What is Capacitive Reactance?
The capacitive reactance is the resistance to electric current in a capacitive circuit. It differs from the perspective of frequency. The resistance does not depend on frequency but capacitive reactance changes with the rise and fall of the supply frequency. The symbol of capacitive reactance is Xc. It is measured in ohms (Ω).
Derivation of Capacitive Reactance
Consider an AC voltage source supplying a capacitor as shown below:
Input voltage can be expressed as sine function as:
Charge across the capacitor is given by:
Also, the current is explained as the time rate of change of charge:
As we know:
Therefore:
When ωt=0:
Since sin(π/2)=1:
Also, ω=2πf:
AC Capacitive Circuit
The waveforms of input voltage & input current versus phase angle are presented below for an AC capacitive circuit:
The current waveform is leading the voltage waveform by 90 degrees in the above waveform presentation. It means that currently leads the voltage by 90 degrees for capacitive circuits.
Vector Diagram
The phaser or vector diagram shows the phase shift between current and voltage in a simpler manner. Current and voltages are presented by two vectors, and the phase angle shows the angular shift between these vectors. The arrowhead represents the direction of angular frequency from voltage to current in a capacitive circuit.
Capacitive Reactance Vs Frequency
Capacitive reactance can be plotted against frequency if the capacitance of the capacitor is known from the below expression:
Therefore:
It is clear from the above expression that capacitive reactance is inversely related to frequency. Therefore, XC will rise when the frequency will be lower and XC will fall as the frequency becomes high.
Capacitive Reactance in Series and Parallel Networks
Capacitive reactance behaves like resistances in parallel and series network combinations. In a series network, all capacitive reactance will be added, and the equivalent reactance shall be the sum of all individual reactance.
In a parallel network, reciprocals of individual reactance are added together to obtain reciprocals of equivalent reactance. The total capacitive reactance shall then be the reciprocal of the above-obtained results.
Below, two examples describe equivalent capacitance and total current calculations for series and parallel capacitive reactance networks.
Equivalent Reactance for the Series Network
Here for illustration, I have considered two capacitors in series having a value of capacitive reactance of 40 and 60 ohms:
Here:
Finding total current:
Equivalent Reactance for Parallel Network
Here for illustration, I have considered two capacitors in parallel having a value of capacitive reactance of 40 and 60 ohms:
Finding total current:
Example 1: Finding Capacitive Reactance for Two Different Frequencies
What will be the capacitive reactance for a 250nF capacitor at two different frequencies of 10kHz and 40kHz:
As:
For f=10kHz:
For f=40kHz:
Example 2: Finding Frequency using Capacitive Reactance
Determine the frequency for a 5.5uF capacitor with a capacitive reactance of 300Ω.
As:
Example 3: Finding Capacitance using Capacitive Reactance
Calculate the capacitance of the capacitor with a reactance of 150 Ω and connected across a 50Hz power source.
As:
Conclusion
Capacitive reactance is the measure of opposition to electric current in a capacitive circuit. It is inversely proportional to both frequency of power supply and capacitance value. In other words, at lower frequencies, capacitive reactance will be high while at higher frequencies, capacitive reactance will be low.
