Electronics/RCL time domain simple

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Figure 1: RCL circuit
Figure 1: RCL circuit

When the switch is closed, a voltage step is applied to the RCL circuit. Take the time the switch was closed to be 0s such that the voltage before the switch was closed was 0 volts and the voltage after the switch was closed is a voltage V. The voltage across the capacitor consists of a forced response v_f and a natural response v_n such that:


The forced response is due to the switch being closed, which is the voltage V for t\ge0. The natural response depends on the circuit values and is given below:

Define the pole frequency \omega_n and the dampening factor \alpha as:



Depending on the values of \alpha and \omega_n the system can be characterized as:

1. If \alpha > \omega_n the system is said to be overdamped. The solution for the system has the form:


2. If \alpha = \omega_n the system is said to be critically damped The solution for the system has the form:

v_n(t)=Be^{-\alpha t}

3. If \alpha < \omega_n the system is said to be underdamped The solution for the system has the form:

v_n(t)=e^{-\alpha t}\big[B_1\cos(\sqrt{\omega_n^2-\alpha^2} t)+B_2\sin(\sqrt{\omega_n^2-\alpha^2} t)\big]

How do you calculate these equations?


Given the following values what is the response of the system when the switch is closed?

1kΩ 0.5H 100nF 1V

First calculate the values of \alpha and \omega_n:


\omega_n=\frac{1}{\sqrt{LC}}\approx 4472

From these values note that \alpha < \omega_n. The system is therefore underdamped. The equation for the voltage across the capacitor is then:


Before the switch was closed assume that the capacitor was fully discharged. This implies that v(t)=0 at the instant the switch was closed (t=0). Substituting t=0 into the previous equation gives:


Therefore B_1=-1. Similarly at the instant the switch is closed, the current in the inductor must be zero as the current can not instantly change. Substituting the equation for v_c(t) into the equation for the inductor and solving at the instant the switch was closed (t=0) gives:


0=100\cdot10^{-9}\big[4359B_2-1000B_1 \big]

Therefore B_2\approx-0.229. Once v_c(t) is known, the voltage across the inductor and resistor (V_{out}) is given by:


You have missed a lot of steps, where are they?

Figure 2: Example 1 Underdamped Response
Figure 2: Underdamped Resonse

Image:Example1 underdamped.png