Circuit Analysis and Signal Response

Introduction

Circuit analysis provides the mathematical foundation for predicting voltages, currents, power, and signal behavior in electronic systems. Although practical circuits may contain thousands of components, their operation can often be understood by applying a small set of principles: Kirchhoff’s laws, element voltage–current relationships, network theorems, and differential-equation models.

This chapter develops methods for both direct-current (DC) and sinusoidal steady-state analysis. It then extends these methods to transient response, resonance, frequency response, and passive filters. The emphasis is not only on obtaining numerical answers, but also on selecting an appropriate method, checking assumptions, interpreting results, and recognizing limitations.

Learning Objectives

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

1. Apply Kirchhoff’s current and voltage laws to electrical networks.
2. Use nodal and mesh analysis to determine unknown circuit variables.
3. Apply superposition, Thévenin, Norton, and maximum-power-transfer methods.
4. Analyze first-order RC and RL transients and interpret time constants.
5. Determine impedance, phase, and power in sinusoidal circuits.
6. Explain resonance and calculate resonant frequency and quality factor.
7. Analyze the frequency response of basic low-pass, high-pass, band-pass, and band-stop filters.
8. Evaluate practical circuit limitations and select suitable analysis methods.

1. Fundamental Circuit Concepts

An electrical circuit consists of interconnected elements that provide paths for charge movement. The basic variables are voltage and current.

• Voltage, measured in volts (V), is the energy difference per unit charge between two points.
• Current, measured in amperes (A), is the rate of charge flow.
• Resistance, measured in ohms (Ω), opposes current.
• Power, measured in watts (W), is the rate of energy transfer.

The passive sign convention assigns positive power to an element that absorbs energy. If current enters the terminal marked positive for voltage, then

where p(t) is instantaneous power in watts, v(t) is voltage in volts, and i(t) is current in amperes. A negative value indicates that the element delivers power under the selected reference directions.

For a resistor, Ohm’s law is

where R is resistance in ohms. The power absorbed by a resistor may be written in equivalent forms:

These equations assume an ohmic resistor with constant resistance. Unit consistency is essential: volts times amperes gives watts.

Kirchhoff’s Laws

Kirchhoff’s current law (KCL) follows from conservation of charge. At any node,

where currents entering a node may be assigned positive signs and currents leaving it negative signs, or vice versa, provided the convention is used consistently.

Kirchhoff’s voltage law (KVL) follows from conservation of energy. Around any closed loop,

The laws apply to lumped circuits in which electromagnetic propagation effects can be neglected relative to circuit dimensions and signal wavelength.

2. Nodal and Mesh Analysis

Nodal Analysis

Nodal analysis determines node voltages relative to a selected reference node, usually called ground. It is especially efficient when a circuit has many current sources or relatively few essential nodes.

The procedure is:

1. Select a reference node and assign it zero volts.
2. Label the remaining node voltages.
3. Write KCL at each unknown-voltage node.
4. Express each branch current using the voltage difference divided by resistance.
5. Solve the resulting simultaneous equations.

Worked Example 1: Nodal Analysis

A node voltage V is connected to ground through a 2 kΩ resistor and a 4 kΩ resistor. A 3 mA current source injects current into the node. KCL gives

The equivalent conductance is

Therefore,

The branch currents are 2 mA through the 2 kΩ resistor and 1 mA through the 4 kΩ resistor, confirming KCL.

A supernode is used when an ideal voltage source lies between two nonreference nodes. The surrounding boundary is treated as one extended node, KCL is written for the supernode, and an additional voltage constraint is included.

Mesh Analysis

Mesh analysis uses loop currents and is most effective for planar circuits containing several voltage sources. A mesh is a loop that contains no other loop inside it.

The procedure is:

1. Assign a mesh current to each independent mesh, usually clockwise.
2. Apply KVL to each mesh.
3. For a resistor shared by two meshes, its voltage drop depends on the difference of mesh currents.
4. Solve the simultaneous equations.

Mesh analysis is based on KVL and requires a planar circuit for direct application. A current source shared by two meshes produces a supermesh; the current-source constraint is then added separately.

3. Network Theorems

Network theorems simplify complicated linear circuits and are particularly useful when the same source network is connected to different loads.

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 source acting alone.

To deactivate independent sources:

• Replace an ideal voltage source by a short circuit.
• Replace an ideal current source by an open circuit.
• Leave dependent sources active.

Superposition applies to voltage and current, but not directly to power because power is nonlinear in voltage and current. For example, the total resistor power is not generally the sum of powers calculated separately from each source.

