DC Circuits and Network Theorems

Introduction

Direct-current resistive circuits provide the experimental foundation for electrical engineering. They connect physical measurements—voltage, current and resistance—to mathematical models based on Ohm’s law, Kirchhoff’s laws and conservation of energy and charge. Network theorems then reduce complicated linear circuits to simpler equivalents that support design, testing and fault diagnosis.

This chapter integrates analytical methods with laboratory verification. Unless stated otherwise, circuits are assumed to be lumped, time-invariant and operating at steady state. Components are treated initially as ideal; practical sections account for resistor tolerance, source resistance, lead resistance, heating and meter loading.

Learning Objectives

After completing this chapter, students should be able to:

• verify Ohm’s law experimentally and identify non-ohmic behavior;
• apply Kirchhoff’s current law (KCL) and voltage law (KVL);
• formulate and solve nodal and mesh equations;
• verify superposition in linear resistive networks;
• determine Thévenin and Norton equivalents analytically and experimentally;
• establish the maximum-power-transfer condition;
• connect voltmeters, ammeters and ohmmeters safely;
• quantify discrepancy and measurement uncertainty; and
• relate laboratory results to engineering design and troubleshooting.

1. Electrical Variables and Sign Conventions

Voltage is electric potential difference, measured in volts (V). Current is the rate of charge flow, measured in amperes (A), and resistance is measured in ohms (Ω).

Here, I is current in amperes, q is charge in coulombs and t is time in seconds.

Under the passive sign convention, current enters the terminal marked positive. The absorbed power is then

where p is power in watts, v is voltage in volts and i is current in amperes. A positive value indicates absorbed power; a negative value indicates delivered power.

For a complete DC network, conservation of energy requires

provided all element powers use one consistent sign convention.

2. Ohm’s Law

For an ohmic conductor whose physical conditions remain approximately constant,

where R is resistance in ohms. Equivalent forms are i=v/R and R=v/i. Dimensional consistency follows because 1\ \Omega=1\ V/A.

Ohm’s law is a constitutive relation, not a universal description of every component. Diodes, lamps, thermistors and many semiconductor devices have nonlinear voltage–current characteristics. Even a nominally ohmic resistor may change resistance when self-heating becomes significant.

For an ideal resistor, substitution into the power equation gives

These forms apply to a resistor under DC steady-state conditions. A resistor’s rated power must exceed its expected dissipation, preferably with an engineering margin.

Series and parallel resistance

Resistors in series carry the same current:

Resistors in parallel share the same voltage:

For exactly two parallel resistors,

The product-over-sum expression is a two-resistor special case, not the general parallel formula.

Experimental verification

For several source settings, measure the resistor voltage and current. A graph of V against I should be approximately linear, with slope R. Linear regression is preferable to calculating resistance from only one point because it uses all observations and reveals offset errors.

3. Kirchhoff’s Laws

3.1 Kirchhoff’s Current Law

KCL follows from conservation of charge. At any lumped circuit node,

Currents entering may be assigned positive and currents leaving negative, or vice versa. The convention must remain consistent. Equivalently, the sum entering equals the sum leaving.

KCL assumes that no unmodeled net charge accumulates at the node. It remains valid in dynamic circuits when capacitor currents and other relevant branch currents are included.

3.2 Kirchhoff’s Voltage Law

KVL states that the algebraic sum of voltages around a closed path is zero:

KVL expresses energy conservation. In ordinary low-frequency, physically compact laboratory networks, induced electromagnetic effects are negligible. If a time-varying magnetic flux links a loop, the induced electromotive force must be included explicitly.

Sign discipline

When traversing a loop:

• crossing an element from − to + is a voltage rise;
• crossing from + to − is a voltage drop;
• resistor polarity follows the selected current direction under the passive sign convention.

Choosing an incorrect reference direction does not invalidate an analysis. A negative result simply means the actual direction is opposite to the assumed direction.

4. Nodal Analysis

Nodal analysis uses node voltages as unknowns and applies KCL. It is especially efficient when a circuit has fewer essential nodes than meshes.

