AC and DC Circuits

Learning Objectives

• Distinguish direct current (DC) from alternating current (AC) by waveform and behavior.
• Calculate RMS (root-mean-square) voltage and current for a sinusoidal AC signal.
• Explain impedance and how it generalizes resistance to AC circuits.
• Calculate real power in an AC circuit using the power factor.
• Compare series and parallel analysis techniques as applied to DC versus AC circuits.
• Identify practical situations where AC or DC is the natural choice.

Quick Answer

A DC (direct current) circuit has current flowing steadily in one direction — the kind supplied by batteries and used inside most electronic devices. An AC (alternating current) circuit has current that periodically reverses direction, following a sinusoidal wave — the kind delivered by wall outlets and power grids worldwide. The distinction matters because they require different analysis tools: DC circuits use plain Ohm's Law (V = IR) with real numbers, while AC circuits use impedance (a complex number combining resistance and reactance) because capacitors and inductors react differently to a constantly changing voltage than they do to a constant one. Power grids use AC because transformers can efficiently step AC voltage up and down for long-distance transmission, something DC cannot do as easily.

DC Circuits: The Basics

In a DC circuit, current flows in a single, constant direction, and voltage doesn't change sign over time (a battery is the classic source).

• Analysis tool: Ohm's Law, V = I × R, with real numbers.
• Techniques: voltage/current division, superposition, Thevenin's and Norton's theorems — all directly applicable.
• Typical sources: batteries, DC power supplies, solar cells.

AC Circuits: The Basics

In an AC circuit, voltage and current vary sinusoidally with time, reversing direction periodically.

v(t) = V_peak × sin(ωt), where ω = 2πf (angular frequency, f in Hz)

Because voltage is constantly changing, capacitors and inductors — which react to rate of change — behave very differently in AC than in DC. This gives rise to reactance:

• Inductive reactance: X_L = ωL (increases with frequency)
• Capacitive reactance: X_C = 1/(ωC) (decreases with frequency)

Combined with resistance, these form impedance, Z = R + jX, a complex number. The AC form of Ohm's Law is:

V = I × Z

RMS Values: Why We Don't Use Peak Voltage

Since AC voltage constantly changes, a single "peak" value doesn't tell you how much real power a load receives. The RMS (root-mean-square) value is the equivalent DC value that would deliver the same average power to a resistive load.

For a sine wave: V_rms = V_peak / √2 ≈ 0.707 × V_peak

Worked Example 1: A household AC supply has a peak voltage of 325.3V. Find the RMS voltage (this is the "230V" you see on nameplates in many countries).

V_rms = V_peak / √2 = 325.3 / 1.414 ≈ 230 V

This is why a "230V AC" outlet is described with a single number even though the instantaneous voltage swings from +325V to −325V — the RMS value is what determines actual heating/power effect, matching a 230V DC source.

Power in AC vs. DC Circuits

DC power: P = V × I (straightforward, since V and I don't change sign or lag each other).

AC power: P = V_rms × I_rms × cos(θ), where θ is the phase angle between voltage and current, and cos(θ) is called the power factor.

The phase angle appears because reactive components (capacitors, inductors) shift current out of phase with voltage — some of the power sloshes back and forth without doing useful work (called reactive power), and only the in-phase component (real power) does actual work.

Worked Example 2: An AC circuit has V_rms = 230V, I_rms = 4A, and a phase angle of 30° between voltage and current. Find the real power delivered.

P = V_rms × I_rms × cos(θ) = 230 × 4 × cos(30°) = 920 × 0.866 ≈ 796.7 W

Compare this to a purely resistive AC load (θ = 0°, cos θ = 1), which would deliver the full 920W — the phase shift caused by reactive components reduces the useful power delivered even though the same RMS voltage and current are present.

Impedance: Resistance's AC Generalization

Worked Example 3: A series AC circuit has a resistor R = 30Ω and an inductor with reactance X_L = 40Ω. Find the total impedance magnitude and the phase angle.

