EE-202 Circuit Theory LAB-2 SECOND-ORDER RLC CIRCUIT’S TRANSIENT RESPONSE TO STEP INPUT AND SINUSOIDS

Bilkent University
EE-202 Circuit Theory
LAB-2
SECOND-ORDER RLC CIRCUIT’S TRANSIENT RESPONSE
TO STEP INPUT AND SINUSOIDS




Çağan İzol
22303351
Section-03


1. Introduction: This experiment investigates the transient response of a second-
order RLC circuit when subjected to step and sinusoidal excitations. The main
objectives are to analyze the circuit’s response both theoretically and
experimentally, compute the quality factor (Q), and compare simulation,
analytical, and hardware results. By studying the transient behavior, we aim to
understand damping effects, resonance properties, and steady-state
performance.

2. Analysis

Mathematical Expectations and Derivations:

, First of all, we take Thevenin equivalent of the circuit to get simpler equations. Then
we apply KVL.

𝑣𝑇 (𝑡) = 𝑅𝑇 𝑖(𝑡) + 𝑣𝐶 (𝑡) + 𝑣𝐿 (𝑡)

We can rewrite i-v characteristics of the inductor and capacitor.

𝑑𝑖(𝑡)
𝑣𝐿 (𝑡) = 𝐿
𝑑𝑡

𝑑𝑣𝐶 (𝑡)
𝑖(𝑡) = 𝐶
𝑑𝑡

Also,

𝑑 2 𝑣𝐶 (𝑡)
𝑣𝐿 (𝑡) = 𝐿𝐶
𝑑2𝑡

Therefore,

𝑑2 𝑣𝐶 (𝑡) 𝑑𝑣𝐶 (𝑡)
𝑣𝑇 (𝑡) = 𝐿𝐶 2
+ 𝑅𝑇 𝐶 + 𝑣𝐶 (𝑡)
𝑑 𝑡 𝑑𝑡

First we deal with the zero-input response

𝑑2 𝑣𝐶 (𝑡) 𝑑𝑣𝐶 (𝑡)
0 = 𝐿𝐶 + 𝑅 𝑇 𝐶 + 𝑣𝐶 (𝑡)
𝑑2 𝑡 𝑑𝑡

Let’s say:

𝑣𝐶 (𝑡) = 𝐾𝑒 𝑠𝑡

Then,

𝐾𝑒 𝑠𝑡 (𝐿𝐶𝑠 2 + 𝑅𝑇 𝐶𝑠 + 1) = 0

Since 𝐾𝑒 𝑠𝑡 can not be zero;

𝐿𝐶𝑠 2 + 𝑅𝑇 𝐶𝑠 + 1 = 0

We will find s from this equation for our circuit.