CHAPTERS 7

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CHAPTERS 7 8

• NETWORKS 1 0909201-01
• 19 October 2004 Lecture 6b
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2005 Quarter One

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test II

• Historic means
• 2004 69.6 (75.4 Adjusted)
• 2003 63
• This Year mean

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networks I

• Todays learning objectives
• build an understanding of
• First (1st) Order Circuits

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admin

• less than 28 hours remain.
• All previous homework 1-6
• due Friday by 5 p.m. for final grade
• new homework
• hw 7 due Thursday (15 Take Home)
• final test this Thurs 315pm
• Rowan Hall Auditorium

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HANDY CHART
ELEMENT CURRENT VOLTAGE
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OP-AMP CIRCUITS WITH C L
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QUIZ Find vo f(vs)
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ANSWER TO QUIZ
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Handier Charts for studying

• see Table 7.9-1
• Characteristics of Energy Storage Elements
• Page 266
• and Table 7.14-2
• Parallel and Series Capacitors and Inductors
• Page 281
• Inductance
• Behaves as a short to constant DC current
• Capacitance
• Behaves as an open circuit to constant DC voltage

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Table 7.9-1 (p. 266) Characteristics of Energy
Storage Elements
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Initial conditions of switched circuits

• switch changes
• t0, at time of switching
• t0-, just prior to switching
• t0, just after switching
• t8, a long time after switching, steady state
• Instantaneously
• Capacitor current can change, voltage cant
• Inductor voltage can change, current cant

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• steady state circuit conditions just before the
  switching change and again a very long period
  after the switching change
• inductor in a steady DC current acts as a short
  circuit with no voltage drop
• vL L (di/dt) if di/dt0 so does v
• capacitor in a constant DC voltage acts as an
  open circuit with no current flow
• iC C (dv/dt) if dv/dt0 so does i

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simple illustrations of switching

• an inductor example
• If R1R21? What is i1 and iL at t(0-) t(0)
• a capacitor and inductor example
• LC1 What is vC and iL at t(0-) t(0)

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HW problem 7.9-2

• page 286

LC2 What is total resistance the 12V source sees
in the circuit at t(0-) t(0), answer is two
(2) numbers Rt(0-) Rt(0),
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Circuit for Example 7.9-1 Switch 1 closes at t
0 and switch 2 opens at t 0, Find iL(0),
vc(0), dVc(0)/dt, and diL(0)/dt, assume switch
2 has been closed for a long time.
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Circuit for Example 7.9-1 at t(0-) Switch 1 is
not closed yet, and switch 2 has been closed for
a long time, all current flows through 1?
resistor. Find iL(0-) and voltage across
capacitor vc(0-) is ?
Show your answers as Learning check 3
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Once you found out what iL(0-) and voltage across
capacitor vc(0-) is, how do these quantities
relate to

• The instant afterwards?
• iL(0-) and iL(0) current in inductor
• vc(0-) and vc(0) voltage across capacitor
• why?

Show your answers as Learning check 4
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Circuit for Example 7.9-1 at t(0) Switch 1 has
just closed, and switch 2 has just opened. To
solve for diL(0)/dt we need a KVL in right hand
mesh, to solve for dvc(0)/dt we need a KCL at
node a.
KVL in right hand mesh vL vC 1iL 0 so
diL(0)/dt -2A/s KCL at node a (vc-10)/2 ic
iL 0 so dvc(0)/dt 12V/s
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What did we learn here?

• at switching time (t0) current in inductor and
  voltage in capacitor remained constant
• but voltage in inductor changed instantaneously
  from 0V to -2V with diL(0)/dt -2A/s
• and
• current through capacitor changed instantaneously
  from 0 to 6 A with
• dvc(0)/dt 12V/s

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LAST STEP What are final conditions at t t(8)
for vc, ic, vL and iL ?
CURRENT IN CAPACITOR is Zero VOLTAGE ACROSS
INDUCTOR is Zero SHOW vc and iL as LEARNING CHECK
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IMPORTANT CONCEPTS FROM CH. 7

• I/V Characteristics of C L.
• Energy storage in C L.
• Writing KCL KVL for circuits with C L.
• Solving op-amp circuits with C or L in feedback
  loop.
• Solving op-amp circuits with C or L at the
  input.

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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

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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

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whats the complete response (CR)?

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

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finding the CR of 1st Order Circuit

• 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(8) Voc or Isc
• Add the natural response (Ke-t/?) to the new
  forced response. Use initial conditions to
  calculate K

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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.
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RC and RL circuits

• RC circuit complete response
• RL circuit complete response

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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

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

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

LC6 Write down vsource at t(0-) t(0)
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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)

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

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

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Figure 8.3-4 (a) A first-order circuit and (b) an
equivalent circuit that is valid after the switch
opens. (c) A plot of the complete response.
LC7 What is Rt(0) and VOC?
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Lets Build the Complete Response for the circuit
1) Find the forced response before the
disturbance. Evaluate at t t(0-) to determine
initial conditions v(0-) or i(0-) in our case
v(0-) 2V 2) Find forced response (steady
state) after the disturbance t t(8) Voc or
Isc in our case VOC 8V 3) Add the natural
response (Ke-t/?) to the new forced response. Use
initial conditions to calculate K in our case Rt
10,000 and C2?F so RtC has value of 20 and
units of milliseconds
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What is meaning of this new equation?

• Lets plot a few points.

When does steady state occur With respect to RtC?
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WITH AN INDUCTOR
t 0
R1
R2
R3
i(t)
L
vs
Why ?
LC8 Give your answer
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Norton equivalent at t0
Why ?
KCL
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SOLUTION
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so complete response is

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

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Figure 8.3-5(a) A first-order circuit and (b) an
equivalent circuit that is valid after the switch
closes. (c) A plot of the complete response.
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Lets Practice getting the CR

• Determine what equation to use
• Determine initial condition
• Determine steady state condition
• Determine Rt
• Solve

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Figure E8.3-1 (p. 308)
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Figure E8.3-2 (p. 309)
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Figure E8.3-3 (p. 309)
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Figure E8.3-4 (p. 309)
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Figure E8.3-5 (p. 309)
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Stability of 1st order circuits

• when ?gt0 the natural response vanishes as t ?8
• THIS IS A STABLE CIRCUIT
• when ?lt0 the natural response grows without bound
  as t?8
• THIS IS AN UNSTABLE CIRCUIT

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forced response summary
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Unit step or pulse signal

• vo(t) A Be-at
• for t gt 0

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Example

• 8.6-2, p. 321-323

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

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Dont forget HW 7 (test takehome)

• due tomorrow - Thursday