Analog Circuit and Signal Foundations

Introduction

Analog electronics concerns voltages and currents that vary continuously over a useful range. Although practical circuits employ discrete components and operate within finite bandwidths, their analysis begins with circuit laws, network models, complex impedance, transient response, transfer functions, and measurement principles.

These foundations are essential because an apparently correct circuit may behave incorrectly when connected to a source, load, probe, or measuring instrument. Engineers must therefore understand both the circuit and the measurement system. They must also recognize where simplified models cease to be valid—for example, when parasitic capacitance, nonlinear behavior, component tolerances, or instrument bandwidth becomes significant.

Learning Objectives

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

1. Apply voltage, current, power, Kirchhoff’s laws, and basic network theorems.
2. Use Thévenin and Norton equivalents within their correct domains of applicability.
3. Analyze sinusoidal circuits using impedance and phasors.
4. Determine first-order RC and RL transient responses.
5. Evaluate loading, voltage division, gain, and decibel quantities.
6. interpret transfer functions, poles, zeros, bandwidth, and phase.
7. Formulate z-, y-, h-, and ABCD two-port equations with correct conventions and units.
8. Characterize analog signals, noise, distortion, and dynamic range.
9. estimate measurement uncertainty and recognize instrument limitations.
10. Operate oscilloscopes, generators, meters, and power supplies safely.

1. Circuit Variables and Reference Conventions

1.1 Voltage and current

Voltage is electrical potential difference, measured in volts (V). For terminals a and b,

Current is the rate of flow of electric charge:

Here, q is charge in coulombs (C), t is time in seconds (s), and i is current in amperes (A).

A voltage polarity and current direction are reference choices. A calculated negative value means that the actual polarity or direction is opposite to the selected reference.

1.2 Power and energy

Under the passive sign convention, current enters the terminal marked positive for voltage. Instantaneous power absorbed by an element is

Power is measured in watts (W). If p > 0, the element absorbs power; if p < 0, it delivers power. Energy transferred between times t₁ and t₂ is

Energy is measured in joules (J). In an ideal resistor,

These equivalent forms apply to a resistor obeying v=iR. For time-varying quantities, v and i in the first expression are instantaneous values.

2. Fundamental Network Laws

2.1 Kirchhoff’s current law

Kirchhoff’s current law (KCL) states that the algebraic sum of currents at a node is zero:

KCL expresses charge conservation. It is applied under the lumped-circuit approximation, where charge accumulation not represented by circuit elements is negligible.

2.2 Kirchhoff’s voltage law

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

KVL follows from energy conservation for lumped circuits. In the presence of time-varying magnetic flux linking a loop, the induced electromotive force must be included explicitly; otherwise, a simple zero-sum statement is incomplete.

2.3 Series and parallel combinations

For resistors in series,

For resistors in parallel,

The two-resistor voltage-divider relationship is

It assumes that no additional load draws current from the divider output. If a load R_L is connected across R₂, replace R₂ with R₂ \parallel R_L.

3. Network Theorems

3.1 Superposition

In a linear circuit containing multiple independent sources, the voltage or current response equals the algebraic sum of the responses produced by each independent source acting alone. When suppressing other independent sources:

• replace an ideal voltage source by a short circuit;
• replace an ideal current source by an open circuit;
• retain dependent sources because they are controlled by circuit variables.

Superposition applies to voltage and current, but not directly to power because power is nonlinear in voltage or current.

3.2 Thévenin and Norton equivalents

A linear two-terminal resistive DC network can be represented at its terminals by a Thévenin voltage source V_Th in series with a resistance R_Th, or by a Norton current source I_N in parallel with R_N.

Here, V_oc is open-circuit voltage and I_sc is short-circuit current, provided the ratio is defined.