Z = R + jX_L = 30 + j40
|Z| = √(30² + 40²) = √(900 + 1600) = √2500 = 50 Ω
θ = arctan(X_L / R) = arctan(40/30) = arctan(1.333) ≈ 53.1°

If this impedance is fed by a 100V RMS source, the current magnitude is I = V/|Z| = 100/50 = 2A, lagging the voltage by 53.1° (since it's an inductive circuit, current lags voltage).

Series and Parallel Analysis: DC vs. AC

The same series/parallel rules apply in AC circuits, but with impedances (complex numbers) replacing plain resistances:

Rule	DC version	AC version
Series total	R_total = R1 + R2 + ...	Z_total = Z1 + Z2 + ... (complex addition)
Parallel total	1/R_total = 1/R1 + 1/R2 + ...	1/Z_total = 1/Z1 + 1/Z2 + ... (complex arithmetic)
Ohm's Law	V = IR	V = IZ (phasors)

Key Terms

Term	Definition
Direct current (DC)	Current that flows in one constant direction
Alternating current (AC)	Current that periodically reverses direction, typically sinusoidal
RMS value	The equivalent DC value that delivers the same average power as the AC waveform
Reactance (X)	Opposition to current change from capacitors (X_C) or inductors (X_L), frequency-dependent
Impedance (Z)	Combination of resistance and reactance, Z = R + jX, the AC generalization of resistance
Power factor	cos(θ), the fraction of apparent power that becomes real (useful) power
Phase angle (θ)	The angular offset between voltage and current waveforms in an AC circuit

Common Mistakes

1. Misconception: "The 230V (or 120V) rating on an outlet is the peak voltage." Why it's wrong: The rated voltage on power grids is always the RMS value, not the peak. The actual peak voltage is about 41% higher (V_peak = V_rms × √2). Correct: For a 230V RMS supply, the peak voltage is about 230 × 1.414 ≈ 325V; insulation and component ratings must account for this higher peak.

2. Misconception: "AC power is simply P = V_rms × I_rms, just like DC power." Why it's wrong: This ignores the phase angle between voltage and current caused by reactive components. Without the cos(θ) term, you calculate "apparent power," not real (useful) power. Correct: Real power in AC circuits is P = V_rms × I_rms × cos(θ); only for a purely resistive load (θ = 0°) does this reduce to the DC-like formula.

3. Misconception: "Impedance is just another name for resistance." Why it's wrong: Resistance is a real number and doesn't depend on frequency. Impedance is a complex number (Z = R + jX) that depends on frequency through the reactance terms X_L = ωL and X_C = 1/(ωC). Correct: Resistance is the real part of impedance; impedance also includes a reactive (imaginary) part that shifts phase and changes with frequency.

Comparison and Connections

Property	DC Circuits	AC Circuits
Current direction	Constant	Periodically reverses
Governing law	V = IR (real numbers)	V = IZ (complex/phasor numbers)
Key quantity	Resistance R	Impedance Z (R + jX)
Power formula	P = VI	P = V_rms × I_rms × cos(θ)
Typical sources	Batteries, DC supplies	Power grid, generators
Transformer compatible	No	Yes (enables efficient long-distance transmission)

Practice Questions

Recall 1: Write the RMS-to-peak relationship for a sinusoidal AC voltage. Answer guidance: V_rms = V_peak / √2 ≈ 0.707 × V_peak.

Recall 2: What quantity replaces resistance in AC circuit analysis, and what are its two components? Answer guidance: Impedance Z = R + jX, where R is resistance and X is reactance (inductive or capacitive).

Understanding 1: Explain why power grids use AC rather than DC for long-distance transmission. Answer guidance: Transformers can efficiently step AC voltage up (for low-loss transmission at high voltage/low current) and back down for safe use, using electromagnetic induction — a mechanism that doesn't work with constant DC voltage.

Understanding 2: Why does an AC circuit with reactive components deliver less real power than V_rms × I_rms would suggest? Answer guidance: Reactive components (capacitors, inductors) shift current out of phase with voltage, so part of the apparent power oscillates back and forth without doing net work; only the in-phase component (scaled by cos θ) is real (useful) power.

