Physics · Ch 4 — Electromagnetic Induction and Alternating Current

AC Circuit Containing a Pure Resistor

4.7.5

AC Circuit Containing a Pure Resistor

Consider a circuit containing a pure resistor of resistance R connected directly across an alternating voltage source $v=V_m\sin\omega t \qquad (4.37)$ . The resulting current i develops a potential drop $V_R=iR \qquad (4.38)$ across R, and Kirchhoff's loop rule (the algebraic sum of potential differences around any closed loop is zero) gives $v-V_R=0$ . Combining equations (4.37) and (4.38), $V_m\sin\omega t = iR$ , so

i = \dfrac{V_m}{R}\sin\omega t = I_m\sin\omega t \qquad (4.39)

where $I_m=V_m/R$ is the peak current in the circuit. Comparing this directly with the applied voltage $v=V_m\sin\omega t$ , the applied voltage and the resulting current are found to be exactly IN PHASE with each other in a purely resistive circuit -- they reach their positive maxima simultaneously, cross zero simultaneously, and reach their negative minima simultaneously, throughout every cycle. On a phasor diagram, this means the voltage phasor $\vec{OA}$ (length $V_m$ ) and the current phasor $\vec{OB}$ (length $I_m$ ) point in exactly the same direction at ev …

Figure 4.40AC circuit with a resistor

What this figure shows. A pure resistor R is connected directly across an alternating source labelled $v=V_m\sin\omega t$ , drawn as a simple single-loop circuit with no other components present. The figure sets up the basic circuit whose Kirchhoff's-law analysis -- $v-iR=0$ , so $i=v/R=(V_m/R)\sin\omega t=I_m\sin\omega t$ -- shows that in a purely resistive AC circuit the current is a scaled-down (by 1/R), but otherwise perfectly IN-STEP …

Figure 4.41Phasor and wave diagram for an AC circuit with R -- current in phase with voltage

What this figure shows. Voltage phasor $\vec{OA}$ (length $V_m$ ) and current phasor $\vec{OB}$ (length $I_m$ ) are drawn overlapping along the SAME direction from the origin (zero angle between them), alongside matching sine-wave graphs of v and i that rise, peak, fall and cross zero at exactly the same instants as each other. The figure is the direct visual statement of the resistive circuit's key result: with no phase angle $\phi$ separating them, voltage and current in a purely resistive AC circuit reach their maxima together, cross zero together, and reach their minima together, througho …

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