DC Circuit Analysis

Introduction and Context

Direct-current (DC) circuit analysis is the foundation of electrical engineering. It provides methods for predicting voltages, currents, power, and energy in circuits supplied by batteries, DC generators, rectifiers, and regulated power supplies. Mechanical engineers encounter these circuits in sensors, actuators, control panels, electric vehicles, instrumentation, and mechatronic systems.

A DC circuit is an interconnected system of electrical sources and components in which steady-state quantities are constant with time. The central analytical task is to determine unknown branch currents and node voltages from known source values, resistances, and circuit connections.

Learning Objectives

After studying this chapter, you should be able to:

1. Apply Ohm’s law and Kirchhoff’s laws to resistive circuits.
2. Reduce series and parallel resistor networks.
3. Use voltage-division and current-division principles.
4. Formulate and solve nodal and mesh equations.
5. Transform sources between equivalent voltage and current forms.
6. Apply superposition, Thévenin, Norton, and maximum-power-transfer theorems.
7. Check results using power balance, units, polarity, and limiting cases.

1. Fundamental Electrical Quantities

Electric current is the rate of charge flow. If charge q moves through a cross-section during time t, then

where I is current in amperes (A), q is charge in coulombs (C), and t is time in seconds (s). Conventional current is taken as the motion of positive charge, from higher to lower potential through a passive resistor. Electron flow in metals is opposite to conventional current.

Voltage is energy transferred per unit charge:

where V is voltage in volts (V) and w is energy in joules (J). A voltage polarity marked + and − means that a positive charge moving from − to + gains energy.

Resistance measures opposition to current. For a uniform conductor,

where R is resistance (Ω), ρ is resistivity (Ω·m), ℓ is conductor length (m), and A is cross-sectional area (m²). This relation assumes uniform material, constant temperature, and approximately uniform current density.

Ohm’s Law

For an ohmic resistor operating at constant physical conditions,

where V is the voltage across the resistor, I is its current, and R is its resistance. The law is a component relationship, not a universal law for devices such as diodes or lamps whose resistance varies with operating point.

The instantaneous power absorbed by a two-terminal element is

Using the passive sign convention, current enters the terminal marked positive. For a resistor,

P is in watts (W). A positive value indicates absorption; a negative value indicates delivery under the selected reference directions.

2. Circuit Topology and Kirchhoff’s Laws

A node is a point at which two or more element terminals are connected. A branch is a single element or series path between nodes. A loop is a closed path. A mesh is a loop that contains no other loop within it.

Kirchhoff’s Current Law

Kirchhoff’s current law (KCL) states that algebraic current at a node sums to zero:

It follows from conservation of charge. Currents entering may be equated to currents leaving. KCL applies regardless of the circuit’s linearity.

Kirchhoff’s Voltage Law

Kirchhoff’s voltage law (KVL) states that the algebraic sum of voltage changes around any closed path is zero:

It follows from conservation of energy and assumes a lumped circuit model in which electromagnetic coupling through the loop is represented by circuit elements. Polarity must be observed consistently when traversing the loop.

3. Series and Parallel Circuits

Elements are in series when they carry the same current and share an exclusive connecting node. For resistors in series,

The equivalent resistance is greater than every individual resistance. The total voltage is divided in proportion to resistance.

Resistors are in parallel when they connect across the same two nodes and therefore have the same voltage. Their equivalent resistance is

For two resistors,

The equivalent resistance of a parallel group is less than the smallest branch resistance.

Voltage Division

For series resistors connected across a source V,

This formula assumes the output is measured across R_k without an external load, or that the load has already been included in the equivalent resistance. A connected load changes the division unless its effect is negligible.

Current Division

For two parallel resistors carrying total current I,

The current in one branch is inversely related to that branch’s resistance. More generally, branch current equals total current multiplied by that branch’s conductance divided by total conductance:

where conductance G = 1/R, measured in siemens (S).

