Electrical Quantities, Components and Circuit Models

Introduction

Electrical circuit analysis begins with a small set of physical quantities and idealized models. Charge describes the electrical property carried by matter; current describes the rate at which charge moves; voltage describes energy difference per unit charge; power describes the rate of energy transfer; and energy describes accumulated work. These quantities provide the language used to analyze resistors, sources, sensors, motors, electronic devices, and power systems.

A circuit model is not the physical circuit itself. It is a representation selected for a particular purpose. An electrical cable may be modeled as an ideal conductor in one analysis, but as a resistance and inductance in another. The quality of an analysis therefore depends both on mathematical technique and on whether the model is appropriate.

Learning Objectives

After studying this chapter, you should be able to:

• define charge, current, voltage, power, and energy with correct units;
• assign reference directions and apply the passive sign convention;
• distinguish instantaneous, average, and accumulated quantities;
• model ideal and practical voltage and current sources;
• describe resistors and recognize basic nonlinear behavior;
• distinguish lumped from distributed models and linear from nonlinear, time-invariant from time-varying models;
• select a suitable circuit model for an engineering application and assess its limitations.

1. Electrical Charge and Current

1.1 Electric charge

Electric charge is a fundamental property of matter. It may be positive or negative and is measured in coulombs (C). The charge of an electron is approximately −1.602 × 10⁻¹⁹ C, while the charge of a proton has the same magnitude and positive sign. In circuit analysis, charge is usually treated as a continuous variable even though matter is quantized at the microscopic level.

The symbol q(t) denotes charge as a function of time. If charge changes by an amount Δq during a time interval Δt, the average current is

where I_avg is in amperes (A), Δq is in coulombs, and Δt is in seconds. Since 1 A = 1 C/s, current is the rate of charge flow.

The instantaneous current is the limiting form of average current:

This definition applies whenever q(t) is differentiable. Conversely, charge transferred between t₁ and t₂ is

The integral has units A·s, which is equivalent to C.

Current is a signed quantity. An arrow placed on a circuit branch specifies a reference direction. A positive calculated value means that actual conventional current agrees with the reference arrow; a negative value means that it flows opposite to the reference direction. Conventional current is defined as the motion of positive charge, so it is opposite to the average motion of electrons in a metallic conductor.

Worked Example 1: Charge from a constant current

A current of 2.5 A flows through a conductor for 8 s. The transferred charge is

Thus, 20 C of charge crosses the chosen reference surface. The result assumes that current is constant during the interval.

2. Voltage and Reference Polarity

2.1 Voltage

Voltage is the difference in electric potential between two points. The voltage from point a to point b is defined as energy change per unit charge:

where v_ab is measured in volts (V), w is energy in joules (J), and q is charge in coulombs. Since 1 V = 1 J/C, a voltage of 5 V corresponds to 5 J of energy change per coulomb according to the selected sign convention.

Voltage is always measured between two points. A single isolated point does not have a meaningful voltage unless a reference point is specified. In circuit diagrams, one node is often designated as ground or reference and assigned 0 V. This is a choice of reference, not a claim that the reference node has no physical electric potential.

A voltage polarity marked + at terminal a and − at terminal b means

If a calculation gives v_ab = −3 V, then point a is actually 3 V below point b relative to the assigned polarity.

2.2 Energy and voltage

For a constant voltage, the energy associated with transferring charge q is

For a variable voltage, the differential relation is dw = v dq, and total energy is obtained by integration. Voltage is not energy: voltage is energy per unit charge.

3. Power and Energy

3.1 Instantaneous power

The instantaneous power associated with a two-terminal element is

where v(t) and i(t) must be defined using compatible voltage polarity and current direction. Power is measured in watts (W), where 1 W = 1 J/s.

Power indicates the rate of energy transfer. The energy transferred from t₁ to t₂ is

The sign of power conveys physical meaning. Under the passive sign convention, p > 0 means the element absorbs power, while p < 0 means the element delivers power to the external circuit. “Absorbs” means that energy enters the element according to the selected reference directions; it does not necessarily mean that the energy is permanently lost.

3.2 Passive sign convention

The passive sign convention is obtained by placing the current reference arrow into the terminal marked positive for the voltage reference. With this assignment,

is positive for absorption. If current enters the negative terminal instead, the power expression for the same physical element is

when v is defined from the positive to the negative terminal.

A source can have positive power under the passive convention if it is being charged, as with a battery receiving energy. A source can have negative power when it supplies energy. The sign is determined by the reference directions and operating condition, not by the label “source” or “load.”

Worked Example 2: Power sign

A 12 V element has a current of 0.40 A entering its positive terminal. Therefore,

The element absorbs 4.8 W. If the same 0.40 A instead leaves the positive terminal, its power is −4.8 W and it delivers 4.8 W to the rest of the circuit.

