CHAPTERS 10

1
CHAPTERS 10 11

• NETWORKS 2 ECE 09.202.01
• 7 November 2006 Lecture 5
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2006 Quarter Two

2
Last week

• I got your note.
• How did Chapter 14 go last week with Dr
  Mandayams lectures?
• Today in Lab we will review HW 2
• It will be due at the end of lab

3
Lets go back to Chapter 10

• Well finish up this chapter and see where we go
  from there

4
Complex exponential forcing function

• the process
• 1. write the excitation (the forcing function) as
  a cosine waveform with a phase angle
• 2. introduce complex excitation (Eulers
  identity)
• e j? cos ? j sin ? (where ? ?t ?)
• 3. use the complex excitation and circuit
  differential equation along with assumed response
  
• 4. determine the constant A Be-j? so that
  desired response is achieved
• xe Ae j(?t ?) Be j(?t ? - ?)

5
steady-state response of circuit to complex
exponential forcing function
An EXAMPLE assume R 2?, L 1H, and vs 10
sin 3t V
step 1) write source as cosine waveform with
phase angle sovs 10 sin 3t V 10 cos
(3t90o), step 2) introduce complex excitation,
so ve 10 e j(3t-90o)
6
continuing our solution

• step 3) introduce complex excitation into
  circuits diff. equation
• ve L(die/dt) Rie
• obtaining
• 10 e j(3t-90o) die/dt 2Rie
• assuming our response is of the form ie Ae
  j(3t-90o)
• step 4) substitute, take the derivative and find
  the constant
• j3Ae j(3t-90o) 2Ae j(3t-90o) 10 e j(3t-90o)
• So j3A 2A 10

7
hw problem 10.4-6

• Find the steady state response if vs(t) cos 2t
  V

8
hw problem 10.4-6
Replace the real excitation by a complex
exponential excitation to get
Let
so
9
hw problem 10.4-6
Substituting
10
hw problem 10.4-6

• And solving

Finally
11
Steinmetz observed this process

• and it led to the phasor concept
• the phasor concept may be used when the circuit
  is linear, the steady-state response is sought
  and all independent sources are sinusoids of the
  same frequency
• a transform is a change in the mathematical
  description of a physical variable to facilitate
  computation
• a phasor is a transformed version of a sinusoidal
  voltage or current waveform and consists of the
  magnitude and phase angle information of the
  sinusoid

12
remember
13
Transforming from the Time Domain ?
to the Frequency Domain

• Write time domain function, y(t) as a cosine
  waveform with phase angle
• Express cosine waveform as complex quantity using
  Eulers identity
• Drop the real part notation
• Suppress the ej?t while noting ? for future use

14
example

• i 5 sin(100t 120o)
• 1st make sure we have cosine function
• i 5 cos(100t 30o)
• Lets convert to the frequency domain
• I Im??
• I 5?30o

15
LC 1

• i 4 cos(20t - 80o)
• Lets convert to the frequency domain
• I Im??

16
LC 2

• i 8 sin(20t - 20o)
• Lets convert to the frequency domain
• I Im??

17
Transforming from the Frequency Domain ?
to the Time Domain

• Write the phasor in exponential form
• Reinsert the factor ej?t
• Reinsert the real part operator Re
• Use Eulers identity to obtain the time function

18
example

• V 24?125o
• Lets convert to the time domain
• v(t) Vm cos(?t ?)
• v(t) 24 cos(?t 125o)

Where do we get the ? value from?
19
LC 3

• V 10?-140o
• Lets convert to the frequency domain
• v(t) Vm cos(?t ?)

20
Why use Phasors?

• The phasor method uses the transformation from
  the time domain to the frequency domain to more
  easily obtain the sinusoidal steady-state
  solution of the differential equation

RC circuit example
RL circuit example
21
RL example solution on board
vs(t) Vmcos 100t V , R 200?, L 2H
22
RC example solution below
i(t) 10 cos 100t A, R1?, C 0.01F
23
hw problem 10.5-1
vs(t) 15cos 4t V , R 6?, L 2H
find v(t) for inductor, steady-state
v(t) _
24
Phasor representation

• transformation from time domain to the frequency
  domain
• converts solution of a differential equation to
  an algebraic one
• we must now learn the phasor relations on R,L
  C elements

25
Resistors

• V RI
• if V 10? 0o, (v10 cos 30 t)
• and R 5
• what does I ?
• I V/R v/R2 cos 30 t

26
R time domain vs. frequency domain
27
LC 4

• current in a resistor is
• i 2 cos 100t A
• Find the steady-state voltage across the resistor
  if its resistance is 10?

28
Inductors

• V j?LI
• In a purely inductive circuit the voltage leads
  the current by exactly 90o (or current lags by
  90o)
• if V 10?50o, and L2,?100 rads/s
• what does I ?
• I V/j?L V/j200 10?50o/200?90o
• I 0.05?-40o

29
L time domain vs. frequency domain
30
LC 5

• current in an inductor is
• i 2 cos 100t A
• Find the steady-state voltage across the inductor
  if its inductance is 10 mH

REMEMBER voltage leads the current by exactly
90o
31
Capacitors

• V (1/j?C)I
• In a purely capacitive circuit the current leads
  the voltage by exactly 90o (or voltage lags by
  90o)
• if V 10?50o, and L2,?100 rads/s
• what does I ?
• I V/ j?L V/ j200 10?50o/200?90o
• I 0.05?-40o

32
C time domain vs. frequency domain
33
LC 6

• current in a capacitor is
• i 2 cos 100t A
• Find the steady-state voltage across the
  capacitor if its capacitance is 1 mF

REMEMBER current leads the voltage by exactly
90o
34
Why use Phasors?

