Circuit Analysis II

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Circuit Analysis II
ET312
Instructor Apisak Worapishet, Mahanakorn
Microelectronics Research Center (MMRC)
e-mailapisak_at_mut.ac.th
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Phasors
Transforming the analysis in the time domain
involving differential equations to the complex
domain where we only have to deal with algebraic
calculation of complex numbers
The concept was introduced in 1893 by
Charles Proteus Steinmetz (1865-1923)
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Power in Sinusoidal Steady State (1)
One-port network with port voltage port
current
Instantaneous power (entering one-port), at time t
(W)

General expressions
Energy (delivering to one-port), during t0 to t
(J)

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Power in Sinusoidal Steady State (2)
Under Sinusoidal Steady State Condition and
may be expressed in PHASOR forms
Time-domain representations
Equivalent PHASOR representations
omitted for brevity
Connections between the two forms
NOTE and are COMPLEX number!
QUESTION Why PHASOR forms? Any constraint?
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Power in Sinusoidal Steady State (3)
Instantaneous power at Sinusoidal Steady State
using time-domain representation



Constant
Sinusoid with angular frequency
Waveforms illustrating and
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Power in Sinusoidal Steady State (4)
Average power (entering one-port),
General expression

is the period of (assuming periodic)
Average power at Sinusoidal Steady State

2nd term in disappears!
NOTE For containing only passive
components This is NOT necessary for !
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Power in Sinusoidal Steady State (5)
Complex power (entering one-port),
complex conjugate


Putting and


PHASOR form for power

Observation

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Power in Sinusoidal Steady State (6)
IMPEDANCE concept in PHASOR analysis
Driving point impedance
Driving point admittance
LOOK similar to Ohms Law !
Assuming with driving point impedance
Complex power in terms of Impedance






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Power in Sinusoidal Steady State (7)
Average power in terms of Impedance


Observations

• Although is kept constant, can be
  varied by
• adjusting the impedances phase

• For containing only passive elements,
  gt
• and hence passitivity conditions for or are

or
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Summary of Power Definitions
Class of instantaneous power, p(t), under SSS

• Complex Power

• Average Power

• Reactive Power

• Apparent Power

Phasor-domain (complex plane) illustrating the
inter-relationship
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Additive Property of Average Power (1)
Consider and with multiple frequencies


NOTE and are related via or
when written in phasor forms
For having two frequency components


thus

Phasor analysis is implicitly performed within
the operator Re !
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Additive Property of Average Power (2)
or






Instantaneous power at Sinusoidal Steady State

1st part
2nd part
3rd part
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Additive Property of Average Power (3)

• Observations
• Instantaneous Power is not the sum of the
  instantaneous
• power due to currents at and
  acting alone
• In contrast, the average power is equivalent to
  the sum
• of the average power due to at
  and
• Superposition holds for the average power
  , not
• the instantaneous power

CAUTION!
Last two observations, NOT NECESSARY Consider the
case when
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Effective or Root-mean-square Value (1)
Consider with a pure resistance
For sinusoidal waveforms and



Definition Effective value for sinusoidal
waveforms
effective current
effective voltage
Power expression in terms of the effective value



factor 2 now removed!
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Effective or Root-mean-square Value (2)
Effective value for NON-sinusoidal waveforms
root-mean-square operation
Assuming and with a period
For any periodic waveforms






Simple and quite convenient to manage!
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Maximum Power Transfer (1)
How to deliver maximum average power from the
source to the load under Sinusoidal Steady State ?
Passive components
gt
gt
and are given, find for
maximum power transfer, ( delivers
maximum power from to )
but


dropped for simplicity
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Maximum Power Transfer (2)
Define and

Find and to maximize (
, given)
_at_

_at_
To obtain maximum power transfer conjugate
matched condition


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Maximum Power Transfer (3)
Average power at under conjugate match

Average power delivered from the source

since

Power efficiency under maximum power transfer


QUESTION Calculate efficiency under other
condition!
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Power Factor (PF) and PF Correction (1)

Definition of PF PF
Implication of PF consider when is
constant
Apparent power is inversely proportional to PF
Phasor-domain (complex plane)
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Power Factor (PF) and PF Correction (2)
Electric Bill Issue!
Consider a (very) simple power transmission model
Power company
House, factory
Average (Real) power used
Power loss during transmission (constant )
To save electric bill, we need PF close to 1!
(less )
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Power Factor (PF) and PF Correction (3)
NOTE PF depends on
PF
Power Factor Correction
Means to save bahts!
How! Consider a case example with loads (a) and
(b) where is constant
same
Phasor-domain
(a)
(b)
PF(a) lt PF(b) lt 1!