The scope must be stated carefully:

• For resistive DC networks, the equivalent uses resistance.
• For sinusoidal steady-state linear networks at a specified frequency, resistance is replaced by the generally complex Thévenin or Norton impedance Z_Th(jω).
• For dynamic networks, a frequency- or operator-dependent driving-point impedance such as Z_Th(s) may be used under zero-state assumptions. Initial energy in inductors and capacitors can require additional equivalent sources. A single constant resistance is not a general transient representation.
• Linear dependent sources remain active when equivalent impedance is determined. A test source may be applied at the terminals.
• Nonlinear networks generally cannot be represented globally by one fixed linear Thévenin equivalent, although a small-signal linearization may be used near a bias point.

If only independent sources are present, R_Th can be found by suppressing them and calculating the resistance seen from the terminals.

Worked Example 1: Resistive equivalent

A 12 V source supplies a 2 kΩ series resistor connected to an output node, with 3 kΩ from that node to ground.

The open-circuit voltage is

With the ideal source suppressed,

Therefore,

For a 4.8 kΩ load,

3.3 Maximum power transfer

For a resistive source equivalent, maximum load power occurs when

For sinusoidal steady state with Z_Th=R_Th+jX_Th, maximum average power is obtained by conjugate matching:

Maximum power transfer gives only 50% efficiency in the purely resistive case and is therefore not usually the design objective in power distribution. It is more relevant to communication and impedance-matching applications.

4. Sinusoidal Signals, Phasors, and Impedance

A sinusoidal voltage can be written as

where V_m is peak amplitude, ω=2πf is angular frequency in rad/s, f is frequency in hertz, and φ is phase.

Its RMS value is V_rms=V_m/√2. Using RMS phasors and the e^jωt convention,

Phasors apply to sinusoidal steady state at one frequency. They do not directly describe switching transients or arbitrary waveforms unless Fourier components are analyzed separately.

4.1 Element impedances

Resistance and reactance are measured in ohms (Ω). Impedance is

Admittance is Y=1/Z=G+jB and is measured in siemens (S).

4.2 AC power

For RMS phasors, complex power is

Here, P is average real power in watts, Q is reactive power in var, and |S| is apparent power in volt-amperes. The power factor is P/|S|=\cos θ, where θ is the voltage-current phase difference.

Worked Example 2: RC low-pass response

A 1 kΩ resistor is in series with a 100 nF capacitor, and the output is measured across the capacitor. At 1 kHz,

Since ωRC=2π(1000)(1000)(100×10^-9)=0.6283,

A 1 V RMS input therefore produces approximately 0.847 V RMS with a phase lag of 32.1°.

5. Transient Response and Stored Energy

Capacitors and inductors store energy:

Capacitor voltage and inductor current cannot change instantaneously unless an impulse of current or voltage, respectively, is applied in an idealized model.

For a stable first-order circuit with a constant input after switching, a state variable x(t) follows

The time constant is τ=R_eqC for a single-capacitor RC circuit and τ=L/R_eq for a single-inductor RL circuit, where R_eq is the resistance seen by the energy-storage element after independent sources are suppressed. Dependent sources must remain active.

Worked Example 3: Capacitor charging

A discharged 10 μF capacitor is charged through 10 kΩ from an ideal 5 V source. The time constant is

Thus,

At t=0.20\text{ s},

The initial current is 5/10\text{ kΩ}=0.50\text{ mA}, and it decays exponentially.

Second-order RLC circuits may be overdamped, critically damped, or underdamped. Their behavior depends on pole locations and cannot generally be reduced to one time constant.

6. Loading and Practical Sources

An ideal voltage source has zero output resistance, while an ideal current source has infinite output resistance. Real sources are represented over a specified operating range by finite output impedance.

For a source V_s with series resistance R_s driving R_L,

Loading error relative to the open-circuit source voltage is

A voltmeter should therefore have input resistance much greater than the source resistance. Conversely, an ammeter should have very low resistance so that its burden voltage minimally changes circuit current.

Figure: A measuring instrument becomes part of the circuit and loads a nonideal source.