• Lets compare algebra to diff-eqs
• Example 10.7-1

35
Table 10.7-1 Time Domain and Frequency Domain
Relationships.
36
Impedance (Z) and Reactance (X)
Z V/I
37
Impedance (Z) and Reactance (X)

• Z (impedance) of an element is the ratio of the
  phasor voltage V to the phasor current I

38
Impedance (Z) and Admittance (Y)

• Y (admittance) of an element is the reciprocal
  of impedance and is analogous to conductance for
  resistive circuits

39
KCL/KVL in the time frequency domains
40
Current and voltage division

• current divider and voltage divider rules hold
  for phasor currents and voltages
• we substitute Impedance (Z) for Resistance (R)
  in the frequency domain

41
Node voltage, mesh current, supernodes, Thevenin
and Norton all work in the frequency domain

• all that work in Net 1 will really begin to pay
  off

42
Phasor diagrams
43
RLC circuit phasor diagram
VLj?LI
VSVRVLVC
NOTE from KVL above
VRRI
VC-jI/?C
44
Complex Impedances of R,L and C

• since Z V/I

for resistors VR RI converts to ZR VR/I R
for inductors VL j?LI converts to ZL VL/I
j?L
for capacitors VC I/j?C converts to ZC VC/I
1/j?C
45
Time domain vs. frequency domain
Ill convert first 2 on the board, calculate ZL1,
ZL2 and IS in frequency domain as Learning Checks
7, 8 and 9
46
Exercise 10.8-1 on Impedance

• write inductors and capacitors impedances and
  current source in phasor form

LC 10
How can we solve this?
47
What is the resonant frequency of an RLC circuit?

• the current I and the voltage across the
  resistor are in phase.
• inductor voltage leads current by 90o
• capacitor voltage lags current by 90o
• for a given L and C there will be a frequency ?
  resulting in

48
resonance in an RLC circuit

• in any RLC circuit where the inductor voltage
  and the capacitor voltage are equal
• since they are out of phase by 180o they cancel
  (sum to zero), resulting in a condition where Vs
  VR and Vs also, is then in phase with the
  current I just like VR
• this condition is known as resonance

49
Chapter 11 key concepts

• a-c steady-state electric power
• instantaneous and average power
• effective value of periodic waveform
• complex power
• power factor
• maximum power transfer
• ideal transformers

50
Sources of electric energy in 1986 and projected
for 2000. Source World Resources, World
Resource Institute, Oxford University Press, NY
2000.
51
Power levels for selected electrical devices or
phenomena.
52
instantaneous power

• product of time domain voltage and current
  associated with one of the circuit elements

53
instantaneous power

• product of time domain voltage and current
  associated with one of the circuit elements
• using trigonometric identities this resolves to

units Watts
54
real (average) power

• the integral of the time function of power over a
  complete period, divided by the period

units Watts
55
Figure 11.3-3An RL circuit with a sinusoidal
voltage source.
Find real (average) power delivered to each
device in the circuit (it is at steady state)
56
Find real (average) power delivered to each
device in the circuit
57
For Learning Check 5
Find real (average) power delivered to resistor
and inductor in the circuit
58
real power was quite easy, instantaneous power is
a bit harder to calculate
lets try an example
59
vs 7 cos 10t V
a) determine instantaneous power delivered by
voltage source to the circuit b) find
instantaneous power delivered to inductor Where
should we begin?
60
vs 7 cos 10t V
j?L j(10)(.3) j3
1/j?C 1/j(0.5) 2/j -j2
1) determine Z for the elements in the
circuit the inductor is in series and the
resistor and capacitor are in parallel, So
61
vs 7 cos 10t V
2) reduce Z for the entire circuit so we can
find i(t) to solve for power p(t) So
LC 6
LC 7
62
vs 7 cos 10t V
3) find i(t) to solve for power p(t) So
Write p(t) solution as LC 8
63
vs 7 cos 10t V
4) Part b) find instantaneous power to inductor
Write p(t) for inductor as solution to LC 9
64
effective values of periodic waveforms

• what is the effective voltage of a 120VAC/60Hz
  outlet in the walls of your home and in this
  building?
• What is Vm?
• What is T?
• What is ??

T 1/f 1/60Hz 16.667msecs
Vm ?
? 2?f 2?60 377rads/sec
65
effective values of periodic waveforms
To understand what Vm might be we need a good
definition of effective value. The effective
value of a current is the steady current (dc)
that transfers the same amount of average (real)
power as the given varying current. The effective
value of a voltage is the steady voltage (dc)
that transfers the same amount of average (real)
power as the given varying voltage.
66
effective values of periodic waveforms
effective value is the square root of the mean
of the squared values also known as (a-k-a) the
root-mean-square
67
effective values of periodic waveforms
Since 120 240VAC are effective values this
means they are root-mean-square (rms) values
So What are the amplitudes of these sinusoidal
waveforms if viewed on a scope?
Vm of 120VAC ? LC10a Vm of 240VAC ?
LC10b
68
NOTE

• In practice, electrical engineers must take
  care to notice (or determine) whether a
  sinusoidal voltage is being expressed in terms of
  its effective value (rms) or its maximum (Vm)
  value
• Generally in
• electronics and communications its Vm
• power applications its Vrms

69
complex power
the time domain the frequency
domain
Sometimes it is much easier to solve problems in
the frequency domain
70
complex power
the time domain the frequency
domain
Complex power delivered to an element is defined
to be S
71
Figure 11.5-2 (p. 479) (a) The impedance
triangle where Z R jX Z (b) The
complex power triangle where S P jQ.
complex power
72
complex power triangle
?S? apparent power
Q reactive power
P real power
73
complex power triangle
note ? ?V - ?I
?S? apparent
j
units Volt-Amps
Q reactive
units VARs
P real
units Watts