# Series RLC Circuit Impedance with Phasor Diagram

A series RLC circuit consists of resistance, inductance, and capacitance in series. Whenever we apply a sinusoidal voltage across the series RLC circuit every voltage and current in the circuit will be also sinusoidal in its steady-state condition. The only difference is in its amplitude and phase angle. It is obvious that there will be no change in the frequency of the signals. In other words, the voltages across the resistance inductance and capacitance will have the same frequency as that of the source.

## Response of Series RLC Circuit

We have shown one simple basic series RLC circuit here in the figure.

The source voltage is sinusoidal and we can represent it as

The impedance of the series RLC circuit is

Therefore the current flowing through the circuit is

Now the expression of the impedance of that RLC circuit can be rewritten as

The polar form of this impedance is

#### Condition of Inductive Circuit

The above-mentioned series RLC circuit shows inductive nature when the inductive reactance is much much more than the capacitive reactance of the circuit.

#### Condition of Capacitive Circuit

The circuit shows its capacitive nature when the capacitive reactance is much much more than the inductive reactance.

The phase difference between the voltage and current is given by

Whenever the inductive reactance is much higher than the capacitive reactance this angle becomes positive. Therefore we can conclude that the current lags the voltage with this angle.

On the other hand, when the inductive reactance is much less than the capacitive reactance this angle becomes negative. In that case, the current leads the voltage with this angle.

Ultimately we can summarise that in the inductive circuit the current lags the source voltage and in the capacitive circuit the current leads the source voltage.

### Phasor Diagram of Series RLC Circuit

#### Phasor Diagram of Inductive Series RLC Circuit

We normally take the direction of the circuit current as the reference axis of the diagram. The voltage drop across the resistance will have the same phase as that circuit current. The voltage drop across the inductive reactance will be perpendicularly upward on the current axis. This is because the voltage drop across a pure inductance has exactly 90° phase advancement in respect of the current. In other words, the current lags the voltage at exactly 90°. The voltage drop across the capacitive reactance will be perpendicularly downward on the current axis.

#### Phasor Diagram of Capacitive Series RLC Circuit

In contrast, across the pure capacitive element, the voltage lags the current by exactly 90°. Alternatively, we can say the current leads the voltage by exactly 90°. The resultant voltage across the entire reactive part of the circuit is the difference between inductive and capacitive voltage drops. The vector sum of the resistive voltage drop and reactive voltage drop is the source voltage of the circuit.

If the inductive voltage drop is more than the capacitive voltage drop, the resultant reactive voltage will be perpendicularly upward.

On the other hand, if the capacitive voltage drop is more than the inductor voltage drop, the resultant reactive voltage will be perpendicularly downward.

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