EEE105 Teori Litar I Chapter 7 Lecture #3. Dr. Shahrel Azmin Suandi Emel:

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1 EEE105 Teori Litar I Chapter 7 Lecture #3 Dr. Shahrel Azmin Suandi Emel: shahrel@eng.usm.my

2 What we have learnt so far? Chapter 7 introduced us to first-order circuit From the last lecture, we have learnt about Source-free RL circuit We want to calculate the i, instead of v (in source-free RC circuit) What we should do?? Find the i(0) = I 0 and the time constant Singularity functions τ = L R Unit step function the most basic Unit impulse function resulted from derivative of unit step Unit ramp function resulted from integration of unit step

3 Remember this!!! For Singularity functions u(t) R δ(t) dt d(u(t)) dt R u(t) dt d(r(t)) dt δ(t) r(t)

4 7.5 Step Response of an RC Circuit When the dc source of an RC circuit is suddenly applied, the voltage or current source can be modeled as a step function The response is known as step response Definition of step response: The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source When do we get step function from a circuit?? When there is a sudden application of a dc voltage or current source

5 Computing capacitor voltage v(t) as the circuit response (1) Refer to the figures on the right Assume that the initial voltage V 0 on the capacitor Voltage of a capacitor cannot change abruptly /instantaneously. Therefore, v(0 )=v(0 + )=V 0 Using KCL we have the following equations C dv dt + v V su(t) R =0 v(0 ): v(0 + ): dv dt + for t>0: dv dt + Before switching After switching v RC = V s RC u(t) v RC = V s RC Rearrange the terms, dv v V s = dt RC Integrating both sides ln v V s V 0 V s = t RC

6 Computing capacitor voltage v(t) as the circuit response (2) Taking the exponential of both side v V s V 0 V s = e t/τ, τ = RC v V s =(V 0 V s )e t/τ v(t) =V s +(V 0 V s )e t/τ, t > 0 Complete Response of RC circuit to a sudden application of a dc voltage source v(t) = ½ V0, t < 0 V s +(V 0 V s )e t/τ, t > 0

7 Computing capacitor voltage v(t) as the circuit response (3) Assuming the capacitor is uncharged initially v(t) = ½ V0 =0, t < 0 V s +(V 0 V s )e t/τ, t > 0 v(t) =V s +(V 0 V s )e t/τ, = V s +(0 V s )e t/τ, = V s V s e t/τ, v(t) =V s (1 e t/τ ), t > 0 v(t) = ½ V0 =0, t < 0 V s (1 e t/τ ), t > 0

8 Computing capacitor voltage v(t) as the circuit response (4) v(t) canberewrittenas Complete Step Response of RC circuit when capacitor is uncharged initially v(t) =V s (1 e t/τ ) u(t) Voltage response i(t), the current through the capacitor i(t) = V s R e t/τ u(t) Current response

9 Finding Step Response besides using derivation methods There are two ways of finding the step response, besides the one using derivation method introduced in previous slides What are they?? Using natural and forced responses Using transient and steady-stage responses The two terms mentioned above can be decomposed to form v(t)

10 Method I Natural and Forced Responses Natural response is the stored energy, and expressed by, where v n Forced response is the independent source, and expressed by v f, where Known as forced response due to the reason that it is produced by the circuit when an external force is applied v v n = V 0 e t/τ v f = V s (1 e t/τ ) is decomposed as follows: v = v n + v f Response that dies out

11 Method II Transient and Steady- State Responses Transient response is the temporary response and is expressed by v t, where Steady-state response is the permanent part response and is expressed by v ss, where Remains after transient response has died out v v t =(V 0 V s )e t/τ v ss = V s is decomposed from these two responses as follows: v = v t + vss

12 Complete Response After Decomposition of Two Responses Natural and transient responses are same under certain conditions Important to know that they are both temporary Forced and Steady-state responses are also same under certain conditions Important to know that they are both will remain in the circuit / permanent initial voltage at t =0 + Final complete response is v(t) =v( )+[v(0) v( )]e t/τ final or steady-state value

13 Final Step to Finding the Step Response of an RC Circuit v(0) Initial capacitor voltage, Can be obtained when t<0 v( ) Final capacitor voltage, Obtained when t>0 Time constant, τ Obtained when t>0 When there is a time delay,, then v(t) =v( )+[v(t 0 ) v( )]e (t t 0)/τ v(t) We must remember that given above applies only to step responses, i.e., when the input excitation is constant t 0