Electrical Quantities, Components and Circuit Models

Introduction

Electrical engineering begins with a small set of physical quantities and idealized models. Charge describes the basic electrical property of matter; current describes the motion of charge; voltage describes energy difference; and resistance, capacitance, and inductance describe how components oppose, store, or release energy. Power and energy connect circuit behavior to practical questions such as heating, battery life, equipment ratings, and safety.

A circuit model is not the physical circuit itself. It is a mathematically useful representation that retains important behavior while neglecting less important details. The quality of a model depends on its purpose, operating range, and required accuracy. For example, a battery may be modeled as an ideal voltage source for a basic calculation, but as a voltage source in series with internal resistance when voltage sag and heating are important.

Learning Objectives

By the end of this chapter, you should be able to:

• define charge, current, voltage, resistance, conductance, capacitance, inductance, power, and energy;
• distinguish direct current (DC) from alternating current (AC);
• apply constitutive relationships such as Ohm’s law and the capacitor and inductor laws;
• interpret characteristic curves and component ratings;
• model ideal and practical sources, loads, and passive components;
• select a suitable model for a stated engineering application; and
• evaluate a simple circuit in terms of voltage, current, power, and energy.

1. Electrical Charge and Current

Electric charge

Electric charge is a fundamental property of matter. It is measured in coulombs (C). The charge on an electron is approximately −1.602 × 10⁻¹⁹ C, while that on a proton is equal in magnitude and positive. In circuit theory, charge is usually treated as a continuous variable even though matter is quantized at the microscopic level.

The symbol q commonly denotes charge. If charge changes with time, the associated current is

where i(t) is current in amperes (A), q(t) is charge in coulombs, and t is time in seconds. Equivalently, 1 A = 1 C/s. The total charge transferred during an interval is

This relationship applies to any time-varying current, provided the current is defined consistently with the chosen reference direction.

Conventional current

In metallic conductors, electrons move from lower electric potential toward higher electric potential, but circuit current is defined in the direction positive charge would move. This convention, called conventional current, is used throughout electrical engineering. Confusing electron flow with conventional current can lead to incorrect polarity conclusions, although the numerical circuit laws remain consistent if one convention is used consistently.

A current reference direction is a choice, not a claim that current must actually flow in that direction. A calculated negative current means that the actual current is opposite to the assumed reference.

2. Voltage and Electrical Potential Energy

Voltage is energy transferred per unit charge. The voltage between points a and b is

where v_ab is voltage in volts (V), w is energy in joules (J), and q is charge in coulombs. Thus, 1 V = 1 J/C. Voltage is always measured between two points; a single node voltage is meaningful only relative to a selected reference node or ground.

For a finite charge transfer under appropriate conditions,

Voltage polarity is indicated by plus and minus signs. If current enters a component at the terminal marked positive, the passive sign convention is being used. This convention makes absorbed power positive when the component absorbs energy.

Kirchhoff’s voltage law

In a lumped circuit, the algebraic sum of voltages around any closed path is zero:

This is a consequence of conservation of energy under the lumped-circuit approximation. It is applicable when electromagnetic propagation and distributed-field effects can be neglected over the circuit dimensions and operating frequency.

3. Direct Current and Alternating Current

A direct current has a constant direction and, in the simplest case, constant magnitude. A direct voltage may be written as v(t) = V_DC. A battery supplying a steady load is often approximated as a DC source, although real batteries exhibit time-dependent voltage and internal resistance.

An alternating quantity changes magnitude and reverses direction periodically. A sinusoidal voltage can be represented as

where V_m is peak voltage in volts, ω = 2πf is angular frequency in rad/s, f is frequency in hertz, and φ is phase angle in radians. The period is T = 1/f.

For a sinusoid with zero average, the root-mean-square value is

and similarly I_rms = I_m/√2. RMS values are useful because a resistor supplied by a sinusoidal voltage dissipates the same average power as it would under a DC voltage of equal RMS value. This equivalence applies to resistive heating, not necessarily to all circuit effects.