7. Decibels and Transfer Functions

7.1 Decibels

For a power ratio,

For voltage or current ratios with equal impedances, or when the relevant impedance factor cancels,

The factor 20 must not be used blindly when the two powers involve different impedances. A gain of 2 corresponds to approximately 6.02 dB, while a voltage ratio of 0.5 corresponds to −6.02 dB. The RC response of Worked Example 2 is 20\log₁₀(0.8467)=-1.45\text{ dB}.

7.2 Transfer functions

For a linear time-invariant system with zero initial conditions, the transfer function is

Here, X(s) and Y(s) are Laplace transforms of input and output, and s=σ+jω. Zeros are roots of the numerator; poles are roots of the denominator. Pole locations determine natural response and stability. A continuous-time causal rational system is asymptotically stable when all poles lie in the open left half of the s-plane.

The frequency response is obtained by evaluating H(jω) when the imaginary axis lies within the region of convergence. Bandwidth is commonly measured between defined cutoff points, often the −3 dB points for power-related low-pass responses, but the definition must always be stated.

8. Two-Port Network Concepts

A two-port network provides a compact representation of amplifiers, filters, transmission sections, and transistor small-signal models. Let V₁ and V₂ be measured from each port’s upper terminal to its lower terminal. Let I₁ and I₂ both be directed into the two-port.

8.1 Impedance and admittance parameters

All z-parameters have units of ohms. They are determined using open-circuit conditions:

Admittance parameters satisfy

All y-parameters have units of siemens and are extracted using short-circuit conditions, such as y₁₁=(I₁/V₁)|_V2=0.

8.2 Hybrid parameters

The h-parameter equations are

Their extraction conditions and units are:

Hybrid parameters are useful in transistor small-signal characterization because they combine input impedance, reverse voltage ratio, forward current ratio, and output admittance.

8.3 ABCD or transmission parameters

With both port currents defined into the network, the standard transmission convention uses -I₂, the current leaving port 2 toward a load:

The units are: A and D dimensionless, B in ohms, and C in siemens. Extraction gives

ABCD matrices are especially convenient for cascades: the overall matrix is the ordered product of the individual matrices. A series impedance Z has matrix [[1,Z],[0,1]], while a shunt admittance Y has matrix [[1,0],[Y,1]]. Cascading a 100 Ω series element followed by a 1 mS shunt element gives

Thus A=1.1, B=100\text{ Ω}, C=1\text{ mS}, and D=1. For a reciprocal two-port under this convention, AD-BC=1.

9. Analog Signals and Signal Quality

An analog signal is represented by a continuously variable physical quantity, although its practical amplitude, rate of change, and bandwidth are finite. Important characteristics include:

• Amplitude: signal magnitude relative to a reference.
• DC offset: average or constant component.
• Frequency and phase: periodic timing properties.
• Bandwidth: frequency range containing significant information or meeting a specification.
• Noise: unwanted random variation.
• Distortion: undesired waveform alteration caused by nonlinearity or frequency-dependent response.
• Dynamic range: ratio between the largest usable signal and the noise floor or minimum detectable signal.
• Signal-to-noise ratio: SNR=P_s/P_n, commonly reported as 10\log₁₀(P_s/P_n) dB.

Clipping occurs when a signal exceeds circuit output limits. Slew-rate limiting occurs when an amplifier cannot change its output rapidly enough. These phenomena are distinct from small-signal bandwidth limitations.

Modern mixed-signal systems convert analog signals to digital data. Before sampling, an anti-aliasing filter should sufficiently attenuate components above half the sampling frequency. Digital processing and AI may classify waveforms or detect anomalies, but they cannot recover information destroyed by clipping, excessive noise, aliasing, or unsafe measurement practice.

10. Measurement Uncertainty

A measured value is incomplete without information about uncertainty. Accuracy describes closeness to the accepted value; precision describes repeatability; resolution is the smallest displayed or detectable increment.

