Capacitance in AC Circuits

When capacitors are connected across a direct current DC supply voltage they become charged to the value of the applied voltage, acting like temporary storage devices and maintain or hold this charge indefinitely as long as the supply voltage is present. During this charging process, a charging current, ( i ) will flow into the capacitor opposing any changes to the voltage at a rate that is equal to the rate of change of the electrical charge on the plates.
This charging current can be defined as: i = CdV/dt. Once the capacitor is “fully-charged” the capacitor blocks the flow of any more electrons onto its plates as they have become saturated. However, if we apply an alternating current or AC supply, the capacitor will alternately charge and discharge at a rate determined by the frequency of the supply. Then the Capacitance in AC circuits varies with frequency as the capacitor is being constantly charged and discharged.
We know that the flow of electrons through the capacitor is directly proportional to the rate of change of the voltage across the plates. Then, we can see that capacitors in AC circuits like to pass current when the voltage across its plates is constantly changing with respect to time such as in AC signals, but it does not like to pass current when the applied voltage is of a constant value such as in DC signals. Consider the circuit below.

AC Capacitor Circuit

In the purely capacitive circuit above, the capacitor is connected directly across the AC supply voltage. As the supply voltage increases and decreases, the capacitor charges and discharges with respect to this change. We know that the charging current is directly proportional to the rate of change of the voltage across the plates with this rate of change at its greatest as the supply voltage crosses over from its positive half cycle to its negative half cycle or vice versa at points, 0o and 180o along the sine wave. Consequently, the least voltage change occurs when the AC sine wave crosses over at its maximum or minimum peak voltage level, (Vm). At these positions in the cycle the maximum or minimum currents are flowing through the capacitor circuit and this is shown below.

AC Capacitor Phasor Diagram

Capacitive Reactance

Capacitive Reactance in a purely capacitive circuit is the opposition to current flow in AC circuits only. Like resistance, reactance is also measured in Ohm’s but is given the symbol X to distinguish it from a purely resistive value. As reactance can also be applied to Inductors as well as Capacitors it is more commonly known as Capacitive Reactance for capacitors in AC circuits and is given the symbol Xc so we can actually say that Capacitive Reactance is Resistance that varies with frequency. Also, capacitive reactance depends on the value of the capacitor in Farads as well as the frequency of the AC waveform and the formula used to define capacitive reactance is given as:
Capacitive Reactance
Xc = 1/2πfc = 1/ωC

Where:
F is in Hertz and C is in Farads.
2πF can also be expressed collectively as the Greek letter Omega, ω to denote an angular frequency.
From the capacitive reactance formula above, it can be seen that if either of the Frequency or Capacitance where to be increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance would reduce to zero acting like a perfect conductor. However, as the frequency approaches zero or DC, the capacitors reactance would increase up to infinity, acting like a very large resistance. This means then that capacitive reactance is “Inversely proportional” to frequency for any given value of Capacitance and this shown below:

Question

1. Find the r.m.s. value of an alternating current whose peak value is 5 amps.
A. 3.53 Amps B. 4.5 Amps C. 2.19 Amps D. 6.50 Amps
2. Which of these is not correct, A.C. circuits are circuits through which an alternating current flows. Such circuits are used extensively in
A. Power transmission B. Radio C. Telecommunication D. Automobile
3. In an A.C. circuit the peak value of the potential difference is 180 V. What is the instantaneous p.d, when it has reached 1/8th of a cycle?
A. 80 Volts B. 90√2 Volts C. 80√2 Volts D. 90 Volts
4. A 240V supply is connected with a resistor of 20 in an A.C. circuit. Find the current in the circuit
A. 15 A B. 23 A C. 12 A D. 34 A
5. At frequency of 50Hz, a capacitor of 5μF is connected to a circuit of 230V supply. Find the capacitive reactance.
A. 630Ω B. 636.62Ω C. 790Ω D. 430.77Ω