FIRST ORDER CIRCUIT Topic v Natural ResponseSource Free

FIRST ORDER CIRCUIT

Topic v Natural Response/Source Free of RL Circuit v Natural Response/Source Free of RC Circuit v Step Response of RL Circuit v Step Response of RC Circuit

First – Order Circuit • A circuit that contains only sources, resistor and inductor is called and RL circuit. • A circuit that contains only sources, resistor and capacitor is called an RC circuit. • RL and RC circuits are called first – order circuits because their voltages and currents are describe by first order differential equations.

First – Order Circuit R i L An RL circuit vs – + Vs R i C An RC circuit

Review (conceptual) • Any first – order circuit can be reduced to a Thévenin (or Norton) equivalent connected to either a single equivalent inductor or capacitor. RTh RN L VTh – + IN C ØIn steady state, an inductor behave like a short circuit. ØIn steady state, a capacitor behaves like an open circuit.

Review (conceptual) • The natural response of an RL and RC circuit is its behavior (i. e. , current and voltage ) when stored energy in the inductor or capacitor is released to the resistive part of the network (containing no independent sources) • The steps response of an RL and RC circuits is its behavior when a voltage or current source step is applied to the circuit, or immediately after a switch state is changed.

Natural Response of an RL circuit Ø Consider the following circuit, for which the switch is closed for t<0, and then opened at t = 0: t=0 Io Ro i L + R V – ØThe dc voltage V, has been supplying the RL circuit with constant current for a long time

Natural Response of an RL circuit Solving the circuit For t ≤ 0, i(t) = Io. Switch is closed. Inductor act as short circuit. ii. At t = 0, assume inductor has initial current, Io iii. The initial energy stored in the inductor is, i.

Natural Response of an RL circuit iv. For t ≥ 0, switch is open and the circuit reduce to

Natural Response of an RL circuit • Applying KVL to the circuit: (1)

Natural Response of an RL circuit • From equation (1), let say; (2) • Integrate both sides of equation (2);

Natural Response of an RL circuit • hence, the current is

Natural Response of an RL circuit • From the Ohm’s law, the voltage across the resistor R is: • the power dissipated in the resistor is:

Time Constant, τ for RL circuit • Time constant, τ determines the rate at which the current or voltage approaches zero. • The time constant of a circuit is the time required for the response to decay to a factor of 1/e or 36. 8% of its initial current • Natural response of the RL circuit is an exponential decay of the initial current. The current response is shown in figure.

Time Constant, τ for RL circuit • Time constant for RL circuit is • Unit : seconds.

• The expressions for current, voltage, power and energy using time constant concept:

Example 1 • The switch has been closed for a long time. At t = 0, the switch is opened. Calculate i (t) for t > 0.

Example 2 • The switch has been closed for a long time. At t = 0, the switch is opened. Calculate i (t) for t > 0.

Natural Response of an RC Circuit • The natural response of RC circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor, C is released to the resistors, R. • Consider the following circuit, for which the switch is closed for t < 0, and then opened at t = 0: + Vo Ro C t=0 + v – R

Natural Response of an RC Circuit Solving the circuit i. For t ≤ 0, switch is closed. Capacitor behaves as open circuit. ii. Voltage will be v(t) = Vo iii. At t = 0, the initial voltage v(0) = Vo iv. The initial value of the energy stored is

Natural Response of an RC Circuit v. For t > 0, switch is opened. The circuit reduces to i + Vo + Ro C v – R

Natural Response of an RC Circuit • Applying KCL to the RC circuit: (1)

Natural Response of an RC Circuit • From equation (1), let say: (2) • Integrate both sides of equation (2): (3) • Therefore: (4)

Natural Response of an RC Circuit • The voltage is: • Using Ohm’s law, the current is: • The power dissipated in the resistor is:

Time Constant, τ for RC circuit • The time constant for the RC circuit equal the product of the resistance and capacitance, • Time constant, seconds

Time Constant, τ for RC circuit • The natural response of RC circuit illustrated graphically in figure.

• The expressions for voltage, current, power and energy using time constant concept:

Example 3 The switch in the circuit below has been closed for a long time, and it is opened at t = 0. i. Find v(t) for t ≥ 0. ii. Calculate the initial energy, W stored in the capacitor.

Summary of Natural Response No RL circuit 1 RC circuit 2 Inductor behaves like a short circuit when being supplied by dc source for a long time Capacitor behaves like an open circuit when being supplied by dc source for a long time 3 Inductor current is continuous i. L(0+) = i. L(0 -) Voltage across capacitor is continuous v. C(0+) = v. C(0 -)

Switching time • For all transient cases, the following instants of switching times are considered. ü t = 0 - : switching between -∞ to 0 or time before. ü t = 0+ : switching at the instant just after time t = 0 s (taken as initial value) ü t = ∞ : switching between t = 0+ to ∞ (taken as final value for step response) • The illustration of the different instance of switching times is: ∞ ∞

Step Response of RL Circuit • The step response is the response of the circuit due to a sudden application of a dc voltage or current source. • Consider the RL circuit below and the switch is closed at time t = 0. i(t) + Vs R + t=0 L v(t) –

Step Response of RL Circuit • After switch is closed, using KVL

Step Response of RL Circuit

Step Response of RL Circuit • the current is; • Or may be written as; • i(0) : initial values of i • i(∞) : final values of i,

Step Response of RL Circuit • The voltage across the inductor is; • Or;

Example 4 The switch is closed for a long time at t = 0, the switch opens. Find the expressions for i. L(t) and v. L(t). t=0 10 V + 2Ω 3Ω i 1/3 H

Example 4

Step Response of RC Circuit • Consider the RC circuit below. The switch is closed at time t=0 + t=0 Is R C i vc(t) –

Step Response of RC Circuit • From the circuit; (1) • Division of Equation (1) by C gives; (2)

Step Response of RC Circuit • Same mathematical techniques with RL, the voltage is: • Or can be written as: • v(0) : initial values of v • v(∞) : final values of v

Step Response of RC Circuit • And the current is: • Or can be written as:

Example 5 The switch has been in position a for a long time. At t = 0, the switch moves to b. Find Vc(t) for t > 0 and calculate its value at t=1 s and t=4 s 3 kΩ 24 V + 5 kΩ a + Vc – 4 kΩ b t=0 0. 5 m. F + 30 V

Example 5