AC in Resistive Circuits

When you plug a lamp or a toaster into a wall outlet, you are connecting a resistive load to an alternating current (AC) source. Unlike direct current (DC), which flows steadily in one direction, AC voltage oscillates sinusoidally — it rises to a positive peak, falls back through zero, plunges to a negative peak, and repeats this cycle many times per second (50 Hz in Europe, 60 Hz in North America). Understanding how a pure resistor responds to this continuously changing voltage is the foundation of all AC circuit analysis. The key insight for a resistive circuit is beautifully simple: a resistor obeys Ohm's Law at every instant in time. Because resistance does not store energy the way capacitors and inductors do, the current through the resistor tracks the voltage across it perfectly — they rise and fall together, reaching their peaks and zeros at exactly the same moments. Physicists say the voltage and current are 'in phase', meaning the phase angle between them is zero. This in-phase relationship has important practical consequences. Power is delivered to the resistor during both the positive and negative half-cycles (since a negative voltage driving a negative current still produces positive power, just like multiplying two negative numbers). The result is continuous, non-zero average power dissipation — exactly the kind of steady heating you want from an electric stove or an incandescent bulb.

Key takeaways
  • The Instantaneous Voltage
  • Applying Ohm's Law Instantaneously
  • Peak vs. RMS Values
  • Power in a Resistive AC Circuit
  • Phasor Representation

The Instantaneous Voltage

The AC source produces a voltage that varies sinusoidally with time: v(t) = V_m sin(ωt), where V_m is the peak (amplitude) voltage and ω = 2πf is the angular frequency in radians per second. At t = 0 the voltage is zero; it climbs to V_m at t = T/4, returns to zero at t = T/2, and so on. This smooth, wave-like variation is why AC systems can be described so cleanly with trigonometry.

Applying Ohm's Law Instantaneously

At every moment in time, the resistor satisfies v(t) = i(t) · R. Dividing the source voltage by the resistance gives i(t) = (V_m / R) sin(ωt) = I_m sin(ωt). Notice the current is also a sine wave with angular frequency ω. The amplitude of the current is I_m = V_m / R — exactly Ohm's Law applied to the peak values. Most importantly, the argument of the sine function is identical for both v and i, which means they are perfectly in phase.

Peak vs. RMS Values

The peak voltage V_m and peak current I_m are the maximum magnitudes of the sinusoids. However, because the wave spends most of its time below the peak, a more useful measure of 'effective' strength is the root-mean-square (RMS) value. For a pure sinusoid, V_rms = V_m / √2 ≈ 0.707 V_m, and similarly I_rms = I_m / √2. These are the values quoted on appliance labels and measured by standard voltmeters. Ohm's Law works perfectly with RMS values too: V_rms = I_rms · R.

Power in a Resistive AC Circuit

Instantaneous power is p(t) = v(t) · i(t) = V_m sin(ωt) · I_m sin(ωt) = V_m I_m sin²(ωt). Using the identity sin²(ωt) = (1 − cos 2ωt)/2, we get p(t) = (V_m I_m / 2)(1 − cos 2ωt). The cosine term oscillates and averages to zero over a full cycle. The average (real) power is therefore P = V_m I_m / 2 = V_rms I_rms = I_rms² R = V_rms² / R. This is always positive — a pure resistor consumes (and dissipates as heat) real power with 100% efficiency, and the power factor is exactly 1.

Phasor Representation

Engineers represent sinusoidal quantities as phasors — rotating vectors whose length equals the RMS (or peak) amplitude and whose angle represents the phase. For a resistive circuit, the voltage phasor V and current phasor I point in exactly the same direction (zero phase difference). On a phasor diagram, both arrows overlap, visually confirming the in-phase relationship. When you later add capacitors or inductors, their phasors will point in different directions, creating the phase shifts that make AC circuit analysis so rich.

The Key Equation

i(t)=VmRsin⁡(ωt)=Imsin⁡(ωt)i(t) = \frac{V_m}{R}\sin(\omega t) = I_m \sin(\omega t)
  • i(t) — Instantaneous current (A (amperes)): The current through the resistor at time t.
  • V_m — Peak voltage (V (volts)): The maximum amplitude of the AC voltage source.
  • R — Resistance (Ω (ohms)): The resistance of the circuit element; assumed purely resistive.
  • \omega — Angular frequency (rad/s): Equal to 2πf, where f is the frequency in hertz.
  • t — Time (s (seconds)): The elapsed time from the chosen reference instant.
  • I_m — Peak current (A (amperes)): Maximum current amplitude; equals V_m / R by Ohm's Law.

Worked Example

Problem. An AC source with peak voltage V_m = 170 V and frequency f = 60 Hz is connected to a 85 Ω resistor. (a) Write expressions for v(t) and i(t). (b) Find the RMS current. (c) Calculate the average power dissipated.

i(t)=2.0sin⁡(377t) A,Irms≈1.41 A,P=170 Wi(t)=2.0\sin(377t)\text{ A},\quad I_{rms}\approx 1.41\text{ A},\quad P=170\text{ W}

Answer. v(t) = 170 sin(377t) V; i(t) = 2.0 sin(377t) A; I_rms ≈ 1.41 A; Average power P = 170 W.

Figure
Diagram
AC Resistive Circuit: Voltage and Current Waveformst↑Amplitude+V_m−V_mT/2T3T/22Tv(t) = V_m sin(ωt)i(t) = I_m sin(ωt)Peaks coincide (in phase)Voltage v(t)Current i(t)
Origins · 1 minGeorg Simon Ohm and how this was discovered
GS
Georg Simon Ohm
1789–1854 · German

Ohm discovered the linear relationship between voltage and current in a conductor, published in his 1827 monograph 'Die galvanische Kette, mathematisch bearbeitet'. This law — V = IR — remains the cornerstone of resistive circuit analysis whether the source is DC or, applied instantaneously, AC. Without Ohm's Law, the entire framework for analyzing AC resistive circuits would not exist. Ohm's work was initially rejected by many German scientists as 'a web of naked fantasies.' He was dismissed from his teaching post in Cologne after publishing his famous paper. It took a decade before European scientists recognized his genius — and he eventually received the Copley Medal from the Royal Society of London in 1841.

  • AC in Capacitive Circuits — Capacitors introduce a 90° phase lead of current over voltage, contrasting sharply with the zero phase shift in resistive circuits — understanding resistance first makes the capacitive case clearer.
  • AC in Inductive Circuits — Inductors cause a 90° phase lag of current behind voltage; comparing this with resistive and capacitive behavior builds the complete picture of AC impedance.
  • RMS Values and AC Power — RMS voltage and current are introduced for resistive circuits and underpin all AC power calculations, including real, reactive, and apparent power.
  • Phasor Diagrams — Phasors are the standard visual tool for comparing phase relationships; resistive circuits provide the simplest case (aligned phasors) from which inductive and capacitive cases diverge.
  • Ohm's Law — AC resistive circuit analysis is a direct extension of Ohm's Law applied instantaneously to time-varying voltages and currents.