3.3 Energy in practical units

Electrical energy is measured in joules, but utility systems commonly use kilowatt-hours:

For a constant power P operating for time Δt, energy is w = PΔt. The units must be compatible; for example, watts multiplied by hours gives watt-hours, not joules unless the hours are converted to seconds.

4. Ideal Circuit Elements

An ideal element is a mathematical model with specified terminal behavior. Idealization makes analysis tractable, but every model has a range of validity.

4.1 Ideal voltage source

An ideal voltage source maintains a prescribed voltage regardless of the current through it. Its current is determined by the external circuit. For a source with voltage V_s,

The source may theoretically deliver or absorb unlimited power. This is physically impossible, so the ideal voltage source is an approximation. It is useful when the source internal resistance is negligible compared with the connected circuit resistance.

An ideal voltage source cannot be connected in parallel with another ideal voltage source of a different voltage without creating an inconsistent model. Parallel sources with equal voltage may be combined only when their other constraints are compatible.

4.2 Ideal current source

An ideal current source maintains a prescribed current regardless of its terminal voltage:

Its voltage is determined by the external circuit, and its theoretical voltage and power limits are unbounded. An ideal current source is a useful approximation when the source’s output resistance is very large compared with the connected load resistance.

Ideal current sources cannot generally be connected in series if they impose different currents.

4.3 Practical sources

Real sources have internal limitations. A practical voltage source is commonly modeled as an ideal voltage source in series with an internal resistance R_s. If current i leaves the positive source terminal, the terminal voltage is

where V_oc is the open-circuit voltage and R_s is the source resistance. This model predicts a voltage drop as load current increases.

A practical current source is commonly modeled as an ideal current source in parallel with an internal resistance R_p. The terminal current is divided between the external load and R_p. The two practical source models can be transformed into one another when their parameters satisfy

and the resistance values are equal. This equivalence applies at the external terminals under the stated linear-resistive assumptions; it does not mean the internal physical construction is identical.

4.4 Resistors

An ideal resistor obeys Ohm’s law:

where R is resistance in ohms (Ω), v is voltage in volts, and i is current in amperes. The relation assumes a linear, time-invariant resistor with a specified constant resistance. Combining Ohm’s law with p = vi gives

These forms apply to a resistor using the passive sign convention. For R > 0, p ≥ 0, so an ideal resistor absorbs power. The absorbed energy is converted primarily into thermal energy.

Resistance is related to material and geometry by

where ρ is resistivity in Ω·m, ℓ is conductor length in m, and A is cross-sectional area in m². This relation assumes a uniform material, uniform cross section, and conditions under which the material resistivity can be treated as constant.

Worked Example 3: Resistor power

A 24 V voltage is applied across a 120 Ω resistor. The current is

The power absorbed is

Equivalently, p = v²/R = 24²/120 = 4.8 W. The resistor must be selected with a power rating greater than the expected dissipation, with an appropriate engineering margin.

5. Nonlinear and Time-Varying Components

5.1 Linear behavior

An element is linear if it satisfies both homogeneity and additivity. If an input x produces output y, homogeneity requires that multiplying x by a constant multiplies y by the same constant. Additivity requires that the response to x₁ + x₂ equals the sum of the responses to x₁ and x₂. Together, these properties imply superposition.

A resistor with constant R is linear because v = Ri. A capacitor and inductor are also linear under ideal constant-parameter models, although they introduce energy storage and differential equations.

5.2 Nonlinear behavior

A nonlinear element does not obey a proportional relationship over its complete operating range. Examples include semiconductor diodes, transistor junctions, incandescent-lamp filaments, magnetic cores near saturation, and varistors.

For a nonlinear two-terminal element, the relationship may be written generally as

A useful local approximation is the incremental resistance:

evaluated at a particular operating point. This is a slope, not necessarily the ratio v/i. The ratio v/i is sometimes called the static or dc resistance. These quantities are equal only for a straight-line relationship through the origin or in special circumstances.

For example, a diode may conduct very little current at low forward voltage and much more current after its junction region becomes strongly forward biased. A piecewise-linear model can approximate this behavior using a threshold voltage and a series resistance, but the approximation is valid only over the intended operating range.

5.3 Time invariance

A model is time-invariant if shifting the time of an input shifts the output by the same amount without changing the element parameters. A resistor with constant R is time-invariant. A resistor whose value changes because of heating, switching, or control action is time-varying:

A component can be linear but time-varying, or nonlinear but time-invariant. These classifications are independent. For example, v = R(t)i is linear in i but time-varying, while v = i³ is nonlinear but time-invariant.

6. Lumped and Distributed Circuit Models

6.1 Lumped-parameter assumption

A lumped circuit model assumes that voltage and current can be defined at discrete elements and that electromagnetic propagation within an element is fast enough to neglect spatial variation. Kirchhoff’s voltage and current laws are then applied to nodes and loops.

