CHAPTER 1: SINUSOIDS AND PHASORS

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CHAPTER 1 SINUSOIDS AND PHASORS

• Sinusoids, Phasors and Phasors Relationships for
  Circuit Element
• Ohms Kirchhoffs Law in the Frequency Domain
• Impedance of Series Parallel Circuit and
  delta-wye transformation

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Sinusoidal
A sinusoidal voltage source supplies a voltage
that varies with time
A general expression for a sinusoidal voltage is
rms root mean square
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Sinusoidal
(a) as a function of ?t, (b) as a function of t
? angular frequency (rad/s) f the number of
cycle per second (Hz) T the time taken to
complete one cycle (s) Vm the maximum value for
the voltage or amplitude of the sinusoidal
voltage. the phase angle of the
sinusoidal function (degree or radians)
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Sinusoidal
p 180o
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Sinusoidal
Let examine the two sinusoids
v2 leads v1 by or v1 lags v2
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Sinusoidal
Example 1 For the following sinusoidal voltage ,
find the value v at time t 0s and t0.5s V 6
cos(100t 60o) Solution At t 0s, At t
0.5s V 6 cos( 0 60o) V 6 cos(50
radian 60o) 3 V 4.26 V
Be careful when adding ?t and phase angel. The
unit ?t is radians. The unit for is degrees. To
add both the values, they should be the same
units.
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Sinusoidal
p 3.142
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Exercise 1
A sinusoidal current has a maximum amplitude of
20A. The current passes through one complete
cycle in 1 ms. The magnitude of the current at
zero time is 10A. a) What is the frequency of
the current in Hz? b) What is the value of the
angular frequency? c) Write the expression for
i(t) in the form Im cos(?t ? ). d) What is the
rms value of the current?
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Exercise 2
A sinusoidal voltage is given by the expression
v 300cos(120pt 30o) a) What is the
frequency in Hz? b) What is the period of the
voltage in milliseconds? c) What is the magnitude
of v at t 2.778 ms? d) What is the rms value of
v?
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Phasors

• A phasor is a complex number that represents the
  magnitude and phase of a sinusoidal voltage or
  current.

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Phasors

• Example

TRY switch from the polar to rectangular form
and vice-versa using the calculator
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Complex Numbers

• Polar z ? ? A x jy Rectangular

• x is the real part
• y is the imaginary part
• z is the magnitude
• ? is the phase

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Complex Number Addition and Subtraction

• Addition is most easily performed in rectangular
  coordinates
• A x jy B z jw
• A B (x z) j(y w)
• Subtraction is also most easily performed in
  rectangular coordinates
• A - B (x - z) j(y - w)

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Complex Number Multiplication and Division

• Multiplication is most easily performed in polar
  coordinates
• A AM ? q B BM ? f
• A ? B (AM ? BM) ? (q f)
• Division is also most easily performed in polar
  coordinates
• A / B (AM / BM) ? (q - f)

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Complex Number
If the sinusoidal voltage represented by the sine
function then subtract the phase,? by 90o.
As example Given v(t) Vm sin (?t 10o).
Transform to phasor.
v(t) Vm sin (?t 10o)
v(t) Vm cos (?t 10o - 90o)
v(t) Vm cos (?t 80o)
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Exercise 3
Given y1 20cos(100t - 30) and y2 40cos(l00t
60). Express y1 y2 as a single cosine
function.
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Impedance

• AC steady-state analysis using phasors allows us
  to express the relationship between current and
  voltage using a formula that looks likes Ohms
  law
• V I Z
• Z is called impedance (units of ohms, O)
• Impedance is (often) a complex number, but is not
  a phasor
• Impedance depends on frequency, ?

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Phasor Relationships for Circuit Elements
Resistor
In the phasor domain, Z R. Voltage-current
relationship is given by V IR .
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Phasor Relationships for Circuit Elements
Inductor
In the phasor domain, Z j?L Voltage-current
relationship is given by V j?LI
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Phasor Relationships for Circuit Elements
Capacitor
In the phasor domain, Voltage-current
relationship is give by
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Impedance
A general expression for impedance , Z
Z R jX O
R resistance , X reactance
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Impedance Summary
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Exercise 4
The current in the 75mH inductor is 4cos(40000t
38o)mA. Calculate a) the inductive
reactance b) the impedance of the inductor c) the
phasor voltage V d) the steady-state expression
v(t).
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Exercise 5
Find the phasor domain equivalent circuit for
the following circuit
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Series Impedance
Z1
Z2
Zeq
Z3

• Zeq Z1 Z2 Z3

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Parallel Impedance
Z3
Zeq

• 1/Zeq 1/Z1 1/Z2 1/Z3

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Exercise 6
Find the input impedance of the circuit. Assume
that the circuit operates at ? 50 rad/s.
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Circuit Analysis Techniques

• While Obeying Passive Sign Convention
• Ohms Law KCL KVL
• Voltage and Current Division
• Series/Parallel Impedance combinations

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Kirchoffs Current Law (KCL)
i1(t)
i5(t)
i2(t)
i4(t)
i3(t)

• The sum of currents entering a node is zero
• Analogy mass flow at pipe junction

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Kirchoffs Voltage Law (KVL)

• The sum of voltages around a loop is zero
• Analogy pressure drop thru pipe loop

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Exercise 7
A 90O, a 32mH inductor and a 5µF capacitor are
connected in series across the terminal of a
sinusoidal voltage, a shown in Fig 9(a) below.
The steady-state expression for the source Vs
is 750cos(5000t 30o)V. a) Construct the phasor
domain equivalent circuit b) Calculated the
steady-state current i(t) by the phasor method
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VOLTAGE DIVIDER
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Exercise 8
Use the concept of voltage division to find the
steady-state expression for vo(t) in the circuit
if the vg 100cos(8000t) V
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CURRENT DIVIDER
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Exercise 9
Determine vo(t) in the circuit using current
divider
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Delta(?)-Wye(Y)
a
Z1
Zb
Zc
Z3
Z2
c
b
Za
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Wye(Y)-Delta(?)
a
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Exercise 10
Find current I in the circuit.
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Exercise 10 (cont)