Imaginary

1
Imaginary

• JS-H 1 16
•   16 But, exerting all my powers to call upon God
  to deliver me out of the power of this enemy
  which had seized upon me, and at the very moment
  when I was ready to sink into despair and abandon
  myself to destructionnot to an imaginary ruin,
  but to the power of some actual being from the
  unseen world, who had such marvelous power as I
  had never before felt in any beingjust at this
  moment of great alarm, I saw a pillar of light
  exactly over my head, above the brightness of the
  sun, which descended gradually until it fell upon
  me.

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Lecture 13 Network Analysis with Capacitors and
Inductors

• Phasors

A real phasor is NOT the same thing used in
Star Trek
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Eulers Identity

• Appendix A reviews complex numbers

Complex exponential (ej?) is a point on the
complex plane
Eulers equation
4
Phasors

• Rewrite the expression for a general sinusoid
  signal

Angle (or argument)
magnitude
Complex phasor notation for the simplification
NB The ejwt term is implicit (it is there but
not written)
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Frequency Domain

• Graphing in the frequency domain helpful in
  order to understand Phasors

pd(? ?0) d(? ?0)
cos(?0t)
Time domain
Frequency domain
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The Electromagnetic Spectrum
Gamma rays
Radio waves
Ultraviolet
Infrared
Xray
Microwaves
wavelength
Visible light
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Phasors

• Any sinusoidal signal can be represented by
  either
• Time-domain form v(t) Acos(?t?)
• Frequency-domain form V(j?) Aej? A ?
• Phasor a complex number expressed in polar form
  consisting of
• Magnitude (A)
• Phase angle (?)
• Phasors do not explicitly include the sinusoidal
  frequency (?) but this information is still
  important

8
Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

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Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

1. Write voltages in phasor notation

10
Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

1. Write voltages in phasor notation
2. Convert phasor voltages from polar to rectangular
   form (see Appendix A)

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Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

1. Write voltages in phasor notation
2. Convert phasor voltages from polar to rectangular
   form (see Appendix A)
3. Combine voltages

12
Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

1. Write voltages in phasor notation
2. Convert phasor voltages from polar to rectangular
   form (see Appendix A)
3. Combine voltages
4. Convert rectangular back to polar

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Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

1. Write voltages in phasor notation
2. Convert phasor voltages from polar to rectangular
   form (see Appendix A)
3. Combine voltages
4. Convert rectangular back to polar
5. Convert from phasor to time domain

NB the answer is NOT simply the addition of the
amplitudes of v1(t) and v2(t) (i.e. 15 15), and
the addition of their phases (i.e. p/4 p/12)
Bring ?t back
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Phasors

• Example 1 compute the phasor voltage for the
  equivalent voltage vs(t)
• v1(t) 15cos(377tp/4)
• v2(t) 15cos(377tp/12)

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Phasors of Different Frequencies

• Superposition of AC signals when signals do not
  have the same frequency (?) the ej?t term in the
  phasors can no longer be implicit

I
v
i2(t)
NB ej?t can no longer be implicit
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Phasors of Different Frequencies

• Superposition of AC signals when signals do not
  have the same frequency (?) solve the circuit
  separately for each different frequency (?)
  then add the individual results

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Phasors of Different Frequencies

• Example 2 compute the resistor voltages
• is(t) 0.5cos2p(100t) A
• vs(t) 20cos2p(1000t) V
• R1 150O, R2 50 O

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Phasors of Different Frequencies

• Example 2 compute the resistor voltages
• is(t) 0.5cos2p(100t) A
• vs(t) 20cos2p(1000t) V
• R1 150O, R2 50 O

• Since the sources have different frequencies (?1
  2p100) and (?2 2p1000) use superposition
• first consider the (?1 2p100) part of the
  circuit
• When vs(t) 0 short circuit

R2
R1
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Phasors of Different Frequencies

• Example 2 compute the resistor voltages
• is(t) 0.5cos2p(100t) A
• vs(t) 20cos2p(1000t) V
• R1 150O, R2 50 O