General method

1. Select a reference node and assign it 0 V.
2. Label every remaining essential-node voltage.
3. Express resistor currents using voltage differences divided by resistance.
4. write one KCL equation for each unknown node.
5. Solve the simultaneous equations and verify current balance.

For a node V_a connected through resistors to nodes V₁,V₂,\ldots,V_n, with net current I_in injected into the node,

This form assumes all branch currents on the left leave the node.

Worked Example: One unknown node

A node V is connected to a 12 V node through 2 kΩ, to ground through 4 kΩ and to a 1 mA source injecting current into the node. KCL gives

Multiplication by 4000 Ω gives

Therefore,

The current toward the 12 V node is (9.333-12)/2\,000=-1.333\ mA, meaning 1.333 mA actually enters from that branch. The ground current is 9.333/4\,000=2.333\ mA. Thus 1.333 mA plus the 1 mA source current equals the 2.333 mA leaving toward ground, verifying KCL.

A voltage source between two nonreference nodes creates a supernode. KCL is written around the supernode, together with the voltage-source constraint.

5. Mesh Analysis

Mesh analysis assigns a current to each independent mesh of a planar circuit and applies KVL. It is efficient when the number of meshes is small.

For two clockwise mesh currents I₁ and I₂ sharing resistance R_s, the voltage drop in the shared resistor, in the direction of mesh 1, is R_s(I₁-I₂). A typical equation pair is

The algebraic source values depend on source polarity relative to the selected loop directions.

Worked Example: Two meshes

Suppose the left mesh contains 2 kΩ, the right mesh contains 3 kΩ and the shared branch is 1 kΩ. A 10 V source is a rise in the clockwise direction of mesh 1. A 4 V source is a drop in the clockwise direction of mesh 2. With currents expressed in mA and resistances in kΩ,

Solving gives

The negative value means the actual right-mesh current is 0.182 mA counterclockwise. The shared-branch current in the mesh-1 direction is

Substitution into both original equations verifies the solution.

An ideal current source shared by two meshes requires a supermesh: write KVL around the outer perimeter and add the current-source constraint.

6. Superposition Theorem

In a linear network containing several independent sources, the voltage or current in any element equals the algebraic sum of the responses produced by each independent source acting alone.

When one source acts alone:

• replace every other ideal independent voltage source by a short circuit;
• replace every other ideal independent current source by an open circuit;
• retain dependent sources because their values depend on circuit variables.

For practical nonideal sources, deactivation leaves their internal resistance in the circuit.

Superposition applies directly to voltages and currents, not power, because power is nonlinear in voltage or current. Component powers must be calculated from the final combined voltage and current.

Worked Example

A node is connected to 10 V through 2 kΩ, to 5 V through 1 kΩ and to ground through 2 kΩ. Direct nodal analysis gives

With only the 10 V source active, the 5 V source becomes a short circuit:

With only the 5 V source active, the 10 V source becomes a short circuit:

Thus V=V₁₀+V₅=5.0\ V, verifying superposition.

7. Thévenin and Norton Theorems

Any linear two-terminal network of sources and resistances can be represented at its port by either:

• a Thévenin voltage source V_th in series with R_th, or
• a Norton current source I_N in parallel with R_N.

For an ordinary resistive network,

and, when R_th\ne0,

Figure: Equivalent Thévenin and Norton representations connected to the same load.

Finding a Thévenin equivalent

1. Remove the load while retaining the original source network.
2. Calculate or measure the open-circuit port voltage: V_th=V_oc.
3. Determine R_th by one valid method:

• deactivate all independent sources and find the resistance seen at the port;
• use R_th=V_oc/I_sc when a safe short-circuit current can be determined; or
• apply a test source and calculate R_th=V_test/I_test, especially when dependent sources are present.

Correct source deactivation is essential. An ideal independent voltage source is replaced by a short circuit, not merely removed or left open. An ideal independent current source is replaced by an open circuit. Dependent sources remain active.