Thévenin’s Theorem

Any linear two-terminal network can be replaced, at its terminals, by an equivalent voltage source Vₜₕ in series with a resistance Rₜₕ.

where V_oc is the open-circuit terminal voltage. With independent sources deactivated,

Alternatively, if the short-circuit current I_sc is known,

For circuits containing dependent sources, a test source is required unless a suitable V_oc/I_sc method is used.

With a load R_L connected, the load current is

and the load voltage is

Norton’s Theorem

The Norton equivalent is a current source I_N in parallel with resistance R_N. The parameters are

and

The Thévenin and Norton forms are related by

These equivalents are valid only at the specified pair of terminals and for linear behavior under the operating conditions used to obtain them.

Maximum Power Transfer

For a resistive Thévenin network, the load receives maximum power when

The maximum load power is then

The efficiency at this condition is 50%, because equal power is dissipated in the load and source resistance. Maximum power transfer is therefore not the same as maximum efficiency.

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

This condition assumes sinusoidal steady state and adjustable load impedance.

4. Capacitors, Inductors, and Transients

A capacitor stores energy in an electric field. Its current–voltage relation is

where C is capacitance in farads. Capacitor voltage cannot change instantaneously in a finite-current circuit. Its stored energy is

An inductor stores energy in a magnetic field. Its voltage–current relation is

where L is inductance in henrys. Inductor current cannot change instantaneously in a finite-voltage circuit. Its stored energy is

First-Order RC Response

For a source-free RC circuit, the capacitor voltage is

where V₀ is the initial capacitor voltage and

is the time constant. R_eq is the resistance seen by the capacitor with independent sources deactivated. After one time constant, the remaining response is e^(−1) ≈ 36.8% of its initial value.

For a step response with final value V_f,

The capacitor voltage is continuous, so V_C(0⁺) = V_C(0⁻).

First-Order RL Response

For a source-free RL circuit,

with

For a general step response,

where I_f is the final steady-state current. The inductor current is continuous at switching: I_L(0⁺) = I_L(0⁻).

Worked Example 2: RC Charging

A 10 V source charges a 10 μF capacitor through a 100 kΩ resistor. The time constant is

For an initially uncharged capacitor,

At t = 1 s,

At approximately 5τ, the capacitor reaches 99.3% of its final value, which is commonly treated as practical completion.

5. Sinusoidal Steady-State Analysis

A sinusoidal voltage may be represented as

where V_m is peak amplitude, ω is angular frequency in rad/s, t is time, and φ is phase. Angular frequency relates to frequency f in hertz by

The root-mean-square value of a sinusoid is

RMS values are used in AC power calculations for resistive heating and average power.

In phasor analysis, resistors, inductors, and capacitors are represented by impedances:

Here j = √(−1). Inductive reactance is positive imaginary; capacitive reactance is negative imaginary. Phasor analysis applies to linear circuits in sinusoidal steady state, not to switching transients.

For series elements, impedances add. For parallel elements, admittances add, where Y = 1/Z. Ohm’s law becomes

For a load with RMS voltage V and impedance Z, average power is

where φ is the phase angle between voltage and current. The factor cos φ is the power factor. Apparent power is S = V_rmsI_rms in volt-amperes, while reactive power is Q = V_rmsI_rms sin φ in reactive volt-amperes.

6. RLC Circuits and Resonance

A series RLC circuit has impedance

Resonance occurs when the net reactance is zero:

Thus,

and

At series resonance, the impedance is ideally R, current is maximum, and voltage and current are in phase. The inductor and capacitor exchange stored energy, while the resistor dissipates energy.

For a series RLC circuit, the quality factor is

A high Q indicates a narrow resonance. The approximate bandwidth between the half-power frequencies is

This relation applies to the standard series or parallel second-order resonator under conventional definitions of Q.

In a parallel resonant circuit, the input impedance is ideally maximum at resonance and source current is minimum. Practical inductor resistance and loading alter the ideal response.

7. Frequency Response and Filters

A transfer function describes the ratio of output phasor to input phasor:

The magnitude |H| describes amplitude scaling, and the argument ∠H describes phase shift. Decibels are defined by

for voltage or current ratios. For power ratios, the coefficient is 10 rather than 20.

RC Low-Pass Filter

For a resistor in series and capacitor connected from output to ground,

The cutoff angular frequency is

and the cutoff frequency is

At f_c, the magnitude is 1/√2 ≈ 0.707 of the low-frequency value, corresponding to −3.01 dB. Frequencies below cutoff pass with relatively little attenuation; higher frequencies are attenuated at approximately −20 dB per decade.

