Series RLC Circuit Equations

Underdamped
Overdamped
Critically Damped
 

The formulas on this page are associated with a series RLC circuit discharge since this is the primary model for most high voltage and pulsed power discharge circuits.

Each of the following waveform plots can be clicked on to open up the full size graph in a separate window.

Underdamped

RLC Underdamped Circuit Condition Equation

RLC Underdamped Circuit Condition Equation

The circuit schematic for the underdamped case is shown below. In this specific model, the resistance is an order of magnitude (10X) less than the value required for a critically damped circuit.

Schematic Diagram for Underdamped Series RLC Circuit Simulation

Schematic Diagram for Underdamped Series RLC Circuit Simulation

The results of the circuit model are shown below. V(1) is the voltage on the 1 mF capacitor as it discharges in an oscillatory mode. V(3) is the voltage on the load resistor, in this case a 0.2 ohm value. The circuit current is graphed in the second, lower plot and reaches its peak value very nearly at t=p/2w0.

Circuit Waveforms for Underdamped Series RLC Circuit Simulation

Circuit Waveforms for Underdamped Series RLC Circuit Simulation

Equation for Series RLC Oscillation Frequency

Equation for Series RLC Oscillation Frequency

Omega

L is the circuit inductance (H)
C is the circuit capacitance (F)
R is the circuit resistance (W).

Equation for Underdamped Series RLC Current vs Time

Equation for Underdamped Series RLC Current vs Time

Current

V0 is the initial voltage on the capacitance (V)

Equation for the Peak Current in a Underdamped Series RLC Discharge Circuit

Equation for the Peak Current in a Underdamped Series RLC Discharge Circuit

Peak Current

Same as above

Overdamped

RLC Overdamped Circuit Condition Equation

RLC Overdamped Circuit Condition Equation

The circuit schematic for the overdamped case is shown below. In this specific model, the resistance is an order of magnitude (10X) more than the value required for a critically damped circuit.

Schematic Diagram for Overdamped Series RLC Circuit Simulation

Schematic Diagram for Overdamped Series RLC Circuit Simulation

The results of the circuit model are shown below. V(1) is the voltage on the 1 mF capacitor as it discharges in a simple RC decay mode. V(3) is the voltage on the load resistor, in this case a 20 ohm value. In this case, once the switch closes and the voltage on the load resistor rises to match the capacitor voltage, both waveforms then essentially overlap and decay at the same rate since the voltage across the inductor is minimal. The circuit current is graphed in the second, lower plot.

Circuit Waveforms for Overdamped Series RLC Circuit Simulation

Circuit Waveforms for Overdamped Series RLC Circuit Simulation

Overdamped Series RLC Circuit Equation for Omega

Overdamped Series RLC Circuit Equation for Omega

Omega

L is the circuit inductance (H)
C is the circuit capacitance (F)
R is the circuit resistance (W).

Equation for Overdamped Series RLC Current vs Time

Equation for Overdamped Series RLC Current vs Time

Current

V0 is the initial voltage on the capacitor (V)

Critically Damped

RLC Critically Damped Circuit Condition Equation

RLC Critically Damped Circuit Condition Equation

The circuit schematic for the critically damped case is shown below. In this specific model, the resistance is exactly equal to the value required for a critically damped circuit.

Schematic Diagram for Critically Damped Series RLC Circuit Simulation

Schematic Diagram for Critically Damped Series RLC Circuit Simulation

The results of the circuit model are shown below. V(1) is the voltage on the 1 mF capacitor as it discharges towards zero with no overshoot. V(3) is the voltage on the load resistor, in this case a 20 ohm value. One can see that the resistor voltage also does not overshoot. The circuit current is graphed in the second, lower plot and reaches its peak value at t=2L/R.This circuit is often desirable (if possible) with high voltage, energy storage capacitors since voltage reversals can frequently decrease the lifetime of the capacitor.

Circuit Waveforms for Critically Damped Series RLC Circuit Simulation

Circuit Waveforms for Critically Damped Series RLC Circuit Simulation

Equation for Critically Damped Series RLC Current vs Time

Equation for Critically Damped Series RLC Current vs Time

Current

V0 is the initial voltage on the capacitor (V)
L is the circuit inductance (H)
R is the circuit resistance (W).

Equation for Critically Damped Series RLC Peak Current

Equation for Critically Damped Series RLC Peak Current

Peak Current

Same as above

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