An ohmmeter must never be connected to an energized circuit. In laboratory work, switch off and isolate the supply, then install a jumper across the terminals previously occupied by an ideal voltage source—or across the output terminals of a zero-volt programmable source, if approved by the equipment procedure. This reproduces the required short-circuit deactivation while keeping the surrounding resistor branches connected.

Worked Example

A 12 V ideal source feeds a 3 kΩ resistor connected to the output node; a 6 kΩ resistor connects the output node to ground.

With the load removed, the voltage divider gives

After deactivating the 12 V source by replacing it with a short circuit, the 3 kΩ and 6 kΩ resistors both connect from the output to ground:

If the source were incorrectly removed and left open, the 3 kΩ branch would be disconnected and an erroneous 6 kΩ result would be obtained.

The Norton current is

For a load R_L, Thévenin analysis gives

8. Maximum-Power-Transfer Theorem

For a Thévenin source driving a positive resistive load,

Differentiating with respect to R_L gives

The stationary point occurs at

For R_th>0, this point is the maximum. The maximum load power is

For the preceding example,

At resistance matching, equal power is dissipated in R_th and R_L, so the efficiency of this simple equivalent is 50%. Maximum power is therefore not maximum efficiency. Power systems normally seek low source resistance and high efficiency, whereas matching is important in communication, sensing and impedance-interface applications.

9. Laboratory Programme

Equipment

Typical equipment includes a current-limited DC supply, digital multimeter (DMM), breadboard, resistors, patch leads, switching links and simulation software. Confirm resistor values and power ratings before energizing the circuit.

Activity 1: Ohm’s law

1. Measure a resistor with the circuit unpowered.
2. Connect it to an adjustable DC source through an ammeter; connect a voltmeter in parallel.
3. Set a conservative current limit.
4. Increase voltage in steps and record V and I.
5. Plot V versus I and estimate the slope.
6. Compare the slope resistance with the ohmmeter value and nominal tolerance.

Activity 2: KCL and KVL

Construct a branched network. Measure every branch current at one node and calculate the signed KCL residual:

Measure all source and resistor voltages around one closed loop and calculate

Small nonzero residuals arise from meter uncertainty, component tolerance, wiring resistance and source variation.

Activity 3: Nodal and mesh verification

Predict node voltages using nodal analysis and branch currents using mesh analysis. Construct the circuit, measure the same quantities and compare prediction with observation. Use one set of actual measured resistor values for a more meaningful comparison.

Activity 4: Superposition

1. Measure a selected branch current or voltage with all independent sources active.
2. Activate one independent source at a time.
3. Deactivate other ideal voltage sources with shorts and ideal current sources with opens.
4. Preserve algebraic signs when adding individual responses.
5. Compare their sum with the all-sources measurement.

Do not add separately calculated powers.

Activity 5: Thévenin and Norton verification

1. Remove the load and energize the original network.
2. Measure the open-circuit voltage V_oc=V_th.
3. Switch off and isolate all supplies. Confirm zero voltage at the port.
4. Deactivate each ideal voltage source by replacing it with a short circuit; do not simply disconnect it and leave an open circuit. Deactivate each ideal current source with an open circuit.
5. Measure the resistance looking into the port to obtain R_th.
6. Restore the original network before re-energizing it.
7. Calculate I_N=V_th/R_th. Direct short-circuit measurement is permitted only when current and power remain within equipment limits.
8. Connect several load resistors and compare their measured port behavior with both equivalents.

Activity 6: Maximum power transfer

Connect a set of load resistors spanning values below and above R_th. For each load, measure V_L and calculate

Plot P_L against R_L. The peak should occur near R_L=R_th, within resistor and measurement tolerances.

10. Measurement Quality and Error Analysis

A real voltmeter has finite input resistance R_V and therefore loads the measured circuit. A real ammeter has nonzero burden resistance R_A, which changes branch current. Resistance measurements may also include lead and contact resistance.