AC systems permit voltage transformation using transformers, efficient generation, and practical distribution at high voltage. DC systems are central to batteries, electronic circuits, photovoltaic systems, and data centers. Modern power systems commonly combine both: AC distribution may feed power electronic converters that produce controlled DC.

4. Resistance and Conductance

Resistance measures opposition to current. For an ohmic resistor, Ohm’s law is

where v is voltage in volts, i is current in amperes, and R is resistance in ohms (Ω). The law describes a linear relationship only when the resistance is constant over the relevant voltage, current, temperature, and frequency range.

The conductance of a component is the reciprocal of resistance:

where G is measured in siemens (S). The current-voltage form is

Conductance is particularly convenient for parallel-connected resistive elements because conductances add directly.

Resistance of a uniform conductor

For a uniform material,

where ρ is resistivity in Ω·m, ℓ is conductor length in meters, and A is cross-sectional area in square meters. The equation assumes uniform material, approximately uniform current distribution, and a temperature at which ρ is known. At high frequency, the skin effect may make current distribution nonuniform, so this simple expression may not predict effective AC resistance accurately.

Characteristic curve and nonlinearity

A resistor’s characteristic curve plots voltage against current. An ideal resistor produces a straight line through the origin. The slope of a v-i plot is resistance. Incandescent lamps, thermistors, diodes, and semiconductor devices are not generally ohmic over their full operating ranges. Their slope may vary with current, voltage, or temperature.

For a nonlinear element, a useful incremental or dynamic resistance near an operating point is

This is a local slope and should not be confused with the ratio v/i, which is sometimes called static resistance.

5. Capacitance

Capacitance is the ability to store energy in an electric field. For a two-terminal capacitor,

where C is capacitance in farads (F), q is charge in coulombs, and v is voltage in volts. For a time-varying voltage,

This relation assumes an ideal capacitor with constant capacitance and negligible leakage, equivalent series resistance, and dielectric loss.

The energy stored in an ideal capacitor is

A capacitor’s voltage cannot change instantaneously if its current is finite, because an instantaneous voltage change would require an impulse of current. Capacitors therefore tend to oppose rapid voltage changes.

For a parallel-plate capacitor with a uniform dielectric,

where ε is permittivity in F/m, A is plate area, and d is plate separation. Fringing fields and nonuniform dielectrics make this an approximation.

Real capacitors have leakage resistance, equivalent series resistance (ESR), maximum voltage, ripple-current, temperature, and frequency limits. A practical model may place a leakage resistor in parallel with an ideal capacitor and an ESR in series with the combination.

6. Inductance

Inductance is associated with energy storage in a magnetic field. For an ideal inductor,

where L is inductance in henries (H). The stored energy is

The relation assumes constant inductance and negligible winding resistance, core loss, saturation, and parasitic capacitance. An inductor’s current cannot change instantaneously under a finite voltage. A sudden current change would require an impulse of voltage.

A practical inductor includes winding resistance and may show nonlinear inductance. In a magnetic core, saturation reduces incremental inductance as current increases. At high frequency, parasitic capacitance and core losses become significant. A practical model may therefore contain a series resistance, nonlinear core behavior, and frequency-dependent loss elements.

7. Power and Energy

Instantaneous electrical power is

where positive power means absorption under the passive sign convention. The unit is the watt (W), with 1 W = 1 J/s. If current enters the positive-voltage terminal, the component absorbs power; if current enters the negative terminal, it delivers power according to the chosen reference.

For a resistor,

These three forms are equivalent for an ohmic resistor with R ≠ 0. They should not be applied indiscriminately to capacitors, inductors, or nonlinear components.

Energy transferred over an interval is

The commercial unit kilowatt-hour is an energy unit: 1 kWh = 3.6 × 10⁶ J.

