total current is 12/(4 + 4) = 1.5 A and the node voltage is 6 V. If a second independent current source is introduced into the same node, the final node voltage is obtained by calculating the voltage due to the voltage source alone and the voltage due to the current source alone, then adding the signed results. The example illustrates the method, but also its limitation: the separate source voltages are intermediate quantities, whereas the physically meaningful power in the load must be calculated from the final 6 V or from the final voltage after both sources are included.

Substitution, Reciprocity, and Compensation

Substitution theorem

The substitution theorem states that any branch of a network may be replaced by another branch, provided the replacement has the same terminal voltage and current as the original branch. The rest of the network then experiences no change in its external behavior. A branch carrying current I and having voltage V may therefore be replaced by an independent voltage source of value V, an independent current source of value I, or a suitable impedance, as long as both terminal variables are preserved.

The theorem is especially useful in incremental analysis and circuit modification. It permits a known branch condition to be replaced by a test source, making the effect of a parameter change easier to examine. It also underlies many equivalent-circuit constructions. The replacement is not arbitrary: preserving only voltage or only current is insufficient unless the network imposes the other variable automatically.

Reciprocity theorem

For a linear bilateral one-port or two-port network containing a single independent excitation, the response observed at one port due to an excitation at another is unchanged when the positions of excitation and response are interchanged, with consistent reference directions. In impedance form for a reciprocal two-port, z12 = z21; in admittance form, y12 = y21. In a simple transfer statement, if a voltage source in branch 1 produces current I2 in branch 2, then the same source placed in branch 2 produces the corresponding current in branch 1, under identical conditions.

Reciprocity requires linearity and bilateral behavior. Unilateral devices, such as idealized active devices with direction-dependent transmission, generally violate it. Dependent sources do not automatically invalidate reciprocity; the complete network must be tested against the relevant reciprocity conditions. Networks containing controlled sources can be reciprocal in special parameter configurations and nonreciprocal in others.

Reciprocity is valuable in electromagnetic measurement, network characterization, antenna studies, and two-port modeling. It can reduce laboratory measurements: a transfer quantity measured in one direction may determine the corresponding quantity in the reverse direction when reciprocity is established.

Compensation theorem

The compensation theorem describes the effect of changing an impedance in a branch of a linear network. If an impedance Z is changed by ΔZ, the change in the current or voltage elsewhere can be found by replacing the changed element with a compensating voltage source whose value is related to the original branch current and the impedance change. With a consistent passive sign convention, the compensating source is commonly represented as

Vc = IΔZ,

with polarity selected to oppose the original branch current in the incremental circuit. The original independent sources are suppressed, and the compensating source is inserted in the affected branch. The resulting incremental response gives the changes in currents and voltages caused by the parameter modification.

The theorem is a circuit-level sensitivity tool. It avoids complete reanalysis when only one impedance changes, although the direction and sign of the compensating source must be handled carefully. For a finite rather than infinitesimal change, the altered branch current used in the compensation formulation must correspond to the stated theorem convention; in practice, direct reanalysis or a sensitivity derivative may be safer when changes are large.

Tellegen’s Theorem: A Conservation Statement

Tellegen’s theorem applies to any lumped network whose branch voltages and currents satisfy Kirchhoff’s voltage and current laws, regardless of whether the elements are linear, bilateral, passive, or time invariant. If vk and ik are the voltage and current of branch k, with all branch currents referenced consistently, then

Σ vk ik = 0.

The algebraic sum of instantaneous powers in all branches is zero. Sources delivering power have negative absorbed power under the passive sign convention, while receiving elements have positive absorbed power. In sinusoidal steady state, the corresponding complex-power relation is

Σ Vk Ik* = 0,

where the asterisk denotes complex conjugation.

Tellegen’s theorem is broader than the other theorems in this chapter because it does not depend on linearity. It is a powerful verification tool: after solving a circuit, the sum of absorbed and delivered powers should be zero, subject to rounding error. It also supports formulations involving energy flow, network simulation, and multiport interconnection.