The approximation is generally appropriate when physical dimensions are much smaller than the relevant electromagnetic wavelength and when propagation delay is insignificant relative to the signal rise time or period. It is often suitable for ordinary low-frequency circuits and short interconnections.

6.2 Distributed-parameter behavior

A transmission line must be treated as distributed when resistance, inductance, capacitance, and conductance are significant continuously along its length. The voltage and current then depend on both position and time, such as v(x,t) and i(x,t). Long cables, high-speed digital interconnects, microwave circuits, and power lines may require this model.

The lumped model is not “wrong” in itself; it is an approximation whose accuracy depends on frequency, dimensions, material properties, and required precision. Ignoring distributed effects can produce incorrect predictions of reflections, delay, impedance, and signal integrity.

7. Reference Directions, Units, and Measurement Practice

A circuit solution should state reference polarities and current arrows before equations are written. Negative results are valid and informative. They indicate disagreement between the selected reference direction and the actual direction, not a failed analysis.

The SI prefixes commonly used in circuit analysis include:

Uppercase and lowercase symbols matter: M means mega, while m means milli. Unit consistency is an essential error check. For example, multiplying volts by amperes gives watts, and multiplying watts by seconds gives joules.

In laboratory work, meters also impose limits. A voltmeter is connected in parallel and should have high input resistance; an ammeter is connected in series and should have low resistance. Connecting an ammeter directly across an ideal voltage source can imply an extremely large current and may damage equipment. Measurement instruments therefore have nonideal behavior and should be included in a refined model when accuracy requires it.

8. Applied Case: Modeling a Battery-Powered Sensor

Background

Consider an environmental sensor powered by a battery. The sensor requires approximately 3.3 V and draws 80 mA during operation. The battery has an open-circuit voltage of 3.7 V and an estimated internal resistance of 2.5 Ω.

Problem

Determine the approximate terminal voltage and power delivered to the sensor when it draws 80 mA. Decide whether the battery model is adequate for a first design estimate.

Analysis and solution

Use the practical voltage-source model:

Substituting V_oc = 3.7 V, i = 0.080 A, and R_s = 2.5 Ω gives

The sensor power is approximately

The internal battery loss is

The source supplies approximately 0.296 W, equal to the sum of load power and internal loss in this simplified model.

Outcome and engineering interpretation

The estimated terminal voltage exceeds 3.3 V, but the margin is only about 0.20 V. As the battery discharges, its open-circuit voltage may decrease and its internal resistance may increase. Temperature, pulse current, wiring resistance, and regulator dropout voltage may further reduce the usable margin. A robust design would therefore examine the minimum battery voltage, worst-case resistance, transient current, and the sensor’s permitted voltage range.

Learning points

1. A real battery is better represented by an ideal source plus internal impedance than by an ideal source alone.
2. Internal resistance causes both voltage sag and power loss.
3. A nominal operating point is not a complete design verification.
4. Model selection should reflect the decision being made: a first estimate may use a resistance, while a detailed battery-management design may require a nonlinear, temperature- and state-of-charge-dependent model.

9. Model Selection, Limitations, and Professional Practice

Circuit models should be selected by identifying the required outputs, operating frequency, physical scale, signal amplitude, temperature range, and acceptable error. An ideal model is valuable when it isolates the dominant behavior. It becomes misleading when neglected effects are comparable to the effect being studied.

Important limitations include source resistance, component tolerance, temperature dependence, parasitic capacitance and inductance, nonlinear saturation, noise, aging, and measurement loading. Component datasheets normally specify ratings and conditions such as maximum voltage, current, power, temperature, and frequency. Exceeding these ratings can create safety hazards, permanent damage, or unreliable operation.

Ethical engineering practice requires transparent assumptions, conservative ratings where appropriate, reproducible calculations, and clear communication of uncertainty. A circuit that works in a nominal simulation may fail in production if tolerances and fault conditions are ignored. Safety-related systems require applicable standards, protective isolation, current limitation, fusing, grounding, and documented verification.

Computer-aided circuit simulators are useful for testing models, sweeping parameters, and examining transient behavior. They do not remove the need to understand the model. Artificial intelligence and automated design tools can assist with parameter extraction or circuit exploration, but engineers remain responsible for validating equations, recognizing unrealistic assumptions, protecting design data, and confirming results experimentally.

Chapter Summary

Charge is measured in coulombs, and current is the time rate of charge flow, i(t) = dq/dt. Voltage is energy change per unit charge and requires a reference polarity between two points. Power is the rate of energy transfer, p = vi, and energy is the time integral of power. Under the passive sign convention, current enters the positive voltage terminal and positive power denotes absorption.