An instrument specification may be expressed as ±(percentage of reading + counts). Suppose a meter reads 5.00 V on a range with 0.01 V resolution and has accuracy ±(0.5% of reading + 2 counts). The limiting uncertainty is

The result may therefore be reported as approximately (5.00±0.05)\text{ V}, after appropriate uncertainty rounding.

For independent standard uncertainties u₁,u₂,\ldots,u_n, a common propagation rule is root-sum-square combination:

This rule assumes the contributions are uncorrelated and represented as standard uncertainties. Worst-case limits are instead added arithmetically when guaranteed bounds are required.

Uncertainty sources include calibration, noise, probe loading, contact resistance, environmental conditions, bandwidth, sampling, component tolerance, and reading technique. Calibration traceability and documented procedures are essential in regulated or safety-critical work.

11. Safe Laboratory Instrumentation

11.1 Digital multimeter

Select the correct function and range before connection. Voltage is measured in parallel; current is measured in series. Connecting a meter configured for current directly across a source can create a short circuit. Observe voltage, current, and measurement-category ratings. IEC measurement categories reflect expected transient environments; a high voltage rating alone does not guarantee suitability for every installation.

11.2 Oscilloscope

Most bench oscilloscope probe ground clips are connected to protective earth. Attaching the clip to a non-grounded live node can short that node to earth, damage equipment, or cause injury. Never defeat protective earthing to float an ordinary oscilloscope. Use a properly rated differential probe, isolated instrument, or approved isolation method when differential high-side measurement is required.

A ×10 passive probe usually provides higher input resistance and lower capacitance than a ×1 probe. It must be compensated using the oscilloscope’s calibration output. Oscilloscope and probe bandwidth must both exceed the frequency content of interest. For a Gaussian-like single-pole response, the approximate rise-time relation is

This is an approximation, not a universal law for all filter shapes.

11.3 Function generator

Many generators have a nominal 50 Ω output impedance. An amplitude specified for a 50 Ω load may become approximately twice as large when connected to a high-impedance load. Confirm whether the display assumes high impedance or 50 Ω termination. Check DC offset and maximum allowable external voltage before connection.

11.4 DC power supply

Before powering a new circuit:

1. inspect polarity, wiring, component orientation, and expected current;
2. set a conservative current limit;
3. set voltage to zero or disable the output;
4. connect the circuit with power off;
5. increase voltage gradually while monitoring current and temperature;
6. discharge energy-storage components before modification.

Series or tracking supply channels must not be connected in configurations that violate isolation or common-mode ratings. High-energy capacitors require discharge resistors and verified zero-energy conditions.

12. Illustrative Application Case: High-Impedance Sensor Interface

Background

This fictional case concerns an industrial chemical sensor modeled near its operating point as a 1.50 V Thévenin source with 100 kΩ output resistance. An engineer connects it directly to a data-acquisition input having 1 MΩ resistance.

Problem

The acquisition system reports less than the expected sensor voltage. The engineer must determine whether the difference indicates sensor failure or loading.

Analysis

The loaded voltage is

The loading error is

The reading is therefore consistent with the circuit model and does not by itself indicate a defective sensor. Cable capacitance would introduce additional frequency-dependent loading and settling time.

Approach and Solution

A low-input-bias-current buffer amplifier with sufficiently high input resistance is inserted between the sensor and acquisition system. If the buffer input resistance is 100 MΩ, the resistive loading error becomes approximately

The engineer also verifies input bias-current error, offset voltage, noise, common-mode range, power-supply range, bandwidth, and input-protection requirements. An anti-alias filter is selected based on the sensor bandwidth and sampling rate.

Outcome

The buffered system gives a value close to the expected open-circuit sensor output, and the remaining uncertainty is documented using sensor tolerance, amplifier offset, meter calibration, and acquisition resolution.

