# Series RLC Circuit (Circuit & Phasor Diagram)

Contents

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Key learnings:
• Series RLC Circuit Definition: An RLC circuit is defined as a circuit where a resistor, inductor, and capacitor are connected in series across a voltage source, influencing the overall phase and magnitude of the circuit’s impedance.
• Phasor Diagram Utility: Phasor diagrams help visualize the phase relationships and magnitudes of voltages and currents in RLC circuits.
• CIVIL Mnemonic: The mnemonic “CIVIL” is a simple way to remember that in a capacitor the current leads voltage, and in an inductor, the voltage leads the current.
• Impedance Analysis: Impedance in an RLC circuit combines resistance, inductive reactance, and capacitive reactance, affecting how the circuit reacts to different frequencies.
• Frequency Response: The behavior of the RLC circuit changes with frequency, where inductive reactance increases with frequency and capacitive reactance decreases, impacting the total impedance.

## What is a Series RLC Circuit?

A series RLC circuit is where a resistor, inductor and capacitor are sequentially connected across a voltage supply. This configuration forms what is known as a series RLC circuit. Below, you’ll find a circuit and phasor diagram illustrating this setup.

## Phasor Diagram of Series RLC Circuit

To create the phasor diagram of a series RLC circuit, combine the individual phasors of the resistor, inductor, and capacitor. It’s important to first grasp how voltage and current relate in each component.

1. Resistor
In a resistor, the voltage and current are synchronized, meaning they are in the same phase with a phase angle difference of zero.
2. Inductor
In an inductor, the voltage leads the current by 90 degrees, which means the voltage reaches its maximum and minimum values 90 degrees before the current does.
3. Capacitor
In a capacitor, the current leads the voltage by 90 degrees. This places the voltage’s peak and trough exactly opposite to those of the current, mirroring the inductor’s phasor diagram.

NOTE: For remembering the phase relationship between voltage and current, learn this simple word called ‘CIVIL’, i.e in capacitor current leads voltage and voltage leads current in inductor.

RLC Circuit
For drawing the phasor diagram of series RLC circuit, follow these steps:

Step – I. In case of series RLC circuit; resistor, capacitor and inductor are connected in series; so, the current flowing in all the elements are same i.e I r = Il = Ic = I. For drawing the phasor diagram, take current phasor as reference and draw it on horizontal axis as shown in diagram.
Step – II. In case of resistor, both voltage and current are in same phase. So draw the voltage phasor, VR along same axis or direction as that of current phasor i.e VR is in phase with I.
Step – III. We know that in inductor, voltage leads current by 90° so draw Vl (voltage drop across inductor) perpendicular to current phasor in leading direction.
Step – IV. In case of capacitor, the voltage lags behind the current by 90° so draw Vc (voltage drop across capacitor) perpendicular to current phasor in downwards direction.
Step – V. For drawing the resultant diagram, draw Vc in upwards direction. Now draw resultant, Vs which is vector sum of voltage Vr and VL – VC.

## Impedance for a Series RLC Circuit

The impedance Z of a series RLC circuit is defined as opposition to the flow of current, due to circuit resistance R, inductive reactance, XL and capacitive reactance, XC. If the inductive reactance is greater than the capacitive reactance, i.e XL > XC, then the RLC circuit has lagging phase angle and if the capacitive reactance is greater than the inductive reactance, i.e XC > XL then the RLC circuit have leading phase angle and if both inductive and capacitive are the same, i.e XL = XC then circuit will behave as purely resistive circuit.

We know that,

Substituting the values VS2 = (IR)2 + (I XL – I XC )2

From this impedance triangle: by using Pythagoras theorem we get;

### Variation in Resistance, Reactance and Impedance with Frequency

In series RLC circuit, three types of impedance are involved-

1. Electrical resistance – Resistance is independent of frequency, so it remains constant with change in frequency.
2. Inductive reactance, XL We know that XL = 2πfL. So, inductive reactance varies directly with frequency. So the graph between frequency and inductive reactance is a straight line passing through the centre as shown by curve a.
3. Capacitive reactance, XC From the formula of capacitive reactance, XC = 1/ 2πfC so, capacitive reactance varies inversely with frequency. Since the net reactance is ( XL – XC). So for drawing curve of ( XL – XC), firstly draw the graph of ( -XC) which is shown by curve b and then draw a curve for net reactance which is shown as curve c.
4. The total impedance of circuit is shown by curve d which is obtained by adding constant resistor value to the net reactance.
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