Capacitive and inductive circuits

Reactance

The ratio of the maximum voltage to the maximum current when a capacitor or inductor is connected to an alternating voltage. The unit of reactance is the ohm.

While inductive and capacitive reactances are similar, in one sense they are opposites. For an inductor, the current peaks ${90}^{\circ }$ after the voltage. For a capacitor the current peaks ${90}^{\circ }$ ahead of the voltage. When we work out the total reactance for an inductor and a capacitor in series, we use the formula $X_{total}=X_L-X_C$

to take this into account. This formula can also be used when there is more than one inductor or more than one capacitor in the circuit. The total reactance is the sum of all of the inductive reactances minus the sum of all the capacitive reactances. The magnitude (number) in the final result gives the ratio of maximum voltage to maximum current in the circuit as a whole. The sign of the final result tells you its phase. If it is positive, the current peaks ${90}^{\circ }$ after the voltage, if it is negative, the current peaks ${90}^{\circ }$ before the voltage.

If a series circuit contains resistors as well, then the situation is more complicated. The maximum current is still proportional to the maximum voltage, but the phase difference between them won't be ${90}^{\circ }$ . The ratio between the maximum voltage and maximum current is called the impedance ( $Z$ ), and its unit is also the ohm. Impedances are calculated using this formula: $Z=\sqrt{X^2+R^2}$

where $X$ is the total reactance of the inductors and capacitors in the circuit, and $R$ is the total resistance of the resistors in the circuit.

It is easier to understand this formula by thinking of a right angled triangle. Resistances are drawn horizontally, reactances are drawn vertically. The hypotenuse of the triangle gives the impedance. This is shown in [link] .

Impedance

The maximum voltage divided by the maximum current for any circuit. The unit of impedance is the ohm.

It is important to remember that when resistors and inductances (or capacitors) are in a circuit, the current will not be in phase with the voltage, so the impedance is not a resistance. Similarly the current won't be exactly ${90}^{\circ }$ out of phase with the voltage so the impedance isn't a reactance either.

This simulation allows you to build some circuits with capacitors and inductors.

Capacitive and inductive circuits

1. Why is the instantaneous value of $\frac{V}{I}$ of little use in an AC circuit containing an inductor or capacitor?
2. How is the reactance of an inductor different to the reactance of a capacitor?
3. Why can the ratio of the maximum voltage to the maximum current in a circuit with a resistor and an inductor not be called a reactance?
4. An engineer can describe a motor as equivalent to a 30 $\Omega$ resistor in series with a 30 mH inductor. If the maximum value of the supply voltage is 350 V, what is the maximum current? Assume that the frequency is 50 Hz.
5. A timer circuit in a factory contains a 200 $\mu$ F capacitor in series with a 10 k $\Omega$ resistor. What is its impedance? Assume that the frequency is 50 Hz.
6. A 3 mH inductor is connected in series with a 100 $\mu$ F capacitor. The reactance of the combination is zero. What is the frequency of the alternating current?