ece 1311ece 1311 first order...

ECE 1311ECE 1311First Order Circuits Chapter 7

Dr. AHM Zahirul AlamECE Dept., IIUM

Dr. AHM Zahirul Alam

2Dr. AHM Zahirul Alam

The Step-Response of a RC Circuit (2)The Step Response of a RC Circuit (2)

Integrating both sides and considering the initial g g gconditions, the solution of the equation is:

⎩⎨⎧

>−+

<=

− 0)(

0)( /

0

0

teVVV

tVtv t

ssτ

Final value at Initial value at Source-freeFinal value at t -> ∞

Initial value at t = 0

Source free Response

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34

Complete Response = Natural response + Forced Response(stored energy) (independent source)

= V0e–t/τ + Vs(1–e–t/τ)

The Step-Response of a RC Circuit (3)p p ( )

Three steps to find out the step response of anThree steps to find out the step response of an RC circuit:

1. The initial capacitor voltage v(0).1. The initial capacitor voltage v(0).2. The final capacitor voltage v(∞) — DC voltage

across C.3. The time constant τ.

/τ/)]( )0([ )( )( tevvvtv −∞−++∞=Note: The above method is a short cut method You may also determine the

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Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws.

The Step-Response of a RC Circuit (4)

Example 5

p p ( )

Find v(t) for t > 0 in the circuit in below. Assume the switch has been open for a long time and is closed at t = 0= 0. Calculate v(t) at t = 0.5.

2t

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Answer: and v(0.5) = V5060)( 2 −= − tetv

The Step-response of a RL Circuit (1)p p ( )

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.

• Initial currenti(0-) = i(0+) = Io

• Final inductor currenti(∞) = Vs/R

• Time constant τ = L/R

)()()( tueRVI

RVti

ts

os τ

−−+=

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44

)()()(RR o

The Step-Response of a RL Circuit (2)p p ( )

Three steps to find out the step response of anThree steps to find out the step response of an RL circuit:

1 Th i iti l i d t t i(0) t t 0+1. The initial inductor current i(0) at t = 0+.2. The final inductor current i(∞).3 The time constant3. The time constant τ.

τ/)]()0([)()( teiiiti −∞−++∞=

Note: The above method is a short cut method You may also determine

)]( )0([)( )( eiiiti ∞++∞

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45

Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws.

The Step-Response of a RL Circuit (4)

Example 6

p p ( )

The switch in the circuit shown below has been closed for a long time. It opens at t = 0. Find i(t) for t > 0.

46

46Answer: teti 102)( −+=

3

•Read Chapters 6

•Required Problems -Ch. 6: 17, 46, 48, 54, 64

•Recommended Problems Ch 6: 13 19 53 65 73 -Ch. 6: 13, 19, 53, 65, 73


Read Chapters 9 & 10 Appendix B Read Chapters 10 & 11Appendix B Required Problems

Ch. 7: 47, 64, 69 Ch. 9: 8, 11, 14, 16, 17 18

Read Chapters 10 & 11 Required Problems

Ch. 10: 19, 36, 42, 56, 80

R d d P bl17, 18

Recommended Problems Ch. 7: 50, 57, 71

Recommended Problems Ch. 10: 16, 39, 66, 77, 76

, ,Ch. 9: 9, 10, 19


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