Worked Example 1: Series–Parallel Reduction

A 24 V source supplies R₁ = 4 Ω in series with a parallel combination of R₂ = 6 Ω and R₃ = 3 Ω. First,

Thus R_eq = 4 + 2 = 6 Ω, and source current is I = 24/6 = 4 A. The voltage across R₁ is 4 × 4 = 16 V, leaving 8 V across the parallel group. Therefore,

Their sum is 4 A, satisfying KCL. Source power is 24 × 4 = 96 W. Resistor powers are 64 W, 10.67 W, and 21.33 W; their sum is also 96 W.

4. Systematic Circuit Analysis

Nodal Analysis

Nodal analysis uses node voltages as unknowns and KCL as the governing equation. Select a reference (ground) node and assign voltages to the remaining nodes relative to it. For a resistor connecting nodes a and b, current leaving node a toward b is

At each nonreference node, write the algebraic sum of currents equal to zero. Nodal analysis is particularly efficient when a circuit has few essential nodes and many current sources.

If a voltage source connects two nonreference nodes, the pair forms a supernode. Write KCL around the supernode and add the voltage constraint imposed by the source.

Worked Example 2: Nodal Analysis

A 2 A current source injects current into node V₁. Node V₁ is connected to ground through 4 Ω and to node V₂ through 2 Ω. Node V₂ is connected to ground through 6 Ω. KCL gives

Multiplying by 12 gives 3V₁ − 6V₂ = 24 and −6V₁ + 8V₂ = 0. From the second equation V₂ = 0.75V₁. Substitution gives 3V₁ − 4.5V₁ = 24, so V₁ = −16 V and V₂ = −12 V under the selected current directions. The negative values are valid: they indicate that the actual node polarities are opposite to the assumed references. The current entering from the source and the resistor currents still satisfy KCL when signs are retained consistently.

Mesh-Current Analysis

Mesh analysis assigns a circulating current to each mesh and applies KVL. For a resistor shared by meshes i and j, its voltage drop in mesh i is R(I_i − I_j). Mesh analysis is efficient for planar circuits with few meshes and many voltage sources.

For a two-mesh network, equations often take the form

The signs depend on source polarities and chosen mesh directions. A current source shared by two meshes produces a supermesh: omit the source branch from the KVL path, write KVL around the outer boundary, and add the current constraint.

5. Source Transformation

A practical voltage source consists of an ideal voltage source V_s in series with resistance R_s. It is equivalent at its external terminals to a practical current source I_s in parallel with the same resistance, where

Conversely,

The polarity and arrow direction must correspond: the current source arrow points from the negative to the positive terminal of the equivalent voltage source. Source transformation preserves terminal voltage-current behavior, not necessarily internal power distribution. It cannot be applied indiscriminately to ideal sources: an ideal voltage source has R_s = 0, while an ideal current source has infinite parallel resistance.

6. Superposition Theorem

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

To deactivate an independent source:

• replace an ideal voltage source with a short circuit;
• replace an ideal current source with an open circuit.

Dependent sources remain active because their values depend on circuit variables. Superposition applies to voltage and current, but not directly to power, because power is nonlinear: the power associated with a total current is not generally the sum of powers associated with individual currents.

7. Thévenin and Norton Theorems

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

• a Thévenin equivalent: a voltage source V_th in series with resistance R_th;
• a Norton equivalent: a current source I_N in parallel with resistance R_N.

The parameters are related by

To find a Thévenin equivalent, remove the load, calculate the open-circuit terminal voltage V_oc, and determine the resistance seen looking into the network with independent sources deactivated. Thus R_th = R_oc for a network containing only resistors and independent sources. Alternatively, calculate short-circuit current I_sc and use

provided the ratio is meaningful and the network is linear. With dependent sources, retain them and use a test source if necessary.

For a load R_L connected to a Thévenin equivalent,

and

These formulas assume a resistive DC load and consistent polarity.

