Series RLC Circuit (Circuit & Phasor Diagram)

What is a Series RLC Circuit?

A series RLC circuit connects a resistor, inductor and capacitor in one path across a voltage source. The same current flows through all three elements. The circuit and phasor diagrams below show this series RLC circuit under sinusoidal steady-state conditions.

Phasor Diagram of Series RLC Circuit

To construct the phasor diagram, choose the common current as the reference and add the resistor, inductor and capacitor voltage phasors. Their relative angles follow from the voltage-current relation of each ideal element.

1. 1. Resistor
      Voltage and current are in phase, so their phase difference is zero.
   2. Inductor
      Voltage leads current by 90 degrees in an ideal inductor.
   3. Capacitor
      Current leads voltage by 90 degrees in an ideal capacitor. Relative to the current reference, capacitor voltage points 90 degrees below the horizontal axis.

The mnemonic CIVIL records the ideal-element phase rule: in a capacitor, current leads voltage; in an inductor, voltage leads current.

RLC Circuit
Use the following steps to draw the series RLC phasor diagram.

Step I. Because the elements are in series, their currents are equal: I _r = I_l = I_c = I. Draw the current reference on the horizontal axis.
Step II. Draw resistor voltage V_R on the same axis because it is in phase with current. The phasor V_R therefore points to the right.
Step III. Draw inductor voltage V_l perpendicular to and 90 degrees ahead of current.
Step IV. Draw capacitor voltage V_c perpendicular to and 90 degrees behind current.
Step V. Keep V_c in the downward direction. Draw source voltage V_s as the vector sum of horizontal resistor voltage V_r and vertical net reactance voltage V_L – V_C.

Impedance for a Series RLC Circuit

Series impedance Z combines resistance R, inductive reactance X_L and capacitive reactance X_C. If X_L > X_C, the net impedance is inductive and source current lags source voltage. If X_C > X_L, the net impedance is capacitive and current leads voltage. When X_L = X_C, the ideal series circuit is at resonance and its input impedance equals R.

We know that,

Substituting the values V_S² = (IR)² + (I X_L – I X_C )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 – The ideal model treats R as constant. A practical resistor’s value and parasitic reactance can vary with frequency.
2. Inductive reactance, X_L – For an ideal inductor, X_L = 2πfL. Inductive reactance therefore increases linearly with frequency, as shown by curve a.
3. Capacitive reactance, X_C – For an ideal capacitor, X_C = 1/(2πfC), so capacitive reactance decreases as frequency rises. Net series reactance is (X_L – X_C). To construct X_L – X_C, first consider the negative capacitive term -X_C. Curve b shows that term, while curve c shows the net reactance.
4. Curve d shows impedance magnitude. It is the square root of R squared plus net reactance squared, not an arithmetic sum of R and reactance.