The transient response of an RLC circuit represents the temporary behavior of voltage and current immediately following a sudden change, such as a switch closing or opening, before the system settles into its steady-state condition. Unlike simpler first-order RC or RL circuits, an RLC circuit contains two energy-storage elements (an inductor L and a capacitor C). The interaction between these two elements allows energy to transfer back and forth between them, governed by a second-order differential equation. 1. Mathematical Modeling
By applying Kirchhoff’s Voltage Law (KVL) to a standard series RLC circuit excited by a constant source voltage Vscap V sub s
, the sum of the voltages across the resistor, inductor, and capacitor must equal the source voltage:
VR(t)+VL(t)+VC(t)=Vscap V sub cap R open paren t close paren plus cap V sub cap L open paren t close paren plus cap V sub cap C open paren t close paren equals cap V sub s
Substituting the characteristic current-voltage relationships (
Ri(t)+Ldi(t)dt+1C∫i(t)dt=Vscap R space i open paren t close paren plus cap L the fraction with numerator d i open paren t close paren and denominator d t end-fraction plus the fraction with numerator 1 and denominator cap C end-fraction integral of i open paren t close paren space d t equals cap V sub s
Differentiating this expression once with respect to time results in the classic second-order linear homogeneous differential equation for the circuit current i(t):
d2i(t)dt2+RLdi(t)dt+1LCi(t)=0the fraction with numerator d squared i open paren t close paren and denominator d t squared end-fraction plus the fraction with numerator cap R and denominator cap L end-fraction the fraction with numerator d i open paren t close paren and denominator d t end-fraction plus the fraction with numerator 1 and denominator cap L cap C end-fraction i open paren t close paren equals 0 2. Key System Parameters The roots of the system’s characteristic equation (
) determine the exact nature of the transient waveform. These roots are shaped by two fundamental physical parameters: Transient Response of Series RLC Circuit using MATLAB
You will see updates in your followed content feed; You may receive emails, depending on your communication preferences. Overview: Transient Analysis of the RLC Circuit (with Examples)