5.4 Circuit Analysis Using Phasors and Complex Impedances

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5.4 Circuit Analysis Using Phasors and Complex
Impedances

1. Replace the time descriptions of the voltage and
   current sources with the corresponding phasors.
   (All of the sources must have the same frequency.)

2. Replace inductances by their complex
impedances ZL j?L. Replace capacitances by
their complex impedances ZC 1/(j?C).
Resistances have complex impedances equal to
their resistances
3. Analyze the circuit using any of the
techniques studied earlier in Chapter 2,
performing the calculations with complex
arithmetic.
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• Find the steady-state current for the circuit as
  follows, and also find the phasor voltage across
  each element and construct a phasor diagram.

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• Find the steady-state voltage uC(t) for the
  circuit as follows, and also find the phasor
  current through each element and construct a
  phasor diagram showing the currents and source
  voltage.

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• To find u1(t) in steady state.

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• To find i(t) in steady state construct a phasor
  diagram showing all three voltages and current
  what is the phasor relationship between us(t) and
  i(t)?

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• To find phasor voltage and the phasor current
  through each element in the circuit.

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Steady-state Response of RLC in Series
Reactance
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• Steady-state Response of RLC in Series

Voltage Triangle
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• Steady-state Response of RLC in Series

Impedances Triangle
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• Power and Power Factor

Power Factor ????
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Three Triangles
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• Example 5.6 Computer the power and reactive
  power taken from the source and each element in
  the circuit.

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• Example 5.7Find the power, reactive power and
  power factor for the source, find the phasor
  current i.

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5.6 THÉVENIN EQUIVALENT CIRCUITS
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5.6 THÉVENIN EQUIVALENT CIRCUITS

• The Thévenin voltage is equal to the open-circuit
  phasor voltage of the original circuit.

• We can find the Thévenin equivalent impedance by
  zeroing the independent sources and determining
  the complex impedance looking into the circuit
  terminals.

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5.6 THÉVENIN EQUIVALENT CIRCUITS
The Thévenin impedance equals the open-circuit
voltage divided by the short-circuit current.
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• Example 5.9 Find Thevenin equivalent circuit for
  the circuit.

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Maximum Average Power Transfer

• If the load can take on any complex value,
  maximum power transfer is attained for a load
  impedance equal to the complex conjugate of the
  Thévenin impedance.

• If the load is required to be a pure resistance,
  maximum power transfer is attained for a load
  resistance equal to the magnitude of the Thévenin
  impedance.

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• Example 5.10 Determine the maximum power
  delivered to a load (a) the load can have any
  complex value (b) the load must be a pure
  resistance.

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5.7 BALANCED THREE-PHASE CIRCUITS

• Much of the power used by business and industry
  is supplied by three-phase distribution systems.

BALANCED THREE-PHASE CIRCUITS Three
equal-amplitude ac voltages have phases that are
1200 apart.
Chapter 17 tells us how three-phase voltages are
generated.
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Wye (Y)-connected
Line a?b?c ?????
Neutral n ?????
Positive phase sequence a ? b ? c
Phase voltage ???VY
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Phase Sequence

• Three-phase sources can have either a positive or
  negative phase sequence.

• The direction of rotation of certain three-phase
  motors can be reversed by changing the phase
  sequence.

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WyeWye Connection

• Three-phase sources and loads can be connected
  either in a wye (Y) configuration or in a delta
  (?) configuration.

• The key to understanding the various three-phase
  configurations is a careful examination of the
  wyewye (Y-Y) circuit.

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Balanced loads ????
sources
Line currents ???
Neutral currents ???? Four-wire connection
Balanced loads All three load impedances are
equal.

• Under balanced three-phase sources and loads,

Therefore, we can omit the neutral wire.
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• WyeWye Connection

In a balanced three-phase system, neutral current
is zero.
we can eliminate the neutral wire. Then, compared
with single phase circuit, only three wires are
needed to connect the sources to the loads , it
is less expensive.
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• WyeWye Connection

Compared with single-phase circuit, the
total instantaneous power in a balanced
three-phase system is constant rather than
pulsating.

• Reactive power

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(???)
A(???)
B(???)
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• A balanced positive-sequence wye-connected 60
  Hz three-phase source has phase voltage UY1000V.
  Each phase of the load consists of a 0.1-H
  inductance in series with a 50-O resistance.
• Find the line currents, the line voltages,
  the power and the reactive power delivered to the
  load. Draw a phasor diagram showing line
  voltages, phase voltages and the line currents.
  Assuming that the phase angle of Uan is zero.

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• Delta (?)-connected Sources

According to KVL,
Thus, the current circulating in the delta is
zero.
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• Wye (Y) and Delta (?)-connected Loads

Two balanced loads are equal.
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• Wye (Y) and Delta (?)-connected Loads

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• Delta - Delta (?- ?) connection

Line current
Assuming line voltage
Then, we get phase current
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Homework 5

• P5.24
• P5.34
• P5.43
• P5.53
• P5.68
• P5.72