Percentage discrepancy can be reported as

This expression is unsuitable when the predicted value is zero. In that case, report absolute error and instrument uncertainty.

Useful uncertainty sources include:

A good laboratory report includes the circuit, sign conventions, raw data, calculations, uncertainty, discrepancies and a technically justified conclusion. Agreement should not be claimed merely because two rounded values appear similar.

11. Illustrative Case: Sensor Interface Design

Background

This fictional case concerns a laboratory sensor represented by an 8 V Thévenin voltage and 2 kΩ output resistance. A data-acquisition input can be configured with different load resistances.

Problem

The engineer must choose between maximum received power and accurate voltage measurement.

Analysis

Maximum power occurs at R_L=2\ k\Omega, producing

and

However, if the purpose is to measure the source’s open-circuit voltage accurately, a much higher input resistance is preferable. With R_L=1\ M\Omega,

The loading error relative to 8 V is approximately 0.20%.

Approach and Outcome

The engineer selects the 1 MΩ input for voltage sensing. Resistance matching is rejected because it would halve the measured voltage and waste half of the generated power in the sensor’s equivalent resistance.

Learning Points

• The optimal load depends on the design objective.
• Maximum power transfer does not imply maximum voltage accuracy or efficiency.
• Thévenin modeling makes loading effects immediately visible.

12. Safety, Professional Practice and Contemporary Tools

Before changing wiring, de-energize the circuit. Never place an ammeter directly across a voltage source. Never use the resistance mode of a DMM on an energized network. Apply current limits and verify resistor power ratings. Electrolytic components, if introduced in later experiments, require correct polarity.

Accurate records are an ethical and professional obligation. Data must not be altered to force agreement with theory. Unexpected results should prompt checks for wiring errors, damaged components, incorrect assumptions or instrument limitations.

Circuit simulators, programmable supplies, source-measure units and automated data acquisition can accelerate testing. AI-based tools may help inspect datasets or suggest fault hypotheses, but generated equations, connection instructions and component ratings require independent engineering verification. Simulation complements rather than replaces physical measurement because models may omit parasitic effects, tolerances and equipment limits.

Chapter Summary

Ohm’s law relates voltage, current and resistance for ohmic components under stable physical conditions. KCL and KVL express conservation of charge and energy and form the basis of nodal and mesh analysis. Superposition separates source contributions in linear circuits but cannot be applied directly to power.

Thévenin and Norton theorems replace a linear two-terminal network with equivalent source-resistance models. To determine equivalent resistance, ideal voltage sources must be replaced by short circuits and ideal current sources by open circuits; dependent sources remain active. Maximum power is delivered to a resistive load when its resistance equals the positive Thévenin resistance, although this operating point is only 50% efficient for the simple equivalent. Reliable laboratory verification requires safe meter connection, explicit sign conventions, suitable uncertainty analysis and honest reporting.

Key Terms

Ohm’s law: The linear relation v=iR for an ohmic element under stable conditions.

Node: A set of points connected by ideal conductors and sharing one potential.

Branch: A circuit path containing one or more elements and carrying one current.

Mesh: A loop containing no other loop in a planar circuit.

KCL: The law stating that the algebraic sum of currents at a node is zero.

KVL: The law stating that the algebraic sum of voltages around a closed path is zero.

Passive sign convention: A convention in which positive current enters an element’s positive-voltage terminal.

Nodal analysis: A method using node voltages and KCL.

Mesh analysis: A method using mesh currents and KVL in planar networks.

Supernode: A nodal-analysis region formed by a voltage source between nonreference nodes.

Supermesh: A mesh-analysis region formed when a current source lies between meshes.

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

Thévenin equivalent: A port model consisting of a voltage source in series with resistance.

Norton equivalent: A port model consisting of a current source in parallel with resistance.

Open-circuit voltage: Port voltage when load current is zero.

Short-circuit current: Port current when terminal voltage is zero.

Source deactivation: Replacement of independent voltage sources by shorts and independent current sources by opens.

