Series RLC Circuit Step Response

Series RLC Circuit Step Response

The series RLC circuit is governed by the second order differential equation below, which is derived from Kirchoff's Voltage Law. Previous to our experiment, we find the damping ratio, natural frequency, damped natural frequency, and DC gain of the circuit. It is useful to note our measured values are R = 1.4 Ω, RL = 1.7 Ω (resistance of inductor), L = 0.999 mH, and C = 0.437 µF.

The damping ratio is ζ = ɑ/ω0, where the neper frequency ɑ = R/2L and the undamped natural frequency ω0 = 1/sqrt(LC). 

Underdamped Series RLC Circuit Step Response

From our measured values, we get the damping ratio to be ζ = 0.0324, which means that the circuit is underdamped since ɑ<ω0. The natural frequencies are calculated above and we get s1,2 = -1551.6 + j48.8E3. The damped natural frequency is calculated at ωd = 47.8E3.

Here we see our circuit, wired according to the diagram using the component values mentioned earlier.The blue wires are used for measuring the capacitor voltage, while the red wires measure the input voltage. The wire colors correspond to the graph colors on the oscilloscope as we'll see next.

Now we run a 2 V step input at 1 Hz and obtain the graph using the oscilloscope feature on WaveForms. We see the typical ringing of a underdamped series RLC circuit. When the input voltage steps down from 2 V to 0 V, the capacitor voltage overshoots by Mp = 1.56/2 = 78%. That is the capacitor steps down the same 2 V as the input, but then overshoots by 78% that amount. The time it takes the capacitor voltage to do this is the rise time tr = (29+30) = 59 µs. 

The period of the oscillations is said to be T = 2ᴨ/ωd = 131.35 µs. If we examine the times when the first two peaks happen, we get that T = 149 µs - 29 µs = 120 µs which has an error of 1%.

Critically Damped RLC Circuit Step Response

If we want to create a critically damped RLC circuit, we must make changes to our components so that ω0 = ɑ. Since the capacitor and inductor are to not be changed, we focus on the resistor, keeping in mind the resistance of the inductor. Finding the critical value for resistor R is shown below.

The value of resistor R needs to be 94.2 Ω in order for the circuit to be critically dampened. However, this is not a readily available value for a resistor. We could use a potentiometer to get a value very close to this. In our case however, we will just use a 100 Ω resistor (97.8 Ω measured) and expect there to be some kind of uncertainty. 

We see our circuit once again with the resistor swapped for the new 100 Ω one.

We now wire the circuit with our new resistor and obtain the graph above. The expected maximum is supposed to occur at 1/ɑ = (2*0.999E-3)/97.8 = 939.8 µs. However, when we look at the graph, we can only see as far as 350 µs. The scaling was adjusted a few times to 'zoom out' of the view, but as we zoom out, the characteristic curvature of the graph is lost.