For sinusoidal AC, the average power in a general load is

where φ is the phase difference between voltage and current. The factor cosφ is the power factor for a sinusoidal steady-state load. For a pure resistor, φ = 0 and power factor is unity. An ideal inductor or capacitor absorbs zero average power over a complete cycle, although it exchanges instantaneous energy with the source.

8. Sources, Loads, and Circuit Models

A source supplies electrical energy, while a load converts or stores it. The distinction depends on the operating condition and reference directions: a battery can absorb power while charging and deliver power while discharging.

Ideal voltage source

An ideal voltage source maintains a specified terminal voltage regardless of current. Its internal resistance is zero. This model is useful for an idealized supply or a stiff network, but an ideal voltage source cannot be connected in parallel with a different ideal voltage source unless their voltages are compatible. Its current is determined by the external circuit.

Ideal current source

An ideal current source maintains a specified current regardless of terminal voltage. Its internal resistance is infinite. It is a useful mathematical model, but a real current source has compliance limits: it can maintain its target current only over a finite voltage range.

Practical voltage source

A common practical model is an ideal voltage source V_s in series with internal resistance R_s. With a load R_L, the load current is

and the load voltage is

The model assumes DC or low-frequency operation and constant resistances. Internal power loss is I²R_s, so the source rating must account for heating.

Practical current source

A practical current source is often modeled as an ideal current source in parallel with a finite resistance. As load voltage approaches the compliance limit, the output current departs from its nominal value. Characteristic curves therefore show a nearly horizontal current-versus-voltage region rather than an indefinitely constant current.

Loads and ratings

A load rating states the range of conditions in which safe operation is expected. Important ratings include maximum voltage, current, power, temperature, frequency, insulation level, and—in capacitors and inductors—ripple current or saturation current. A rating is not necessarily the normal operating value. Continuous operation at a maximum rating may reduce reliability, especially when cooling, ambient temperature, tolerances, and transients are considered.

9. Kirchhoff’s Current Law and Interconnection Models

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

It follows from conservation of charge in the lumped-circuit model. For example, if 2 A enters a node and 0.7 A leaves through one branch, the remaining branch current leaving is 1.3 A.

Ideal wires are assigned zero resistance and identical potential throughout. Real conductors have resistance, inductance, thermal limits, contact resistance, and electromagnetic coupling. The ideal wire model is appropriate only when voltage drop and parasitic effects are negligible relative to the required accuracy.

A load can also be represented by its input resistance, impedance, or a more detailed equivalent circuit. The simplest model may be adequate for power budgeting, while a frequency-dependent model is required for filters, motors, communications equipment, and switching converters.

10. Worked Examples

Example 1: Charge and average current

A charge of 0.24 C passes through a conductor in 12 ms. The average current is

Thus, the average current is 20 A. This calculation does not imply that the instantaneous current was constant.

Example 2: Practical source and load

A 12 V source has internal resistance 0.40 Ω and supplies a 5.6 Ω load. The current is

The load voltage is

The internal loss is

The load power is P_L = V_LI = (11.2)(2.00) = 22.4 W. The source delivers 24.0 W, and 1.60 W is dissipated internally, confirming power balance.

Example 3: Capacitor energy

A 100 μF capacitor is charged to 24 V. Its stored energy is

Although the energy is small, a high-voltage capacitor can present a shock and arc hazard because it may retain charge after disconnection.

11. Applied Case: Selecting a Model for a Battery-Powered Sensor

Background

Consider an illustrative wireless environmental sensor powered by a 3.7 V lithium-ion cell. The electronics require 80 mA during measurement and 5 mA in sleep mode. A regulator and wiring introduce additional losses.

Problem

The designer must estimate operating voltage, battery power, and whether an ideal-source model is sufficiently accurate.