Application 1: A DC circuit has V = 24V and I = 2A. Find the power delivered. Answer guidance: P = V × I = 24 × 2 = 48 W.

Application 2: An AC circuit has V_rms = 120V and I_rms = 3A with a purely resistive load. Find both the peak voltage and the real power delivered. Answer guidance: V_peak = 120 × √2 ≈ 169.7V; since it's purely resistive, θ = 0°, so P = V_rms × I_rms = 120 × 3 = 360 W.

Analysis 1: A series RL circuit has R = 6Ω and X_L = 8Ω at a given frequency. If the frequency doubles, how does the impedance change (assuming R stays fixed), and what happens to the current for a fixed RMS voltage? Answer guidance: X_L = ωL doubles to 16Ω, so |Z| = √(6²+16²) = √(36+256) = √292 ≈ 17.09Ω (up from √(36+64)=10Ω). Current I = V/|Z| decreases as impedance rises, so for a fixed voltage, current drops and the phase angle becomes more inductive (larger lag).

Analysis 2: Compare the design implications of choosing DC versus AC for a battery-powered handheld device versus a national electricity grid. Why is the "right" choice different in each case? Answer guidance: Handheld devices use DC because batteries are inherently DC sources, circuits are low-voltage/short-distance (no transmission loss concern), and DC avoids the complexity of reactive components and rectification circuitry. The grid uses AC because it needs to transform voltage up for efficient long-distance transmission (lower I²R losses at high voltage) and back down for safe consumer use — a capability only practical with AC and transformers.

FAQ

Q1: Can Ohm's Law be used directly in AC circuits? Yes, in the form V = IZ, using impedance instead of resistance and phasor (complex) representations of voltage and current instead of simple real numbers.

Q2: Why is RMS used instead of average voltage for AC? Because a symmetric sine wave averages to zero over a full cycle, which would incorrectly suggest zero power delivered. RMS specifically captures the heating/power effect by using the square root of the mean of the squared values, which is always positive and matches an equivalent DC value.

Q3: What does it mean for current to "lag" or "lead" voltage? In an inductive circuit, current lags voltage (peaks later in time) because inductors oppose changes in current. In a capacitive circuit, current leads voltage (peaks earlier) because capacitors oppose changes in voltage. In a purely resistive circuit, voltage and current are perfectly in phase.

Q4: Why does frequency matter in AC circuits but not in DC circuits? DC has zero frequency (f=0), so reactance terms (X_L = ωL, X_C = 1/ωC) become irrelevant — X_L = 0 (inductor acts as a short) and X_C = infinity (capacitor acts as an open). In AC, frequency directly changes these reactances and hence total impedance.

Q5: Is household electronics DC or AC internally? Almost always DC internally — even though wall power is AC, devices contain a rectifier and power supply that convert AC to a stable DC voltage before it reaches the internal circuitry (processors, LEDs, etc.), which require constant-direction current to function correctly.

Quick Revision

• DC: constant-direction current; AC: periodically reversing, typically sinusoidal current.
• DC uses Ohm's Law V=IR with real numbers; AC uses V=IZ with complex impedance.
• RMS voltage/current: V_rms = V_peak/√2 for a sine wave — this is what's used for power calculations and nameplate ratings.
• Impedance Z = R + jX combines resistance (R) and reactance (X); reactance depends on frequency.
• Inductive reactance X_L = ωL increases with frequency; capacitive reactance X_C = 1/(ωC) decreases with frequency.
• Real AC power: P = V_rms × I_rms × cos(θ), where θ is the phase angle (power factor = cos θ).
• Purely resistive AC loads have θ=0°, so P = V_rms × I_rms exactly, matching the DC-like formula.
• AC allows efficient voltage transformation (via transformers) for power transmission; DC does not.
• Series/parallel impedance combination rules mirror DC resistance rules but use complex arithmetic.
• Most electronic devices convert incoming AC to internal DC via rectification.

Related Topics

Prerequisites: Ohm's Law, Series and Parallel Circuits, basic complex numbers (helpful but not mandatory).

Related Topics: Frequency Response, Filters, Network Theorems.

Next Topics: Frequency Response, Filters.