## Site Search

# Phase Shift in a RC Element

If a sinusoidal voltage is applied to an ohmic resistance *R*, the resulting current is in phase with the voltage, i.e. the corresponding vectors *I* and *U* of the rms values point in the same direction (illustrations on the left below). A capacitor acts as a pure reactance *X*_{C}, where the voltage trails the current by a phase angle of j = 90° so that the two corresponding vectors form a right angle (illustrations in the centre below). If an ohmic resistance and capacitor are connected in series to form an *RC element*, the resulting overall resistance includes both active and reactive components. In this case, the voltage lags behind the current by a phase angle j that lies somewhere between 0° and 90° depending on the frequency, resistance and capacitance (illustrations on the right below). The voltage vector *U* here is formed through geometric addition of the partial voltages *U*_{R} (in phase with the current) and *U*_{C} (lagging behind the current by 90°).

As in the case of DC circuits, Ohm's law also applies here to the rms values of the current and voltage.

The impedance *Z* is the term for the *apparent resistance* of an AC circuit. It is composed of both the active resistance and the reactance. These values need to be added *geometrically* in the same way as the various voltages. This is demonstrated in the following illustration which shows a *triangle of resistances* for an RC element.

The impedance *Z* can be easily calculated from the triangle of resistances, which being right-angled, results in the following relationship:

Taking the square root of both sides of the equation and replacing *X*_{C} with the relationship ascertained earlier leads to the following:

The phase angle j can also be determined from the triangle of resistances as follows:

Given known values of w or *f*, *R* and j, this equation can be used to determine the capacitance by resolving in terms of *C*. The following relationship is then obtained:

Alternatively (as described in the next experiment), the phase shift between the capacitor voltage *U*_{C} and supply voltage *U* can be measured. The capacitance is then derived from the following formula:

**Example**: a resistor *R* = 220 W connected in series with a capacitor *C* = 10 µF are subjected to an alternating voltage *U* = 220 V with a frequency *f* = 50 Hz. Calculate the capacitor's reactance, impedance, current *I*, both partial voltages *U*_{R} and *U*_{C} as well as the phase angle j between *U* and *I*.

Result: