# Damped RLC Circuit Equation

I'm trying to measure the inductance of a toroid with the use of a 555 in astable mode. Hopefully, by putting the toroid in parallel with a known capacitor and driving the circuit with a low duty cycle (~10%), I should be able to measure the frequency of the underdamped oscillation, as well as the damping factor, and thereby deduce the undamped resonant frequency and hence inductance. However, I'm not too sure as to how to solve the differential equation. I know how to solve for LC, RC circuits and so forth, but this is a little different.

simulate this circuit – Schematic created using CircuitLab

My interpretation goes as follows: The 555's output, which goes through the diode, charges the inductor and capacitor. The 555's frequency is a lot lower than the time constant of the components so at the start it should be at a steady state, where:

$I_L = I(0) = I_R$

$I_C = 0$

$V_C = V_R = V(0)$

$V_L = 0$

At t = 0, the 555 should switch off, so the only voltage & current in the circuit should be as a result of the inductor and capacitor oscillating (the diode is there to isolate the RLC from the 555). However, there's an immediate issue: I can't apply Kirchoff's laws since now the circuit is effective a series RLC, which means that the current through all components should be the same: but the current through the capacitor can't change instantly, right? As a result, I'm not sure how to derive the differential equation, which means I'm not sure if the standard result for resonant frequency and damping coefficient are the same as normal.

Note that I have tested this with an oscilloscope and it has produced an underdamped oscillation, so I'm that it should work in theory.

I've looked elsewhere but I can't seem to find the derivation for a non-driven RLC circuit.

• Assuming the duty cycle and PWM frequency are appropriate, and it seems that is the case from your description of the oscilloscope trace, the damping coefficient can be calculated from the logarithmic decrement and the undamped natural frequency, wn, can then be determined from the measured damped frequency. L can then be obtained from wn=1/SQRT(LC) – Chu Mar 26 '15 at 16:39
• Yes, but are the equations identical for those for all damped harmonic motion circuits? – Alex Freeman Mar 26 '15 at 16:42
• Yes, It's a linear 2nd order system. – Chu Mar 26 '15 at 16:44
• Okay, but how is the damping factor linked to the inductance and resistance? Doesn't this depend on the circuit topology? – Alex Freeman Mar 26 '15 at 16:52
• Not on the topology, it's a 'normal' series RCL circuit. The undamped natural frequency is given by wn=1/SQRT(LC), and the damping coefficient is: zeta=0.5 Rsqrt(C/L) – Chu Mar 26 '15 at 17:21

These are equations copied from the same page for the under-damped case. Given R, C along with measured $\omega_d$, L can be solved from these equations. (You don't even need to measure the attenuation factor $\alpha$.)