RC High-Pass Filter

For a capacitor in series and resistor connected to ground, with output across the resistor,

The same cutoff frequency applies. Low frequencies are attenuated, while high frequencies approach unity gain.

Band-Pass and Band-Stop Filters

A band-pass filter passes a range of frequencies and attenuates frequencies below and above that range. A band-stop, or notch, filter rejects a selected band while passing frequencies outside it. Second-order RLC networks can provide sharper selectivity than a single RC section.

The ideal transfer equations must be modified when the source has nonzero resistance, the load is finite, or active circuitry follows the filter. Loading changes the effective R and therefore shifts the cutoff frequency. Buffer amplifiers are often used to reduce this interaction.

8. Applied Case: Signal Conditioning for a Sensor Interface

Background

Consider an illustrative sensor interface in which a temperature sensor produces a slowly varying voltage, but the connecting cable picks up high-frequency electromagnetic interference. The sensor output has a source resistance of 2 kΩ, and the receiving microcontroller has a high-impedance analog input.

Problem

The interface should attenuate interference above approximately 160 Hz while preserving the slowly varying temperature signal. A first-order RC low-pass filter is selected, with the capacitor connected at the microcontroller input.

Analysis and Design

The source resistance is approximately the filter resistance, so select C = 100 nF. The cutoff frequency is

Substituting R = 2 kΩ and C = 100 nF gives

This cutoff is too high. To obtain approximately 160 Hz, choose

Thus,

A standard 470 nF capacitor gives approximately

Practical Solution

The selected capacitor should have suitable voltage rating, tolerance, temperature stability, and leakage characteristics. If the microcontroller input samples rapidly, an additional buffer or carefully designed anti-aliasing filter may be required. The filter should be tested with the sensor’s actual source impedance rather than an ideal voltage source.

Outcome and Learning Points

The RC network reduces high-frequency interference while retaining low-frequency sensor information. The case illustrates that filter design depends on source impedance, component availability, loading, tolerances, and the required signal bandwidth—not only on the nominal cutoff equation.

9. Limitations, Safety, and Professional Practice

Circuit equations are models. Real components have parasitic resistance, inductance, capacitance, leakage, finite ratings, and frequency-dependent behavior. Important limitations include:

• Ideal voltage sources cannot supply unlimited current.
• Capacitors and inductors have equivalent series resistance and self-resonant frequencies.
• Resistors generate thermal noise and may change value with temperature.
• Measurement instruments load the circuit and introduce bandwidth limits.
• Grounding and shielding affect noise and signal integrity.
• Tolerances shift time constants, cutoff frequencies, and resonant frequencies.

In laboratory and industrial work, verify voltage, current, power, insulation, and fault-current ratings before energizing a circuit. Use current-limited supplies, appropriate fuses, safe probing procedures, and proper isolation. Circuit simulations are valuable for prediction, but measurements are necessary to identify modeling errors and nonideal behavior.

Contemporary circuit design increasingly uses automated symbolic analysis, numerical simulation, and optimization tools. Such tools can accelerate design, but engineers remain responsible for selecting correct models, checking boundary conditions, validating units, and interpreting results. Artificial intelligence may assist with topology exploration or parameter fitting, but it must not replace independent verification, safety review, or documented engineering judgment.

Chapter Summary

Circuit analysis is based on conservation of charge and energy, expressed through KCL and KVL. Nodal analysis uses node voltages and is efficient for current-source networks, while mesh analysis uses loop currents and is convenient for planar voltage-source networks. Superposition separates the effects of independent sources in linear circuits. Thévenin and Norton theorems replace a linear two-terminal network with simpler equivalent forms. Maximum power is delivered to a resistive load when the load equals the Thévenin resistance, although this condition gives only 50% efficiency.

Capacitors and inductors store energy and produce transient responses characterized by RC and RL time constants. In sinusoidal steady state, impedance permits resistor, inductor, and capacitor networks to be analyzed algebraically using phasors. RLC circuits exhibit resonance at a frequency determined by L and C, with selectivity described by Q and bandwidth. Transfer functions quantify signal amplitude and phase, while RC and RLC filters select desired frequency ranges. Practical design must account for loading, tolerances, parasitics, safety, and measurement limitations.

Key Terms

Voltage: Electrical potential difference between two points.

Current: Rate of charge flow through a circuit.