Tellegen’s theorem should not be confused with a statement that every individual element conserves power. An ideal resistor absorbs real power, while an ideal inductor or capacitor exchanges energy without average real-power absorption in sinusoidal steady state. The theorem concerns the complete network balance.

Thévenin and Norton Equivalents

Thévenin’s theorem

Any linear two-terminal network can be represented, at its terminals, by an ideal voltage source Voc in series with an impedance Zth. The voltage Voc is the terminal voltage when no load is connected. Zth is the impedance seen looking into the network after independent sources have been deactivated, provided no difficulty is caused by dependent sources.

For a resistive network,

Rth = Voc/Isc,

when the short-circuit current Isc is finite and the terminal relation is well defined. The Thévenin equivalent connected to a load ZL gives

IL = Voc/(Zth + ZL),

and

VL = ZL Voc/(Zth + ZL).

This representation is particularly useful when the load changes repeatedly while the source network remains fixed. Once Voc and Zth are known, different load currents, voltages, and powers can be obtained without re-solving the original network.

Norton’s theorem

The Norton equivalent is an ideal current source IN in parallel with an impedance ZN. The Norton current is the short-circuit current at the terminals, and ZN = Zth. The source transformation relation is

IN = Voc/Zth = Isc.

For a load ZL in parallel with ZN,

IL = IN ZN/(ZN + ZL).

Thévenin and Norton forms are mathematically equivalent, but one may be more convenient than the other. A parallel load arrangement often favors Norton form, whereas a series load arrangement often favors Thévenin form.

Finding equivalent impedance with dependent sources

When dependent sources are present, they must remain active while determining Zth. A common method is to suppress independent sources, apply a test voltage Vt or test current It at the terminals, and calculate the resulting response:

Zth = Vt/It.

The test source may be DC, AC, or an s-domain source according to the model. If a dependent source is controlled by a variable affected by the terminal excitation, the test-source method preserves that control relationship. Simply combining visible resistors after suppressing independent sources can produce an incorrect result.

Equivalence and its boundaries

A Thévenin or Norton equivalent preserves terminal behavior for all loads within the model assumptions. It does not preserve internal power distribution, internal stress, noise behavior, thermal characteristics, or nonlinear operation unless those features are explicitly included. For example, an equivalent source resistance may predict the load voltage accurately while failing to represent the maximum current capability of a battery or converter.

Maximum Power Transfer and Design Trade-offs

For a resistive Thévenin source with voltage Vth and resistance Rth, the load receives maximum power when

RL = Rth.

The load power is

PL = Vth² RL/(Rth + RL)².

Differentiation with respect to RL gives the maximum at RL = Rth. At this point, the load voltage is Vth/2, the load current is Vth/(2Rth), and the maximum load power is Vth²/(4Rth). The source resistance also absorbs the same power, so the efficiency is 50 percent. This condition is often undesirable in power-delivery systems but useful in signal and measurement systems where power transfer, rather than efficiency, is the priority.

For AC networks, maximum average power transfer occurs when the load impedance is the complex conjugate of the Thévenin impedance:

ZL = Zth*.

If Zth = Rth + jXth, then ZL = Rth − jXth, assuming Rth is positive. The reactive parts cancel at the source-load interface, and the load receives maximum average real power. In practical systems, exact conjugate matching may be relaxed because matching networks introduce loss, bandwidth limitations, sensitivity to component tolerances, and reduced efficiency away from the design frequency.

Maximum power transfer is not the same as maximum efficiency, maximum voltage, or maximum reliability. In a power distribution system, a load much larger than the source resistance generally gives better efficiency, although the delivered power may be below its theoretical maximum. Engineering design therefore requires a stated objective before applying the theorem.

A Unified Method for Theorem Selection

A useful analytical workflow begins by identifying the desired quantity and the structure of the circuit.

If several independent sources contribute to one voltage or current, superposition is appropriate. If one branch is known by its terminal voltage and current and needs to be conceptually replaced, substitution is appropriate. If a parameter has changed and the incremental response is needed, compensation can reduce the analysis. If a transfer relationship between two locations is being interchanged, reciprocity should be tested. If a complete solution requires a conservation check, Tellegen’s theorem provides an independent verification.

