1. Chapter 7 First-Order Circuits
電路學(一)
Chapter 7
First-Order Circuits

2. First-Order Circuits Chapter 7
7.1 The Source-Free RC Circuit
7.2 The Source-Free RL Circuit
7.3 Singularity Functions
7.4 Step Response of an RC Circuit
7.5 Step Response of an RL Circuit
7.6 Applications

3. 7.1 The Source-Free RC Circuit (1)
A first-order circuit is characterized by a first-order differential equation.
By KCL
Ohms law
Capacitor law
Apply Kirchhoff’s laws to purely resistive circuit results in algebraic equations.
Apply the laws to RC and RL circuits produces differential equations.

4. 7.1 The Source-Free RC Circuit (2)
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
Time constant
Decays more slowly
Decays faster
The time constant  of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.
v decays faster for small t and slower for large t.

5. 7.1 The Source-Free RC Circuit (3)
The key to working with a source-free RC circuit is finding:
where
The initial voltage v(0) = V0 across the capacitor.
The time constant  = RC.

6. 7.1 The Source-Free RC Circuit (4)
Example 1
In the figure, let vC(0) = 15 V. Find vC, vx, and ix for t > 0.
8  ix + vC – + vx –
5 
0.1 F
12 

7. 7.1 The Source-Free RC Circuit (5)
Example 2
Refer to the circuit below, determine vC, vx, and io for t ≥ 0.
Assume that vC(0) = 30 V.
Please refer to lecture or textbook for more detail elaboration.
Answer: vC = 30e–0.25t V ; vx = 10e–0.25t ; io = –2.5e–0.25t A

8. 7.1 The Source-Free RC Circuit (6)
Example 3
The switch in circuit has been closed for a long time, and it is opened at t = 0. Find v(t) for t ≥ 0. Calculate the initial energy stored in the capacitor.

9. 7.1 The Source-Free RC Circuit (7)
Example 4
The switch in circuit below is opened at t = 0, find v(t) for t ≥ 0.
Please refer to lecture or textbook for more detail elaboration.
Answer: V(t) = 8e–2t V

10. 7.2 The Source-Free RL Circuit (1)
A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent)
By KVL
Inductors law
Ohms law

11. 7.2 The Source-Free RL Circuit (2)
A general form representing a RL
where
The time constant  of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.
i(t) decays faster for small t and slower for large t.
The general form is very similar to a RC source-free circuit.

12. 7.2 The Source-Free RL Circuit (3)
Comparison between a RL and RC circuit
A RL source-free circuit
where
A RC source-free circuit
where

13. 7.2 The Source-Free RL Circuit (4)
The key to working with a source-free RL circuit is finding:
where
The initial voltage i(0) = I0 through the inductor.
The time constant  = L/R.

14. 7.2 The Source-Free RL Circuit (5)
Example 5
Assume that i(0) = 10 A, calculate i(t) and ix(t) in the circuit. .
Equivalent circuit
Loop analysis

15. 7.2 The Source-Free RL Circuit (6)
Example 6
Find i and vx in the circuit.
Assume that i(0) = 5 A.
Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = 5e–53t A

16. 7.2 The Source-Free RL Circuit (7)
Example 7
The switch in the circuit has been closed for a long time. At t = 0, the switch is opened. Calculate i(t) for t > 0.

17. 7.2 The Source-Free RL Circuit (8)
Example 8
For the circuit, find i(t) for t > 0.
Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = 2e–2t A

18. 7.2 The Source-Free RL Circuit (9)
Example 9
Assume the switch in the circuit was open for a long time, find io, vo, and i for all time.

19. 7.3 Singularity Functions (1)
The singularity functions奇異函數are functions that either are discontinuous or have discontinuous derivatives.

20. 7.3 Singularity Functions (2)
The unit step function單位步階函數 u(t) is 0 for negative values of t and 1 for positive values of t.

21. 7.3 Singularity Functions (3)
Represent an abrupt change for:
voltage source.
for current source:

22. 7.3 Singularity Functions (4)
The unit impulse function單位脈衝函數 (t) is zero everywhere except at t = 0, where is undefined.

23. 7.3 Singularity Functions (5)
The unit ramp function單位斜率函數 r(t) is zero for negative t and has a unit slope for positive values of t.

24. 7.3 Singularity Functions (6)
Example 10
Express the voltage pulse in the figure in terms of the unit step. Calculate its derivative and sketch it.

25. 7.3 Singularity Functions (7)
Example 11
Express the current pulse in the figure in terms of the unit step. Find its integral and sketch it.

26. 7.3 Singularity Functions (8)
Example 12
Express the sawtooth function in the figure in terms of the singularity functions.

