NETWORKS 2: 090920201

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CHAPTER 10

• NETWORKS 2 0909202-01
• 1 November 2005 Lecture 2
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2005 Quarter Two

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Review Chapter 10 key concepts

• Todays learning objectives
• terminology for sinusoidal sources
• Phase angles in progress
• analyzing sinusoidal sources
• steady-state response of an RL circuit to a
  sinusoidal forcing function
• complex exponential forcing function
• the phasor concept

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Figure 10.3-4 Triangle for A and B -
Trigonometry is helpful here
sin ? B/C, cos ? A/C, tan ? sin/cos B/A
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Phase angle (?)

• if a current and voltage waveform are out of
  phase with each other this difference is referred
  to as a phase angle (leading or lagging).
• It (?) can be expressed as a difference in
  degrees or radians
• It can also be observed in the functions that
  represent these waveforms as a time difference

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Circuit response to a forcing function

• If we have a circuit which experiences a forcing
  function of vs V0cos?t
• it will experience a forced response of the
  form
• vf A cos ? t B sin ? t
• this is equivalent to

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Figure 10.3-5 The AB triangle
v(t) A cos?t B sin?t
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LC 1

• From the previous example, Find the current
  restated in the second form if it is equal to the
  value below in the first form?

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Sinusoidal Sources

• T period of oscillation
• f 1/T (frequency)
• Units of Hertz
• Cycles per second
• ? 2?f 2?/T
• Angular frequency
• Radians per second
• ? phase angle
• Offset from zero
• May be degrees or rads

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representing signals by equations

• typical problem encountered by ECEs
• solution is straightforward
• 1. measure the amplitude (A)
• 2. determine the period (T), calculate frequency
  ? 2?f
• 3. Pick a time (t) and measure v (or i), observe
  ascending or descending and finally, calculate
  phase angle ?

When v(t) is ascending at time t1
When v(t) is descending at time t1
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sample problem analytic representation of
sinusoidal voltage waveforms v1(t) and v2(t) of
the form v(t)Acos(?t?)
2A30V, T0.2 sec Determine Vm and ?
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continuing sample problem

• 2A 30V, A Vmax 15V
• ? 2?/T 2?/0.2 10?
• At circled point v1(t) is going up (ascending
  with time) and values are
• v1(t) 10.6066V
• t 0.15 sec
• So

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continuing sample problem
If you calculated this on your calculator you
should find that ?-5.4977873 radians is the
answer. This equates to how many degrees? -315o
45o
Write the final analytic equation for v1(t) in
this form as LC2
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do the same for v2(t)
At circled point v2(t) is going down (descending
with time) and values are v2(t) 10.6066V t
0.15 sec
Write the final analytic equation for v2(t) in
this form as LC3
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some practice problems

• a voltage is v6cos(4t30o)
• (a) find the period of oscillation
• (b) find phase relation to associated current i
  8cos(4t-70o)
• (a) T2?/? since ?4 T2?/4 ?/2
• (b) write answer as LC4

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more practice

• a voltage is v3cos4t4sin4t
• (a) find the voltage in the form below
• REMEMBER

?tan-1(B/A) when Agt0 and ?180o tan-1(B/A) when
Alt0 Write your answer as LC5
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steady-state response of circuit to sinusoidal
forcing function
When vs(t) Vm cos ?t, the complete response
looks like this i(t) Ke-t/? Imax cos (?t ?)
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Over time the transient term dies out

• the steady-state response that is left is
  i(t) Imax cos (?t ?)
• applying our Net 1 knowledge about KVL in this
  loop we find
• Vmax cos ?t L(di/dt) Ri
• substituting i(t) above into this we find after
  many calculations

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Example
If vs(t)10 cos 3t V and L1H, R4? what is
i(t)? Provide your answer as LC6
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Net II and complex math

• net I relied heavily on Algebra, some trig and
  1st order differential equations
• net II requires complex math as well as 1st and
  2nd order differential equations
• lets review some..

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Complex numbers
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Complex numbers
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Rectangular, Exponential and Polar Notation

• In the Complex Plane where x is the real part of
  the complex number and y is imaginary part of the
  complex number we can express the complex number
  c in three distinct ways
• Rectangular c a jb
• Exponential c rej?
• Polar c r??

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Lets express c in all forms

• If c 4 j3
• What is it in exponential form?
• What is r?
• r (a2 b2)1/2
• r (42 32)1/2 (16 9)1/2 5
• What is ??
• ? tan-1(b/a) when agt0
• ? tan-1(3/4) 36.9o
• Write c rej? as LC7
• Write c in polar form as LC8

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Operations with complex numbers

• on board
• REMEMBER
• Easiest to subtract and add complex numbers in
  rectangular form
• Easiest to multiply and divide in polar form

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practice notation examples
Find a and b if Provide your answer as
LC9 Find A and ? if Provide your answer as
LC10
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How can we apply this?

• to make solutions of net II problems easier
• Well, first a bit of tougher learning then a
  greater simplification that will make our lives
  enormously easier.

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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 ? - ?)

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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)
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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)

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hw problem 10.5-6

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

0.1F
5?
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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

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remember
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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

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example

• i 5 sin(100t120o)
• Lets convert to the frequency domain
• I Im??

Provide your answer as LC3
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LC 4

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

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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

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example

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

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LC 5

• V 10?-140o
• Lets convert to the frequency domain
• I Im??

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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
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RL example solution on board
vs(t) Vm cos 100t V , R 200?, L 2H
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RC example solution below
i(t) 10 cos 100t A, R1?, C 0.01F
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hw problem 10.6-1

• v(t) 15 cos 4t V, R6?, L2H
• find v(t) for inductor, steady-state

v(t) -
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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

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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

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R time domain vs. frequency domain
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LC 6

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

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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

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L time domain vs. frequency domain
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LC 7

• 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
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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

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C time domain vs. frequency domain
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LC 8

• 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
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Why use Phasors?

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

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

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

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KCL/KVL in the time frequency domains
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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

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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

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Phasor diagrams