If the load changes while the source network does not, Thévenin or Norton reduction is usually the most efficient approach. If the design question asks for the load value that maximizes delivered power, reduce the source network first and then apply the maximum-power condition. In circuits with dependent sources, use a test source to obtain the driving-point impedance rather than relying on source deactivation alone.

Applied Case Study: Sensor Interface and Selective Loading

Background

Consider an illustrative precision sensor module represented at its output terminals by a Thévenin voltage of 2.0 V and a Thévenin resistance of 6 kΩ. The module is to drive either a data-acquisition input or a calibration instrument. The instrument input resistance is selectable.

Situation and problem

The designer must choose an input resistance that provides adequate signal voltage while considering loading and power transfer. A junior engineer proposes matching the input resistance to 6 kΩ because this maximizes power transfer. The instrumentation engineer argues for an input resistance of at least 60 kΩ to reduce signal attenuation.

Analysis

For an input resistance RL, the load voltage is

VL = 2.0 RL/(6 + RL) V,

where resistances are in kΩ. With RL = 6 kΩ, VL = 1.0 V, and the load receives maximum theoretical power. However, the source resistance also dissipates the same power, and the sensor output is reduced to one-half of its open-circuit value.

With RL = 60 kΩ,

VL = 2.0(60)/(6 + 60) ≈ 1.818 V.

The signal attenuation is only about 9.1 percent. The power delivered to the input is lower than the maximum-power value, but the measurement voltage is substantially higher. If the acquisition system has high input impedance and sufficient noise performance, the 60 kΩ or a still larger input resistance is preferable.

Suppose an AC version of the sensor has Zth = 4 kΩ + j3 kΩ at the operating frequency. A conjugate-matched input would be ZL = 4 kΩ − j3 kΩ. Such a matching network could maximize average real power, but it may be inappropriate for a wideband measurement because the reactive compensation is frequency dependent.

Approach and solution

The correct design choice depends on the objective. For powering a matched RF receiver at a specified frequency, conjugate matching may be justified. For voltage measurement, the priority is high input impedance, low loading, bandwidth, and acceptable noise. The Thévenin model makes this trade-off visible without repeatedly analyzing the internal sensor circuit.

Outcome and learning points

The case demonstrates that maximum power transfer is not a universal design rule. Equivalent circuits enable rapid comparison of alternatives, but the chosen criterion must correspond to the application. In measurement systems, maximizing voltage fidelity often matters more than maximizing power. In addition, an AC equivalent must be interpreted over its frequency range, not treated as a frequency-independent resistor.

Common Errors, Limitations, and Professional Considerations

A frequent error in superposition is replacing a current source by a short circuit or a voltage source by an open circuit. The correct ideal replacements are voltage-source short circuit and current-source open circuit. Another error is suppressing dependent sources. They remain active unless the problem explicitly changes the circuit model.

In Thévenin analysis, open-circuit voltage and short-circuit current must be measured at the same terminal pair and with compatible reference directions. A voltage source connected directly across an ideal current source, or a network with no finite driving-point impedance, may require special treatment. The ratio Voc/Isc is not always a safe substitute for a test-source calculation.

Power calculations require attention to sign conventions and RMS values. In AC analysis, average real power is not obtained by multiplying arbitrary peak magnitudes; the phase relationship must be included. Tellegen verification is meaningful only when branch voltage and current references are consistent.

Equivalent circuits also have practical limits. Components may be nonlinear, temperature dependent, frequency dependent, noisy, or subject to saturation. A small-signal Thévenin model around one operating point may be invalid after a large signal change. Safety and regulatory requirements may prohibit operating a source at its theoretical maximum-power point if it creates excessive heating or violates current limits. Engineers must therefore combine theorem-based analysis with thermal, reliability, electromagnetic-compatibility, and protection assessments.

Digital Tools, Sensitivity, and Emerging Practice

Circuit simulators implement nodal, modified nodal, and sparse-matrix methods that embody the same network principles. A useful professional workflow is to derive a Thévenin or Norton model analytically, verify it using simulation, and then test it over component tolerances and frequency. Parameter sweeps reveal whether a design is robust or whether a nominally optimal match is highly sensitive.