27. 7.3 Singularity Functions (9)
Example 13
Given the signal
express g(t) in terms of step and ramp functions.

28. 7.3 Singularity Functions (10)
Example 14
Evaluate the following integrals involving the impulse function:

29. 7.4 The Step-Response of a RC Circuit (1)
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 condition:
v(0-) = v(0+) = V0
Applying KCL, or
Where u(t) is the unit-step function

30. 7.4 The Step-Response of a RC Circuit (2)
Integrating both sides and considering the initial conditions, the solution of the equation is:
Final value at t -> ∞
Initial value at t = 0
Source-free Response
完整響應
自然響應
激勵響應
Complete Response = Natural response + Forced Response
(stored energy) (independent source)
= V0e–t/τ Vs(1–e–t/τ)

31. 7.4 The Step-Response of a RC Circuit (3)
Three steps to find out the step response of an RC circuit:
The initial capacitor voltage v(0).
The final capacitor voltage v() — DC voltage across C.
The time constant .
steady-state
response
穩態響應
transient response
暫態響應

32. 7.4 The Step-Response of a RC Circuit (4)
Example 15
The switch has been in position A for a long time. At t = 0, the switch move to B. Find v(t) for t > 0 in the circuit and calculate v(t) at t = 1 s and 4 s.

33. 7.4 The Step-Response of a RC Circuit (5)
Example 16
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.
Calculate v(t) at t = 0.5.
Please refer to lecture or textbook for more detail elaboration.
Answer: and v(0.5) = V

34. 7.4 The Step-Response of a RC Circuit (6)
Example 17
In the figure, switch has been close for a long time and is opened at t = 0. Find i and v for all time.

35. 7.5 The Step-response of a RL Circuit (1)
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 current
i(0-) = i(0+) = Io
Final inductor current
i(∞) = Vs/R
Time constant t = L/R

36. 7.5 The Step-Response of a RL Circuit (2)
Three steps to find out the step response of an RL circuit:
The initial inductor current i(0) at t = 0+.
The final inductor current i().
The time constant .
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.

37. 7.5 The Step-Response of a RL Circuit (4)
Example 18
Find i(t) in the circuit for t > 0. Assume that the switch has been closed for a long time.

38. 7.5 The Step-Response of a RL Circuit (5)
Example 19
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.
Please refer to lecture or textbook for more detail elaboration.
Answer:

39. 7.5 The Step-Response of a RL Circuit (6)
Example 20
At t = 0, switch 1 is closed, and switch 2 is closed 4 s later. Find i(t) for t > 0. Calculate i for t = 2 s and t = 5 s.

40. 7.6 First-Order Op Amp Circuits (1)
Example 21
For the op amp circuit, find vo for t > 0, given that that v(0) = 3 V. Let Rf = 80 k, R1 = 20 k, and C = 5 μF.

41. 7.6 First-Order Op Amp Circuits (2)
Example 22
Determine v(t) and vo(t) in the circuit.

42. 7.6 First-Order Op Amp Circuits (3)
Example 23
Find the step response vo(t) for t > 0 in the op amp. Let vi = 2u(t) V, R1 = 20 k, Rf = 50 k, R2 = R3 = 10 k, and C = 2 μF.

43. 7.7 Applications (1) Delay Circuits
Example 24
Consider the delay circuit, and assume that R1 = 1.5 M, 0 < R2 < 2.5 M. (a) Calculate the extreme limits of the time constant of the circuit. (b) How long does it take for the lamp to glow for the first time after the switch is closed? Let R2 assume its largest value.

44. 7.7 Applications (2)
Photoflash Unit

45. 7.7 Applications (3)
Example 25
An electric flashgun has a current-limiting 6-k resistor and 2000-μF electrolytic capacitor charged to 240 V. If the lamp resistor is 12 , find (a) the peak charging current, (b) the time required for the capacitor to fully charge, (c) the peak discharging current, (d) the total energy stored in the capacitor, and (e) the average power dissipated by the lamp.

46. 7.7 Applications (4) Relay Circuits
Example 26
The coil of a certain relay is operated by a 12-V battery. If the coil has a resistance of 150  and an inductance of 30 mH and the current needed to put in is 50 mA, calculate the relay delay time.

47. 7.7 Applications (5) Automobile Ignition Circuit
Example 27
A solenoid with resistance 4  and an inductance 6 mH is used in an automobile ignition circuit. If the battery supplies 12-V, determine: the final current through the solenoid when the switch is closed, the energy stored in the coil, and the voltage across the air gap, assume that the switch takes 1 μs to open.