Maximum power transfer: The condition that maximizes load power; for positive DC resistances, R_L=R_th.

Meter loading: Modification of a circuit’s behavior by the measuring instrument.

Review Questions

Short Answer Questions

1. State Ohm’s law and its principal physical limitation.
2. Explain the passive sign convention.
3. How is an ideal independent voltage source deactivated?
4. Why can superposition not be applied directly to power?
5. What does a negative mesh-current result mean?

Long Answer Questions

1. Derive the nodal-analysis equations for a general three-node resistive network.
2. Explain KCL and KVL in terms of conservation laws and discuss their practical assumptions.
3. Describe analytical and experimental methods for finding a Thévenin equivalent.
4. Derive the maximum-power-transfer condition for a positive resistive Thévenin source.
5. Discuss meter loading, resistor tolerance, self-heating and supply resistance as sources of laboratory discrepancy.

Analytical / Application Questions

1. A 15 V source supplies 2 kΩ in series with a node connected to 3 kΩ to ground. Find the Thévenin equivalent at the node-ground port and determine the load for maximum power.
2. Design an experiment to verify superposition in a two-source network, including source-deactivation procedures and signed data processing.
3. A sensor has V_th=5\ V and R_th=10\ k\Omega. Compare the load voltage and power for 10 kΩ and 1 MΩ inputs, then recommend a load for voltage measurement.

Multiple Choice Questions with Answers

Question 1

A 12 V supply is connected across a 3 kΩ resistor. Assuming an ideal ohmic resistor, what current flows?

A. 0.25 mA

B. 4 mA

C. 36 mA

D. 4 A

Correct Answer: B. 4 mA

Question 2

At a node, 2 mA and 3 mA enter while one branch carries 1 mA away. What current must leave through the remaining branch?

A. 2 mA

B. 4 mA

C. 5 mA

D. 6 mA

Correct Answer: B. 4 mA

Question 3

Which equation correctly represents KVL for a closed loop?

A. The algebraic sum of loop voltages is zero.

B. The algebraic sum of branch resistances is zero.

C. Every branch current is equal.

D. Every element absorbs equal power.

Correct Answer: A. The algebraic sum of loop voltages is zero.

Question 4

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

A. It is replaced by a short circuit.

B. Its current is doubled.

C. It is replaced by an open circuit.

D. It is retained unchanged.

Correct Answer: C. It is replaced by an open circuit.

Question 5

Why should powers due to individual sources not be added using superposition?

A. Power is independent of resistance.

B. Power is nonlinear in voltage and current.

C. KCL prohibits adding powers.

D. Independent sources cannot deliver power.

Correct Answer: B. Power is nonlinear in voltage and current.

Question 6

An ideal voltage source is deactivated while determining R_th. What must replace it?

A. An open circuit

B. A short circuit

C. A current source of equal magnitude

D. A resistor equal to the load

Correct Answer: B. A short circuit

Question 7

A network has V_th=10\ V and R_th=5\ k\Omega. What is its Norton current?

A. 0.5 mA

B. 2 mA

C. 5 mA

D. 50 mA

Correct Answer: B. 2 mA

Question 8

For a positive resistive Thévenin network with R_th=4\ k\Omega, which load receives maximum power?

A. 1 kΩ

B. 2 kΩ

C. 4 kΩ

D. 8 kΩ

Correct Answer: C. 4 kΩ

Question 9

A Thévenin source has V_th=8\ V and R_th=2\ k\Omega. What is the maximum load power?

A. 4 mW

B. 8 mW

C. 16 mW

D. 32 mW

Correct Answer: B. 8 mW

Question 10

Which procedure is safe and technically correct before measuring a circuit’s resistance with a DMM?

A. Keep the circuit energized and use the highest resistance range.

B. Connect the DMM in current mode across the circuit.

C. De-energize and isolate the circuit, confirm zero voltage, then measure.

D. Short the DMM probes across the energized source.

Correct Answer: C. De-energize and isolate the circuit, confirm zero voltage, then measure.