Transient Response of First Order Circuits: Source-Free RC and RL Circuits, Study notes of Electronic Circuits Analysis

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ELL 100 - Introduction to Electrical Engineering

LECTURE 9:

TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS

(NATURAL RESPONSE)

SOURCE-FREE RC CIRCUITS EXAMPLE Fluid-flow analogy: water tank emptying through a small pipe Electrical circuit: Capacitor discharging through resistance

SOURCE-FREE RC CIRCUITS APPLICATIONS High pass filter Low pass filter

SOURCE-FREE RC CIRCUITS APPLICATIONS Oscillators 555 Timer circuits Timers Camera Flash

SOURCE FREE RC CIRCUITS APPLICATIONS Computer Circuits Digital and Time delay circuits

SOURCE FREE RC CIRCUITS APPLICATIONS Pacemakers Timing device in automobile intermittent wiper system

TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS

• A first-order circuit is characterized by a first-order differential equation.
• Example :
  • a circuit comprising a resistor and capacitor (RC circuit)
  • a circuit comprising a resistor and an inductor (RL circuit) Applying Kirchhoff’s laws to RC or RL circuit results in differential equations involving voltage or current, which are first-order.

TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS EXCITATION There are two ways to excite the circuits.

• Initial conditions of the storage elements– Source-Free Circuits (Energy stored in the capacitor, Energy stored in the inductor)
• Independent sources – Forced Excitation circuits (DC sources, Sinusoidal sources, Exponential Sources)

SOURCE-FREE RC CIRCUIT

• A source-free RC circuit occurs when its DC source is suddenly disconnected.
• The energy already stored in the capacitor(s) is released to the resistor(s) & dissipated.
• RC source-free circuit is analyzed from its

initial voltage v (0) = V

0 and time constant τ

SOURCE-FREE RC CIRCUIT DERIVATION

• Assume the voltage v ( t ) across the capacitor.
• Since the capacitor is initially charged, Assume that at time t = 0, the initial voltage is, with the corresponding value of the energy stored as 0 v (0)  V 2 0 1 (0) 2 w  CV

SOURCE-FREE RC CIRCUIT DERIVATION => Integrating both sides, we get => => But from the initial conditions, v ( 0 ) = A = V 0

Hence, (Exponentially Decaying) 0 dv v dt RC   dv 1 dt v RC   ln ln A t v RC    ln v t A RC   / ( ) t RC v t A e   / 0 ( ) V t RC v t e  

SOURCE-FREE RC CIRCUIT VOLTAGE RESPONSE

• As t increases, the voltage decreases exponentially towards zero. The rapidity with which the voltage decreases is expressed in terms of the time constant , denoted by τ.

SOURCE-FREE RC CIRCUIT TIME CONSTANT Graphical determination of the time constant τ from the response curve. t v(t)/V 0 τ 0. 2 τ 0. 3 τ 0. 4 τ 0. 5 τ 0.

SOURCE-FREE RC CIRCUIT TIME CONSTANT