NETWORKS 1: 09092010304

1
CHAPTER 8

• NETWORKS 1 0909201-03/04
• 10 December 2003 Lecture 8b
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2003

2
admin

• hw 7 due today, hw 8 due at final
• test review 5.15pm thurs. at end of lab
• last lab 6 due by end of next weeks normal lab
  day (no later than 5 PM)
• final exam Next Mon 15 Dec 245pm
• Rowan Hall Auditorium
• take home portion
• Assignment 8 (15)
• Tool Kit (10)

3
networks I

• Todays learning objectives
• master first order circuits
• build knowledge of the complete response
• use Thevenin and Norton equivalents to simplify
  analysis of first order circuits
• calculate the natural (transient) response and
  forced (steady-state) response

4
new concepts from ch. 8

• response of first-order circuits
• to a constant input
• the complete response
• stability of first order circuits
• response of first-order circuits
• to a nonconstant (sinusoidal) source

5
What does First Order mean?

• circuits that contain capacitors and inductors
  can be defined by differential equations
• circuits with ONLY ONE capacitor OR ONLY ONE
  inductor can be defined by a first order
  differential equation
• such circuits are called First Order Circuits

6
whats the complete response (CR)?

• Complete response transient response steady
  state response
• OR.
• Complete response natural response forced
  response

7
finding the CR of 1st Ord. Cir

• Find the forced response before the disturbance.
  Evaluate at t t(0-) to determine initial
  conditions v(0) or i(0)
• Find forced response (steady state) after the
  disturbance t t(?) Voc or Isc
• Add the natural response (Ke-t/?) to the new
  forced response. Use initial conditions to
  calculate K

8
Figure 8.0-1 (p. 290)A plan for analyzing
first-order circuits. (a) First, separate the
energy storage element from the rest of the
circuit. (b) Next, replace the circuit connected
to a capacitor by its Thévenin equivalent
circuit, or replace the circuit connected to an
inductor by its Norton equivalent circuit.
9
RC and RL circuits

• RC circuit complete response
• RL circuit complete response

10
simplifying for analysis

• Using Thevenin and Norton Equivalent circuits
  can greatly simplify the analysis of first order
  circuits
• We use a Thevenin with a Capacitor
• and a Norton with an Inductor

11
Thevenin Equivalent at t0
i(t)
-
12
Norton equivalent at t0
13
1st ORDER CIRCUITS WITH CONSTANT INPUT
14
Example (before switch closes)

• If vs 4V, R1 20kohms,
• R2 20 kohms
• R3 40 kohms
• What is v(0-) ?

15
as the switch closes

• THREE PERIODS emerge..
• 1. system change (switch closure)
• 2. (immediately after) capacitor or inductor in
  system will store / release energy (adjust and/or
  oscillate) as system moves its new level of
  steady state (a.k.a. transient or natural
  response) . WHY???
• 3. new steady state is then achieved (a.k.a. the
  forced response)

16
Thevenin Equivalent at t0
i(t)
-
KVL
17
SOLUTION OF 1st ORDER EQUATION
18
SOLUTION CONTINUED
19
SOLUTION CONTINUED
20
so complete response is

• complete response v(t)
• forced response (steady state) Voc
• natural response (transient)
• (v(0-) Voc) e -t/RtC) NOTE ? RtC

21
Figure 8.3-4 (p. 301) (a) A first-order circuit
and (b) an equivalent circuit that is valid after
the switch opens. (c) A plot of the complete
response, v(t), given in Eq. 8.3-8.
22
WITH AN INDUCTOR
t 0
R1
R2
R3
i(t)
L
vs
Why ?
23
Norton equivalent at t0
Why ?
KCL
24
SOLUTION
25
so complete response is

• complete response i(t)
• forced response (steady state) Isc
• natural response (transient)
• (i(0-) isc) e t(Rt/L)) NOTE ? L/Rt

26
Figure 8.3-5 (p. 302) (a) A first-order circuit
and (b) an equivalent circuit that is valid after
the switch closes. (c) A plot of the complete
response, i(t), given by Eq. 8.3-9.
27
Figure E8.3-1 (p. 308)
28
Figure E8.3-2 (p. 309)
29
Figure E8.3-3 (p. 309)
30
Figure E8.3-4 (p. 309)
31
Figure E8.3-5 (p. 309)
32
Stability of 1st order circuits

• when ?0 the natural response vanishes as t ??
• THIS IS A STABLE CIRCUIT
• when ?as t??
• THIS IS AN UNSTABLE CIRCUIT

33
forced response summary
34
Unit step or pulse signal

• vo(t) A Be-at
• for t 0

35
Example

• 8.6-2, p. 321-323

36
Figure 8.6-12 (p. 322) The circuit considered
in Example 8.6-2
37
Figure 8.6-13 (p. 322) Circuits used to
calculate the steady-state response (a) before t
0 and (b) after t 0.
38
HANDY CHART
ELEMENT CURRENT VOLTAGE
39
IMPORTANT CONCEPTS FROM CHAPTER 8

• determining Initial Conditions
• determining T or N equivalent to simplify
• setting up differential equations
• solving for v(t) or i(t)

40
Dont forget HW 8 (test review)

• Thursday 5.15 pm 11 Dec after lab