Learning Points

• Measuring equipment forms part of the circuit.
• High source resistance requires very high input resistance and low input bias current.
• A DC loading calculation does not capture cable capacitance or dynamic settling.
• Automated or AI-assisted diagnostics remain dependent on correct physical modeling and calibrated measurements.

13. Limitations, Professional Practice, and Contemporary Developments

Circuit models are approximations. Real resistors possess parasitic inductance and capacitance; capacitors exhibit leakage and equivalent series resistance; inductors exhibit winding resistance and core loss. Source and load impedances may vary with frequency, signal level, temperature, and bias.

Professional practice requires model validation, uncertainty records, calibration control, safe voltage and energy limits, electrostatic-discharge precautions, and truthful reporting. Engineers must not suppress anomalous measurements merely because they disagree with simulation. Automated oscilloscopes, remote laboratories, programmable instruments, and AI-based anomaly detection can improve productivity, but responsible use requires traceable calibration, reviewable settings, cybersecurity controls, and human verification.

Chapter Summary

Analog circuit analysis begins with consistently defined voltage, current, power, KCL, and KVL. Superposition applies to linear responses, while Thévenin and Norton models reduce linear two-terminal networks within stated conditions. Resistive DC equivalents use resistance; AC and dynamic networks generally require frequency- or operator-dependent impedance and consideration of initial energy.

Phasors and complex impedance simplify sinusoidal steady-state analysis. Transient behavior follows stored energy and initial conditions, with stable first-order circuits exhibiting exponential response. Loading explains why sources, loads, and instruments cannot be analyzed independently. Decibels express ratios, while transfer functions describe gain, phase, poles, zeros, stability, and bandwidth.

Two-port networks may be represented using z, y, h, or ABCD parameters. Each representation has specific current conventions, units, and extraction conditions. Reliable engineering additionally requires awareness of analog signal quality, uncertainty, instrument limitations, and laboratory safety. Sound measurement practice is inseparable from sound circuit theory.

Key Terms

Admittance: Reciprocal of impedance, measured in siemens.

Analog signal: Signal represented by a continuously variable physical quantity.

Bandwidth: Specified frequency range over which a system meets a criterion.

Complex power: RMS phasor product S=VI^*=P+jQ.

Decibel: Logarithmic unit used to express power or amplitude ratios.

Dynamic range: Ratio between the largest usable signal and the minimum usable signal or noise floor.

Impedance: Complex ratio of phasor voltage to phasor current.

Loading: Change in circuit behavior caused by connection of a finite-impedance load or instrument.

Norton equivalent: Current-source and parallel-impedance representation of a linear two-terminal network under specified conditions.

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

Phasor: Complex representation of sinusoidal magnitude and phase at one frequency.

Pole: Value of the complex variable at which a transfer function becomes unbounded.

Resolution: Smallest increment that an instrument can display or distinguish.

Thevenin equivalent: Voltage-source and series-impedance representation of a linear two-terminal network under specified conditions.

Time constant: Parameter governing the exponential rate of a first-order response.

Transfer function: Zero-initial-condition ratio of output transform to input transform for an LTI system.

Two-port network: Four-terminal network represented through paired port variables.

Uncertainty: Quantified doubt associated with a measurement result.

Zero: Value of the complex variable at which a transfer-function numerator becomes zero.

Review Questions

Short Answer Questions

1. State the passive sign convention and explain the meaning of negative calculated power.
2. Why must dependent sources remain active when finding equivalent resistance or impedance?
3. Distinguish accuracy, precision, and resolution.
4. Why is the factor 20 used for some voltage ratios expressed in decibels?
5. Why can an oscilloscope ground clip create a safety hazard?

Long Answer Questions

1. Explain Thévenin and Norton equivalents for resistive DC, sinusoidal AC, and dynamic transient networks, including their limitations.
2. Derive the first-order response formula from a standard RC charging differential equation.
3. Explain how poles and zeros influence a circuit’s transient and frequency responses.
4. Compare z-, y-, h-, and ABCD two-port parameter sets, including equations, units, extraction conditions, and preferred applications.
5. Develop a safe procedure for using a function generator, oscilloscope, multimeter, and current-limited power supply on a new circuit.