Impedance: Complex opposition to sinusoidal current, measured in ohms.

Admittance: Reciprocal of impedance.

KCL: Conservation of charge at a circuit node.

KVL: Conservation of energy around a closed path.

Nodal analysis: Circuit method based on unknown node voltages.

Mesh analysis: Circuit method based on loop currents.

Superposition: Addition of individual linear-source responses.

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

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

Time constant: Parameter determining the speed of a first-order transient.

Phasor: Complex representation of a sinusoidal quantity.

Reactance: Imaginary component of impedance.

Resonance: Condition in which inductive and capacitive reactances cancel.

Quality factor: Measure of resonator selectivity.

Transfer function: Ratio of output to input in the frequency domain.

Cutoff frequency: Frequency at which a standard filter magnitude falls to 0.707 of its passband value.

Power factor: Cosine of the phase angle between voltage and current.

Review Questions

Short Answer Questions

1. State Kirchhoff’s current law and explain its physical basis.
2. When is nodal analysis generally more convenient than mesh analysis?
3. How are independent voltage and current sources deactivated for superposition?
4. Define the time constant of an RC circuit.
5. What is meant by the cutoff frequency of a first-order filter?

Long Answer Questions

1. Explain nodal analysis, including the treatment of a supernode, and illustrate the method with a suitable circuit.
2. Derive the Thévenin and Norton load-current relationships and explain when each equivalent is useful.
3. Explain capacitor and inductor continuity conditions during switching.
4. Develop the impedance expression for a series RLC circuit and discuss its behavior below, at, and above resonance.
5. Compare low-pass, high-pass, band-pass, and band-stop filters in terms of their frequency-selective behavior.

Analytical / Application Questions

1. A 12 V Thévenin source has an internal resistance of 6 Ω. Determine the load resistance for maximum power transfer and calculate the maximum load power.
2. A 5 kΩ resistor charges a 2 μF capacitor from a 10 V ideal source. Determine the time constant and capacitor voltage after one time constant.
3. Design an RC low-pass filter with a 1 kHz cutoff using a 10 kΩ resistor. Calculate the required capacitance and comment on one practical nonideality that may affect the result.

Multiple Choice Questions with Answers

Question 1

Which law states that the algebraic sum of currents at a node is zero?

A. Ohm’s law

B. Kirchhoff’s current law

C. Kirchhoff’s voltage law

D. Faraday’s law

Correct Answer: B. Kirchhoff’s current law

Question 2

A 3 mA current enters a node connected to ground through 2 kΩ and 4 kΩ resistors. What is the node voltage?

A. 2 V

B. 3 V

C. 4 V

D. 6 V

Correct Answer: C. 4 V

Question 3

For superposition analysis, an ideal independent voltage source is replaced by a:

A. Open circuit

B. Short circuit

C. Resistor equal to its load

D. Current source

Correct Answer: B. Short circuit

Question 4

A Thévenin equivalent consists of:

A. A current source in series with resistance

B. A voltage source in parallel with resistance

C. A voltage source in series with resistance

D. A capacitor in parallel with an inductor

Correct Answer: C. A voltage source in series with resistance

Question 5

Maximum power is transferred from a resistive Thévenin network when:

A. R_L = 0

B. R_L = Rₜₕ

C. R_L = 2Rₜₕ

D. R_L → ∞

Correct Answer: B. R_L = Rₜₕ

Question 6

A 100 kΩ resistor and a 10 μF capacitor have an RC time constant of:

A. 0.1 s

B. 1 s

C. 10 s

D. 100 s

Correct Answer: B. 1 s

Question 7

The impedance of an ideal inductor in sinusoidal steady state is:

A. R

B. 1/(jωC)

C. jωL

D. 1/(jωL)

Correct Answer: C. jωL

Question 8

At series RLC resonance, the net inductive and capacitive reactances are:

A. Both maximum

B. Equal and additive

C. Equal and cancelling

D. Both zero for all frequencies

Correct Answer: C. Equal and cancelling

Question 9

For a first-order RC low-pass filter, the cutoff frequency is:

A. RC/(2π)

B. 2πRC

C. 1/(2πRC)

D. 2π/(RC)

Correct Answer: C. 1/(2πRC)

Question 10

At the cutoff frequency of a standard first-order voltage filter, the magnitude is approximately:

A. 0.1 of passband value

B. 0.5 of passband value

C. 0.707 of passband value

D. 1.414 of passband value

Correct Answer: C. 0.707 of passband value