Symbolic tools can calculate transfer functions, driving-point impedances, and derivatives such as ∂VL/∂R. Automatic differentiation and optimization routines can support tolerance analysis and component selection, but they do not replace physical interpretation. A numerical result can be misleading if a source was incorrectly suppressed, a dependent-source control variable was omitted, or an AC quantity was interpreted with the wrong convention.

Modern power-electronic converters and active front ends may present nonlinear or controlled input impedances. In such systems, Thévenin and Norton equivalents are often local small-signal models rather than global descriptions. Stability studies may require frequency-dependent output impedance, passivity analysis, and interaction assessment between interconnected converters. The classical theorems remain foundational, but their application must respect the operating point and frequency range.

Best Practices for Analysis and Design

1. State the terminal pair, reference directions, operating domain, and linearity assumptions before applying a theorem.

2. Use superposition for voltage or current, not by directly adding separately calculated powers.

3. Retain dependent sources and use a test source when finding impedance in an active network.

4. Check equivalent circuits by comparing open-circuit voltage, short-circuit current, and at least one finite-load response.

5. Use Tellegen’s theorem as a power-balance check after a detailed solution.

6. Choose maximum power transfer only when delivered power is the actual design objective.

7. Include frequency, tolerance, thermal, noise, and safety constraints in professional decisions.

8. Treat simulation as verification and exploration, not as a substitute for a correctly formulated model.

Chapter Summary

Network theorems transform circuit analysis from a collection of simultaneous equations into a set of purposeful analytical strategies. Superposition separates the contributions of independent sources in linear networks, while substitution permits a branch to be replaced when its terminal voltage and current are preserved. Reciprocity relates forward and reverse transfer in reciprocal networks, and compensation gives an incremental method for evaluating impedance changes. Tellegen’s theorem provides a general power-balance relation and an important verification tool.

Thévenin and Norton theorems reduce any suitable linear two-terminal network to a source and an impedance, allowing efficient evaluation of changing loads. Dependent sources require special care and commonly necessitate a test-source method for determining equivalent impedance. Maximum power transfer follows from the equivalent circuit, but its condition must not be confused with maximum efficiency or best voltage regulation. In professional practice, theorem selection depends on the design objective, model validity, frequency range, and physical constraints.

Key Terms

Linearity: The property that a network obeys homogeneity and additivity between excitation and response.

Superposition theorem: The theorem stating that a linear-network response equals the algebraic sum of responses due to individual independent sources.

Source deactivation: Replacement of an ideal voltage source by a short circuit or an ideal current source by an open circuit.

Substitution theorem: The principle that a branch may be replaced when its terminal voltage and current remain unchanged.

Reciprocity theorem: The relation stating that excitation and response may be interchanged in a linear bilateral reciprocal network.

Compensation theorem: A method for finding response changes caused by an impedance variation using a compensating source.

Tellegen’s theorem: The theorem that the algebraic sum of branch powers in a network is zero.

Two-terminal network: A network accessible through one pair of terminals.

Thévenin equivalent: An ideal voltage source in series with an equivalent impedance representing a linear two-terminal network.

Norton equivalent: An ideal current source in parallel with an equivalent impedance representing a linear two-terminal network.

Open-circuit voltage: The terminal voltage when no external load is connected.

Short-circuit current: The terminal current when the network terminals are directly shorted.

Driving-point impedance: The ratio of voltage to current measured at the same pair of terminals.

Transfer impedance: The ratio relating a voltage at one location to a current at another location.

Dependent source: A source whose value is controlled by a voltage or current elsewhere in the circuit.

Test source: An applied voltage or current used to determine the impedance of an active network.

Maximum power transfer: The condition under which a specified load receives the greatest average power from a source network.

Conjugate matching: AC matching in which the load impedance is the complex conjugate of the source Thévenin impedance.

Terminal equivalence: Equality of external voltage-current behavior at specified terminals under stated assumptions.

Small-signal model: A linearized model describing behavior near a selected operating point.