Ideal voltage sources impose voltage and ideal current sources impose current, whereas practical sources include internal resistance and therefore exhibit loading effects. A resistor obeys v = Ri under a linear, constant-parameter model, with power p = i²R = v²/R. Nonlinear devices require more general characteristic relations, and incremental resistance describes local slope behavior.

Circuit models may be lumped or distributed, linear or nonlinear, and time-invariant or time-varying. These classifications describe assumptions, not permanent labels for all operating conditions. Correct reference directions, SI units, realistic component ratings, and explicit assumptions are essential foundations for reliable circuit analysis.

Key Terms

Charge: Electrical property measured in coulombs.

Current: Rate of flow of electric charge, measured in amperes.

Voltage: Energy difference per unit charge between two points.

Potential difference: Another name for voltage between specified points.

Power: Rate of energy transfer, measured in watts.

Energy: Accumulated ability to perform work, measured in joules.

Reference direction: Chosen positive direction for a circuit variable.

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

Ideal voltage source: Model that maintains prescribed voltage regardless of current.

Ideal current source: Model that maintains prescribed current regardless of voltage.

Internal resistance: Resistance used to represent losses within a practical source.

Resistance: Opposition to current, measured in ohms.

Ohm’s law: Relation v = Ri for an ideal linear resistor.

Linear element: Element satisfying homogeneity and additivity.

Nonlinear element: Element whose terminal relation is not globally proportional.

Incremental resistance: Local slope dv/di of a nonlinear characteristic.

Time-invariant model: Model whose parameters do not change with time.

Lumped model: Model in which circuit variables are represented at discrete elements.

Distributed model: Model in which voltage and current vary continuously with position.

Review Questions

Short Answer Questions

1. Define electric current and state its SI unit.
2. Why must voltage be specified between two points?
3. State the passive sign convention.
4. What is the difference between ideal and practical voltage sources?
5. Define incremental resistance for a nonlinear element.

Long Answer Questions

1. Explain charge, current, voltage, power, and energy, including their units and defining equations.
2. Compare ideal voltage-source and ideal current-source models, including their limitations.
3. Derive the three resistor power expressions from p = vi and v = Ri.
4. Explain the distinction between linear/nonlinear and time-invariant/time-varying behavior, using examples.
5. Discuss when a lumped circuit model should be replaced by a distributed model.

Analytical / Application Questions

1. A 10 V source supplies a 250 Ω resistor. Calculate current and resistor power, and state whether the resistor absorbs or delivers power.
2. A practical 12 V source has internal resistance 1 Ω and supplies a 5 Ω load. Determine load current, load voltage, and internal power loss.
3. A sensor draws 100 mA from a battery modeled as 4.0 V in series with 3 Ω. Determine the terminal voltage and explain two reasons why the actual value may differ.

Multiple Choice Questions with Answers

Question 1

A current of 3 A flows for 4 s. What charge is transferred?

A. 0.75 C

B. 1.33 C

C. 7 C

D. 12 C

Correct Answer: D. 12 C

Question 2

Under the passive sign convention, positive power for an element means that the element:

A. absorbs energy

B. has zero voltage

C. delivers energy

D. has zero current

Correct Answer: A. absorbs energy

Question 3

A 20 V voltage is applied to a 10 Ω resistor. What power does it absorb?

A. 2 W

B. 20 W

C. 40 W

D. 200 W

Correct Answer: C. 40 W

Question 4

Which model maintains a prescribed voltage regardless of its current?

A. Ideal current source

B. Ideal voltage source

C. Ideal resistor

D. Practical current source only

Correct Answer: B. Ideal voltage source

Question 5

The SI unit equivalent to one watt is:

A. C/s

B. J/C

C. J/s

D. V/s

Correct Answer: C. J/s

Question 6

For a practical voltage source modeled by an ideal source V_oc in series with R_s, increasing load current generally causes terminal voltage to:

A. increase by iR_s

B. decrease by iR_s

C. remain exactly V_oc

D. become zero immediately

Correct Answer: B. decrease by iR_s

Question 7

Which relation describes an ideal linear resistor?

A. v = Ri

B. v = R/i

C. i = Rv²

D. v = i + R

Correct Answer: A. v = Ri

Question 8

The quantity dv/di evaluated at a point on a nonlinear characteristic is called:

A. open-circuit voltage

B. static power

C. incremental resistance

D. source resistance only

Correct Answer: C. incremental resistance

Question 9

Which statement correctly describes a distributed circuit model?

A. All circuit dimensions are ignored

B. Voltage and current may vary with position and time

C. It can contain no resistance

D. It is valid only for dc circuits

Correct Answer: B. Voltage and current may vary with position and time

Question 10

A resistor carries 0.5 A and has resistance 8 Ω. What power does it absorb?

A. 2 W

B. 4 W

C. 8 W

D. 16 W

Correct Answer: A. 2 W