## Illustration 31.7: RC Circuits and Phasors

Please wait for the animation to completely load.

Assume an ideal power supply. The top graph shows the voltage as a function of time across the source **(red)**, the resistor **(blue)**, and the capacitor **(green)** **(voltage is given in volts and time is given in seconds)**. Restart.

In order to analyze circuits with impedances that change as a function of frequency, we can use a phasor representation for the voltages across and the current through the various circuit elements. This allows us to take into account the phase difference between the voltages across the capacitor, the resistor, and the power supply.

To begin analyzing the circuit in the animation, we should first notice that Kirchhoff's law holds at any instant of time. Pause the animation and pick a time and find the voltage across the source, the resistor, and the capacitor. Verify that the voltage across the resistor plus the voltage across the capacitor equals the source voltage at that time. Notice, however, that if you add up the peak voltages, the peak resistor voltage plus the peak capacitor voltage is not equal to the peak source voltage.

You must account for the phase difference in the voltages. One way to account for the phase differences is to describe the voltage and current with phasors. Below the circuit are an animation and graph that show the phasor representation of the circuit elements (this allows us to show the phase difference), where the capacitor voltage is π/2 behind the resistor since its voltage lags the current. Notice that the phasors rotate at an angular speed of ω = 2 π f. The projection of the phasor vector on the y axis is plotted in the lower graph. Pause the animation. Note that the phasor animation matches the circuit graph. Try another frequency and verify that the two plots are the same (except at t = 0 because of the initial condition of the capacitor). Therefore, we can use phasor diagrams to show the phase angle between the source voltage, the resistor voltage, and the capacitor voltage. For more on phasors, see Illustration 31.6 and Explorations 31.5 and 31.6.

Illustration authored by Anne J. Cox.

Script authored by Wolfgang Christian and Anne J. Cox.