Analysis

If the cell is initially modeled as an ideal 3.7 V source, the measurement power at the electronics is approximately P = VI = (3.7)(0.080) = 0.296 W. However, a real cell has internal resistance. If its internal resistance is 0.25 Ω, the voltage drop during measurement is

This drop is modest for this load, but it becomes more important at higher current or near the battery’s minimum state of charge. A regulator may also require a minimum input voltage and dissipate power. The battery model should therefore include internal resistance, voltage variation with state of charge, temperature dependence, and protection-circuit limits.

Approach and outcome

For an early energy estimate, a nominal-voltage model is adequate. For regulator selection and end-of-discharge prediction, a practical battery model is required. Software simulation can use a voltage source, series resistance, and a state-of-charge-dependent voltage table. Laboratory measurements should verify current pulses, temperature rise, and actual terminal voltage.

Learning points

1. A model should be selected for the engineering question, not merely for convenience.
2. Ratings and transient behavior may matter more than nominal values.
3. Internal resistance converts stored chemical energy into heat and causes load-dependent voltage sag.
4. Measurements are necessary when model uncertainty can affect safety, reliability, or product performance.

12. Limitations, Safety, and Professional Practice

The lumped-circuit model neglects spatial electromagnetic effects. It becomes less reliable when conductor dimensions are not small compared with the electromagnetic wavelength, when switching edges are very fast, or when distributed transmission-line behavior is important. Real components also exhibit tolerances, parasitic elements, temperature dependence, aging, and manufacturing variation.

Engineers should distinguish nominal, typical, maximum, and minimum values in datasheets. Designs should include appropriate derating, fault protection, fuses or circuit breakers, insulation coordination, grounding, creepage and clearance, and protection against stored energy. Capacitors and inductors require special attention during shutdown because stored energy can remain after the source is removed.

Electrical work must comply with applicable standards, laboratory procedures, and occupational safety rules. Never infer that a low nominal voltage is automatically safe: available current, energy storage, wet conditions, and fault paths affect hazard. Measurement instruments also affect circuits through input resistance, capacitance, bandwidth, and grounding connections.

Contemporary systems increasingly combine passive components with power electronics, sensors, embedded control, and digital twins. Automated simulation and AI-assisted design can help explore component values and detect inconsistent assumptions, but the engineer remains responsible for validating models, checking units, reviewing boundary conditions, and confirming results experimentally. An apparently plausible simulation is not evidence that a circuit is safe.

Chapter Summary

Charge is measured in coulombs, and current is the time rate of charge flow. Voltage is energy per unit charge, while resistance and conductance describe the relationship between voltage and current in resistive elements. Capacitance stores energy in an electric field and relates current to voltage change; inductance stores energy in a magnetic field and relates voltage to current change. Power is the rate of energy transfer, and energy is the time integral of power.

DC has fixed direction, whereas AC varies with time and commonly reverses direction periodically. RMS values permit meaningful comparison of sinusoidal resistive heating with DC. Ideal voltage and current sources provide simplified models; practical sources include internal resistance and compliance limits. Characteristic curves reveal whether devices are linear or nonlinear, and ratings define safe operating boundaries. Kirchhoff’s laws express conservation of charge and energy within the lumped-circuit approximation. Effective engineering requires choosing a model appropriate to the purpose, checking units and power balance, accounting for parasitics and ratings, and validating important assumptions by measurement.

Key Terms

Charge: Fundamental electrical property measured in coulombs.

Current: Rate of charge flow, measured in amperes.

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

Potential difference: Voltage between two specified points.

Resistance: Opposition to current, measured in ohms.

Conductance: Reciprocal of resistance, measured in siemens.

Capacitance: Charge stored per unit voltage, measured in farads.

Inductance: Flux-linkage-related property that opposes current change, measured in henries.

Power: Rate of energy transfer, measured in watts.

Energy: Ability to perform work or accumulated power transfer, measured in joules.

DC: Electrical quantity with constant direction, commonly constant magnitude.

AC: Electrical quantity that varies with time and commonly reverses direction.

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

Ideal source: Source maintaining its specified voltage or current without internal limitation.

Internal resistance: Resistance representing voltage drop and loss within a practical source.