Review Questions

Short Answer Questions

1. What conditions must hold for superposition to be applicable?

2. Why are dependent sources retained during source deactivation?

3. State the terminal equivalence represented by a Thévenin circuit.

4. What does Tellegen’s theorem express in terms of branch powers?

5. Distinguish maximum power transfer from maximum efficiency.

Long Answer Questions

1. Explain the superposition theorem, including source deactivation rules, treatment of dependent sources, and the reason it cannot be applied directly to power.

2. Derive the relationship between Thévenin and Norton equivalents and discuss how each form assists load analysis.

3. Explain reciprocity and compensation theorems, including their assumptions and engineering applications.

4. Derive the maximum-power-transfer condition for a resistive Thévenin network and discuss its efficiency implication.

5. Describe how a test source is used to determine equivalent impedance when dependent sources are present.

Analytical / Application Questions

1. A source network has Vth = 10 V and Rth = 5 Ω. Determine the load resistance for maximum power, the maximum load power, and the load voltage at that condition.

2. A sensor has a Thévenin equivalent of 1.5 V and 10 kΩ. Compare the output voltage for 10 kΩ and 100 kΩ input resistances, and recommend one for voltage measurement with justification.

3. An AC network has Zth = 8 + j6 Ω. Determine the conjugate-matched load and explain two practical reasons why a designer might deliberately choose a different load.

22. Multiple Choice Questions with Answers

Question 1

When applying superposition in a linear circuit, an ideal independent voltage source that is not being considered is replaced by:

A. An open circuit

B. A short circuit

C. A resistor of infinite value

D. A dependent voltage source

Correct Answer: B. A short circuit

Question 2

Which statement about superposition and power is correct?

A. Power can always be added from separate source cases

B. Power is linear in voltage and current

C. Voltage or current may be superposed first, then total power calculated

D. Superposition applies only to resistive power

Correct Answer: C. Voltage or current may be superposed first, then total power calculated

Question 3

A dependent source is retained during source deactivation because it:

A. Has zero internal resistance

B. Is controlled by circuit variables

C. Is always an independent source

D. Must be replaced by an open circuit

Correct Answer: B. Is controlled by circuit variables

Question 4

For a linear two-terminal network, the Thévenin impedance can be found in the presence of dependent sources by:

A. Deleting all sources

B. Using only the visible resistors

C. Applying a test source and calculating the terminal voltage-current ratio

D. Setting the open-circuit voltage equal to zero

Correct Answer: C. Applying a test source and calculating the terminal voltage-current ratio

Question 5

Tellegen’s theorem states that:

A. All branch currents are equal

B. The sum of branch powers is algebraically zero

C. Every network must be reciprocal

D. Maximum power occurs at equal voltage and current

Correct Answer: B. The sum of branch powers is algebraically zero

Question 6

A resistive Thévenin source has Rth = 12 Ω. The load receives maximum power when its resistance is:

A. 3 Ω

B. 6 Ω

C. 12 Ω

D. 24 Ω

Correct Answer: C. 12 Ω

Question 7

At the maximum-power-transfer condition for a purely resistive source, the efficiency from source equivalent to load is:

A. 25 percent

B. 50 percent

C. 75 percent

D. 100 percent

Correct Answer: B. 50 percent

Question 8

If Zth = 4 + j5 Ω, the load for maximum average AC power is:

A. 4 + j5 Ω

B. −4 + j5 Ω

C. 4 − j5 Ω

D. 5 − j4 Ω

Correct Answer: C. 4 − j5 Ω

Question 9

The Norton equivalent of a linear two-terminal network consists of:

A. A voltage source in series with an impedance

B. A current source in parallel with an impedance

C. A current source in series with an impedance

D. A voltage source in parallel with an impedance

Correct Answer: B. A current source in parallel with an impedance

Question 10

A high input resistance is generally preferred when a Thévenin source is used for voltage measurement because it:

A. Maximizes source power dissipation

B. Reduces loading and preserves terminal voltage

C. Always produces conjugate matching

D. Makes the source resistance zero

Correct Answer: B. Reduces loading and preserves terminal voltage