Analytical / Application Questions

1. A 2.0 V source with 200 kΩ output resistance is measured first by a 1 MΩ oscilloscope and then by a 10 MΩ probe. Calculate both readings and loading errors.
2. Design an RC low-pass filter with a nominal cutoff frequency of 2 kHz using a 10 nF capacitor. Choose a standard resistor value and calculate the resulting cutoff frequency.
3. A cascaded network contains a shunt admittance of 2 mS followed by a series impedance of 50 Ω. Determine its ABCD matrix in the stated order and verify whether its determinant indicates reciprocity.

Multiple Choice Questions with Answers

Question 1

Under the passive sign convention, an element has v=8\text{ V} and current i=-0.5\text{ A}. What does the power indicate?

A. The element absorbs 4 W

B. The element delivers 4 W

C. The element absorbs 16 W

D. The element has zero power

Correct Answer: B. The element delivers 4 W

Question 2

Which statement correctly describes a Thévenin equivalent for a linear reactive network in sinusoidal steady state?

A. It must always use a constant resistance independent of frequency

B. It uses an ideal current source in series with a resistance

C. It generally uses a voltage source in series with a frequency-dependent complex impedance

D. It is valid only if the network contains no capacitors

Correct Answer: C. It generally uses a voltage source in series with a frequency-dependent complex impedance

Question 3

A 10 V source with 2 kΩ series resistance drives an 8 kΩ load. What is the load voltage?

A. 2 V

B. 8 V

C. 10 V

D. 12.5 V

Correct Answer: B. 8 V

Question 4

Using the e^jωt convention, what is the impedance of a capacitor?

A. jωC

B. 1/(jωC)

C. jωL

D. 1/R

Correct Answer: B. 1/(jωC)

Question 5

A discharged capacitor charges toward 12 V with a time constant of 1 s. What is its voltage at t=1\text{ s}?

A. Approximately 4.42 V

B. Approximately 6.00 V

C. Approximately 7.59 V

D. Approximately 12.00 V

Correct Answer: C. Approximately 7.59 V

Question 6

A voltage gain of 0.5 corresponds to what decibel gain when the impedance factor cancels?

A. −6.02 dB

B. −3.01 dB

C. 3.01 dB

D. 6.02 dB

Correct Answer: A. −6.02 dB

Question 7

With I₁ and I₂ directed into a two-port, which ABCD equation uses the standard transmission convention?

A. [V₁\ I₁]^T=[A\ B;C\ D][V₂\ I₂]^T

B. [V₁\ I₁]^T=[A\ B;C\ D][V₂\ -I₂]^T

C. [I₁\ I₂]^T=[A\ B;C\ D][V₁\ V₂]^T

D. [V₁\ V₂]^T=[A\ B;C\ D][I₁\ I₂]^T

Correct Answer: B. [V₁\ I₁]^T=[A\ B;C\ D][V₂\ -I₂]^T

Question 8

Which h-parameter has units of siemens?

A. h₁₁

B. h₁₂

C. h₂₁

D. h₂₂

Correct Answer: D. h₂₂

Question 9

A meter reads 4.00 V and has accuracy ±(1% of reading + 1 count) on a range with 0.01 V resolution. What is the limiting uncertainty?

A. ±0.01 V

B. ±0.04 V

C. ±0.05 V

D. ±0.40 V

Correct Answer: C. ±0.05 V

Question 10

Why must an ordinary earth-referenced oscilloscope ground clip not be attached to an arbitrary live node?

A. It increases the oscilloscope bandwidth

B. It may short the node to protective earth

C. It changes the probe from ×10 to ×1

D. It prevents the trigger from operating

Correct Answer: B. It may short the node to protective earth