Characteristic curve: Graph showing a component’s electrical response, commonly voltage versus current.

RMS value: Effective value of an AC quantity for specified power equivalence.

Compliance voltage: Voltage range over which a practical current source maintains its specified current.

Parasitic element: Unintended resistance, capacitance, or inductance associated with a real component.

Rating: Specified operating limit or condition for safe and reliable use.

Review Questions

Short Answer Questions

1. Define electric current and state its SI unit.
2. What does voltage represent physically?
3. State the ideal capacitor current-voltage relationship.
4. Distinguish between an ideal and a practical voltage source.
5. Why can a capacitor be hazardous after a circuit is switched off?

Long Answer Questions

1. Explain resistance, conductance, Ohm’s law, and the limitations of a linear resistor model.
2. Compare DC and sinusoidal AC, including period, frequency, peak value, RMS value, and applications.
3. Explain how capacitors and inductors store energy and why their voltages or currents cannot change instantaneously under finite conditions.
4. Discuss ideal and practical voltage and current sources, including characteristic curves and compliance limitations.
5. Explain the significance of component ratings, parasitic effects, tolerances, and derating in engineering design.

Analytical / Application Questions

1. A 24 V practical source has a series resistance of 0.8 Ω and supplies a 11.2 Ω load. Calculate the load current, load voltage, internal power loss, and load power.
2. A 470 μF capacitor is charged to 18 V. Calculate its stored energy and explain two safety precautions for discharging it.
3. An engineer models a battery as an ideal 3.7 V source for a low-current sensor but observes large voltage sag during a radio transmission pulse. Explain why the model fails and propose a more suitable model and measurements.

Multiple Choice Questions with Answers

Question 1

Which quantity is defined as the time rate of change of charge?

A. Voltage

B. Current

C. Power

D. Conductance

Correct Answer: B. Current

Question 2

A resistor carries 3 A when the voltage across it is 12 V. What is its resistance?

A. 0.25 Ω

B. 4 Ω

C. 9 Ω

D. 36 Ω

Correct Answer: B. 4 Ω

Question 3

Which relationship describes an ideal capacitor?

A. v = iR

B. p = i²R

C. i = C dv/dt

D. v = L di/dt

Correct Answer: C. i = C dv/dt

Question 4

What is the energy stored in an ideal inductor?

A. 1/2 Li²

B. 1/2 Cv²

C. vi

D. i²/R

Correct Answer: A. 1/2 Li²

Question 5

A sinusoidal voltage has a peak value of 100 V. Its RMS value is approximately:

A. 50 V

B. 70.7 V

C. 100 V

D. 141.4 V

Correct Answer: B. 70.7 V

Question 6

Under the passive sign convention, a component absorbs positive power when:

A. Current enters its negative-voltage terminal

B. Voltage and current are both zero

C. Current enters its positive-voltage terminal

D. Its resistance is infinite

Correct Answer: C. Current enters its positive-voltage terminal

Question 7

A practical voltage source is commonly modeled as:

A. An ideal voltage source in parallel with zero resistance

B. An ideal current source in series with zero resistance

C. An ideal voltage source in series with internal resistance

D. A capacitor in parallel with an inductor

Correct Answer: C. An ideal voltage source in series with internal resistance

Question 8

What does Kirchhoff’s current law express at a circuit node?

A. Conservation of charge

B. Conservation of temperature

C. Conservation of resistance

D. Conservation of frequency

Correct Answer: A. Conservation of charge

Question 9

A 10 Ω resistor carries 2 A. Its power dissipation is:

A. 5 W

B. 20 W

C. 40 W

D. 100 W

Correct Answer: C. 40 W

Question 10

Which statement about an ideal current source is correct?

A. It maintains specified current regardless of terminal voltage

B. It always has zero internal resistance

C. It cannot deliver power

D. Its current is determined only by the load resistance

Correct Answer: A. It maintains specified current regardless of terminal voltage