• Since the sources have different frequencies (?1
  2p100) and (?2 2p1000) use superposition
• first consider the (?1 2p100) part of the
  circuit

R1 R2
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Phasors of Different Frequencies

• Example 2 compute the resistor voltages
• is(t) 0.5cos2p(100t) A
• vs(t) 20cos2p(1000t) V
• R1 150O, R2 50 O

• Since the sources have different frequencies (?1
  2p100) and (?2 2p1000) use superposition
• first consider the (?1 2p100) part of the
  circuit
• Next consider the (?2 2p1000) part of the
  circuit

R2
R1
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Phasors of Different Frequencies

• Example 2 compute the resistor voltages
• is(t) 0.5cos2p(100t) A
• vs(t) 20cos2p(1000t) V
• R1 150O, R2 50 O

• Since the sources have different frequencies (?1
  2p100) and (?2 2p1000) use superposition
• first consider the (?1 2p100) part of the
  circuit
• Next consider the (?2 2p1000) part of the
  circuit
• Add the two together

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Impedance

• Impedance complex resistance (has no physical
  significance)
• Allows us to use network analysis methods such as
  node voltage, mesh current, etc.
• Capacitors and inductors act as
  frequency-dependent resistors

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Impedance Resistors

• Impedance of a Resistor
• Consider Ohms Law in phasor form

Phasor
Phasor domain
NB Ohms Law works the same in DC and AC
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Impedance Inductors

• Impedance of an Inductor
• First consider voltage and current in the
  time-domain

NB current is shifted 90 from voltage
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Impedance Inductors

• Impedance of an Inductor
• Now consider voltage and current in the
  phasor-domain

Phasor
Phasor domain
Phasor
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Impedance Capacitors

• Impedance of a capacitor
• First consider voltage and current in the
  time-domain

Phasor
Phasor
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Impedance Capacitors

• Impedance of a capacitor
• Next consider voltage and current in the
  phasor-domain

Phasor domain
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Impedance
Impedance of resistors, inductors, and capacitors
Phasor domain
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Impedance
Impedance of resistors, inductors, and capacitors
Phasor domain
AC resistance
reactance
Not a phasor but a complex number
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Impedance

• Practical capacitors in practice capacitors
  contain a real component (represented by a
  resistive impedance ZR)
• At high frequencies or high capacitances
• ideal capacitor acts like a short circuit
• At low frequencies or low capacitances
• ideal capacitor acts like an open circuit

Practical Capacitor
Ideal Capacitor
NB the ratio of ZC to ZR is highly frequency
dependent
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Impedance

• Practical inductors in practice inductors
  contain a real component (represented by a
  resistive impedance ZR)
• At low frequencies or low inductances ZR has a
  strong influence
• Ideal inductor acts like a short circuit
• At high frequencies or high inductances ZL
  dominates ZR
• Ideal inductor acts like an open circuit
• At high frequencies a capacitor is also needed to
  correctly model a practical inductor

Ideal Inductor
Practical Inductor
NB the ratio of ZL to ZR is highly frequency
dependent
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Impedance

• Example 3 impedance of a practical capacitor
• Find the impedance
• ? 377 rads/s, C 1nF, R 1MO

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Impedance

• Example 3 impedance of a practical capacitor
• Find the impedance
• ? 377 rads/s, C 1nF, R 1MO

R
C
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Impedance

• Example 4 find the equivalent impedance (ZEQ)
• ? 104 rads/s, C 10uF, R1 100O, R2 50O, L
  10mH

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Impedance

• Example 4 find the equivalent impedance (ZEQ)
• ? 104 rads/s, C 10uF, R1 100O, R2 50O, L
  10mH

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Impedance

• Example 4 find the equivalent impedance (ZEQ)
• ? 104 rads/s, C 10uF, R1 100O, R2 50O, L
  10mH

R1
L
NB at this frequency (?) the circuit has an
inductive impedance (reactance or phase is
positive)