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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 XC, 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 UR (in phase with the current) and UC (lagging behind the current by 90°). 

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

As in the case of DC circuits, Ohm's law also applies here to the rms values of the current and voltage.CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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.

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

Taking the square root of both sides of the equation and replacing XC with the relationship ascertained earlier leads to the following:

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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:

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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 UR and UC as well as the phase angle j between U and I.
Result: 

CAPACITOR IN ALTERNATING CURRENT AC CIRCUITS !

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