8. Maximum Power Transfer

The load receives maximum power from a linear resistive network when

For the Thévenin representation, load power is

At R_L = R_th,

The efficiency at maximum power transfer is 50%, because equal power is dissipated in R_L and R_th. Consequently, maximum power transfer is not the same as maximum efficiency. In practical power systems, loads are often designed for high efficiency rather than this condition. For AC networks, the corresponding condition involves complex-conjugate impedance matching; that result is outside the purely resistive scope of this chapter.

9. Applied Case: DC Sensor Interface for a Mechanical System

Background

Consider an illustrative case in which a mechanical-position sensor is represented by a 12 V source with 2 Ω internal resistance. A controller input has an effective resistance of 10 Ω. Engineers must estimate input voltage, current, and power before selecting wiring and protection.

Analysis

The sensor and controller form a Thévenin circuit: V_th = 12 V, R_th = 2 Ω, and R_L = 10 Ω. Therefore,

The controller receives P_L = I²R_L = 10 W. Internal sensor dissipation is I²R_th = 2 W, and source power is 12 W, confirming power balance.

Suppose a long cable adds 3 Ω in series. The effective source resistance becomes 5 Ω, so current becomes 12/(5 + 10) = 0.8 A and controller voltage becomes 8 V. The result may fall below the controller’s required operating voltage. A designer could increase source voltage, use a lower-resistance cable, place signal conditioning near the sensor, or specify an input with greater resistance.

Learning Points

The Thévenin model separates source behavior from load behavior, makes cable effects visible, and supports rapid what-if analysis. It also demonstrates why a nominal sensor voltage is insufficient: internal and wiring resistance determine the actual delivered voltage.

10. Verification, Limitations, and Professional Practice

Reliable circuit analysis includes independent checks. Confirm that current directions and voltage polarities are stated; use SI units consistently; verify KCL at nodes and KVL around loops; compare total source power with total absorbed power; and test limiting cases such as R_L approaching zero or infinity.

The ideal models used here have limits. Real batteries have state-of-charge-, temperature-, and frequency-dependent behavior. Resistors have tolerances and temperature coefficients. Wires have resistance, connectors create contact resistance, and sources may have current or voltage limits. At high frequencies, parasitic capacitance and inductance invalidate a purely resistive DC model. Measurement instruments also load circuits: a voltmeter should have high input resistance, while an ammeter should have very low resistance and must be connected in series.

Safety is a professional requirement. De-energize circuits before changing connections, use appropriately rated meters and fuses, prevent short circuits, and respect battery fault currents. In industrial practice, analysis should be documented with reference polarity, assumptions, tolerance margins, and verification results. Computer-aided circuit solvers are useful, but engineers must still inspect the topology, model validity, units, and plausibility of computed results.

Chapter Summary

DC circuit analysis predicts voltages, currents, and power using component laws and conservation principles. Ohm’s law describes an ohmic resistor, while KCL expresses charge conservation and KVL expresses energy conservation. Series and parallel reduction, voltage division, and current division simplify common networks. Nodal analysis uses KCL and node voltages; mesh analysis uses KVL and loop currents. Source transformation replaces a practical voltage source with an equivalent current source. Superposition separates linear responses from independent sources. Thévenin and Norton theorems reduce any linear two-terminal network to simple equivalents, enabling efficient load analysis. Maximum power is delivered when load resistance equals Thévenin resistance, although this condition gives only 50% efficiency. Every solution should be checked through units, polarity, conservation, power balance, and realistic component limitations.

Key Terms

Current: Rate of charge flow, measured in amperes.

Voltage: Energy transferred per unit charge, measured in volts.

Resistance: Opposition to current, measured in ohms.

Conductance: Reciprocal of resistance, measured in siemens.

Ohm’s law: Relationship V = IR for an ohmic element.

Node: Electrically common connection point in a circuit.

Branch: Circuit element or path between nodes.

Loop: Any closed path in a circuit.

Mesh: A loop containing no internal loop.

KCL: Kirchhoff’s law requiring algebraic currents at a node to sum to zero.

KVL: Kirchhoff’s law requiring algebraic voltages around a closed path to sum to zero.

Voltage division: Distribution of series voltage in proportion to resistance.

Current division: Distribution of parallel current in proportion to conductance.

Nodal analysis: Method based on node voltages and KCL.

Mesh analysis: Method based on mesh currents and KVL.

Superposition: Addition of individual linear responses from independent sources.

Thévenin equivalent: Voltage source in series with equivalent resistance.

Norton equivalent: Current source in parallel with equivalent resistance.

Maximum power transfer: Condition R_L = R_th for a resistive DC network.

Passive sign convention: Reference convention in which current enters the positive-voltage terminal.

Review Questions

5 Short Answer Questions

1. State Ohm’s law and identify its applicability condition.
2. What is the difference between a node, branch, loop, and mesh?
3. State the source-deactivation rules used in superposition.
4. How are Thévenin and Norton equivalents related?
5. Why does maximum power transfer not imply maximum efficiency?

5 Long Answer Questions

1. Explain KCL and KVL from the principles of charge and energy conservation, respectively.
2. Derive the voltage-divider relation for a series network and discuss the effect of a connected load.
3. Describe a complete nodal-analysis procedure, including treatment of a supernode.
4. Explain how to obtain a Thévenin equivalent when dependent sources are present.
5. Compare nodal, mesh, superposition, Thévenin, and Norton methods in terms of appropriate use.

3 Analytical / Application Questions

1. A 30 V source feeds a 5 Ω resistor in series with a parallel combination of 10 Ω and 15 Ω. Determine total current, branch currents, and power in each resistor.
2. A linear network has V_th = 18 V and R_th = 6 Ω. Determine load current and load power for R_L = 3 Ω, 6 Ω, and 12 Ω. Identify which load receives maximum power.
3. A sensor with a 24 V Thévenin voltage and 4 Ω internal resistance is connected through a 6 Ω cable resistance to a 20 Ω controller input. Determine controller voltage and power, and explain one design change that would improve the controller voltage.

Multiple Choice Questions with Answers

Question 1

A 12 V source is connected to a 4 Ω resistor. What current flows?

A. 0.33 A

B. 3 A

C. 8 A

D. 48 A

Correct Answer: B. 3 A

Question 2

Two resistors of 6 Ω and 3 Ω are connected in parallel. Their equivalent resistance is:

A. 2 Ω

B. 3 Ω

C. 9 Ω

D. 18 Ω

Correct Answer: A. 2 Ω

Question 3

Which replacement deactivates an ideal independent voltage source in superposition?

A. An open circuit

B. A 1 Ω resistor

C. A short circuit

D. A current source

Correct Answer: C. A short circuit

Question 4

In a parallel circuit, all branches have the same:

A. Current

B. Resistance

C. Power

D. Voltage

Correct Answer: D. Voltage

Question 5

A Thévenin equivalent has V_th = 20 V and R_th = 5 Ω. With R_L = 5 Ω, load current is:

A. 1 A

B. 2 A

C. 4 A

D. 5 A

Correct Answer: B. 2 A

Question 6

Maximum power is delivered to a resistive load when:

A. R_L = 0

B. R_L = R_th

C. R_L = 2R_th

D. R_L is infinite

Correct Answer: B. R_L = R_th

Question 7

The algebraic sum of currents at a node is zero according to:

A. Ohm’s law

B. KVL

C. KCL

D. Maximum-power theorem

Correct Answer: C. KCL

Question 8

A practical voltage source V_s with series resistance R_s transforms to a current source of:

A. V_sR_s

B. R_s/V_s

C. V_s + R_s

D. V_s/R_s

Correct Answer: D. V_s/R_s

Question 9

For a resistor, a positive value of p = vi using the passive sign convention indicates that the element:

A. Absorbs power

B. Generates charge

C. Has zero current

D. Must be a source

Correct Answer: A. Absorbs power

Question 10

A 10 V Thévenin source with 2 Ω resistance supplies a 3 Ω load. The load voltage is:

A. 4 V

B. 6 V

C. 8 V

D. 10 V

Correct Answer: B. 6 V