1. CHAPTER 1: SINUSOIDS AND PHASORS
Sinusoids, Phasors and Phasors Relationships for Circuit Element
Ohms & Kirchhoff’s Law in the Frequency Domain.
Impedance of Series & Parallel Circuit and delta-wye transformation

2. BEE1123[CH1]: Sinusoids and Phasor
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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BEE1123[CH1]: Sinusoids and Phasor

3. BEE1123[CH1]: Sinusoids and Phasor
Sinusoidal
Vm sin ωt: (a) as a function of ωt, (b) as a function of t
ω = 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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BEE1123[CH1]: Sinusoids and Phasor

4. BEE1123[CH1]: Sinusoids and Phasor
Sinusoidal
π = 180o
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BEE1123[CH1]: Sinusoids and Phasor

5. BEE1123[CH1]: Sinusoids and Phasor
Sinusoidal
Let examine the two sinusoids
v2 leads v1 by or v1 lags v2
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BEE1123[CH1]: Sinusoids and Phasor

6. BEE1123[CH1]: Sinusoids and Phasor
Sinusoidal
Example 1
For the following sinusoidal voltage , find the value v at time t= 0s and t=0.5s V = 6 cos(100t + 60o)
Solution
At t = 0s, At t = 0.5s
V = 6 cos( o) V = 6 cos(50 radian + 60o)
= 3 V = 4.26 V
Be careful when adding ωt and phase angel. The unit ωt is radians. The unit for is degrees. To add both the values, they should be the same units.
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BEE1123[CH1]: Sinusoids and Phasor

7. BEE1123[CH1]: Sinusoids and Phasor
Sinusoidal
Radians
Degree
π = 3.142
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BEE1123[CH1]: Sinusoids and Phasor

8. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 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(ωt + ).
d) What is the rms value of the current?
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BEE1123[CH1]: Sinusoids and Phasor

9. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 2
A sinusoidal voltage is given by the expression
v = 300cos(120t + 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 = ms?
d) What is the rms value of v?
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BEE1123[CH1]: Sinusoids and Phasor

10. BEE1123[CH1]: Sinusoids and Phasor
Phasors
A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current.
COMPLEX NUMBER
POLAR
EXPONENTIAL
RECTANGULAR
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BEE1123[CH1]: Sinusoids and Phasor

11. BEE1123[CH1]: Sinusoids and Phasor
Phasors
Example:
COMPLEX NUMBER
POLAR
RECTANGULAR
* TRY
switch from the polar to rectangular form and vice-versa using the calculator
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BEE1123[CH1]: Sinusoids and Phasor

12. Complex Numbers Polar: z   = A = x + jy : Rectangular imaginary axis
real axis
imaginary axis
x is the real part
y is the imaginary part
z is the magnitude
 is the phase
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BEE1123[CH1]: Sinusoids and Phasor

13. 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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BEE1123[CH1]: Sinusoids and Phasor

14. 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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BEE1123[CH1]: Sinusoids and Phasor

15. BEE1123[CH1]: Sinusoids and Phasor
Complex Number
If the sinusoidal voltage represented by the sine function then subtract the phase, to 90o.
As example:
Given v(t) = Vm sin (ωt +10o). Transform to phasor.
v(t) = Vm sin (ωt +10o)
v(t) = Vm cos (ωt + 10o - 90o)
v(t) = Vm cos (ωt – 80o)
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BEE1123[CH1]: Sinusoids and Phasor

16. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 3
Given y1= 20cos(100t - 30°) and y2 = 40cos(l00t + 60°). Express y1 + y2 as a single cosine function.
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BEE1123[CH1]: Sinusoids and Phasor

17. BEE1123[CH1]: Sinusoids and Phasor
Impedance
AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law:
V = I Z
Z is called impedance (units of ohms, Ω)
Impedance is (often) a complex number, but is not a phasor
Impedance depends on frequency, ω
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BEE1123[CH1]: Sinusoids and Phasor

18. Phasor Relationships for Circuit Elements
Resistor
In the phasor domain,
Z = R.
Voltage-current relationship is given by V = IR .
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BEE1123[CH1]: Sinusoids and Phasor

19. Phasor Relationships for Circuit Elements
Inductor
In the phasor domain,
Z = jωL
Voltage-current relationship is given by V = jωLI
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BEE1123[CH1]: Sinusoids and Phasor

20. Phasor Relationships for Circuit Elements
Capacitor
In the phasor domain,
Voltage-current relationship is give by
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BEE1123[CH1]: Sinusoids and Phasor

21. BEE1123[CH1]: Sinusoids and Phasor
Impedance
A general expression for impedance , Z
Z = R ± jX Ω
R = resistance , X = reactance
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BEE1123[CH1]: Sinusoids and Phasor

22. BEE1123[CH1]: Sinusoids and Phasor
Impedance Summary
Element
Impedance
Resistance
Reactance
Resistor
ZR = R
= R  0 R
Inductor
ZL = jL
= L  90 L
Capacitor
ZC = 1 / jC
= 1/C  -90
1 / jC
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BEE1123[CH1]: Sinusoids and Phasor

23. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 5
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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BEE1123[CH1]: Sinusoids and Phasor

24. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE
Find the phasor domain equivalent circuit for the following circuit
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BEE1123[CH1]: Sinusoids and Phasor

25. BEE1123[CH1]: Sinusoids and Phasor
Series Impedance Z1 Z2
Zeq Z3
Zeq = Z1 + Z2 + Z3
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BEE1123[CH1]: Sinusoids and Phasor

26. BEE1123[CH1]: Sinusoids and Phasor
Parallel Impedance Z1 Z2 Z3
Zeq
1/Zeq = 1/Z1 + 1/Z2 + 1/Z3
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BEE1123[CH1]: Sinusoids and Phasor

27. BEE1123[CH1]: Sinusoids and Phasor
Exercise
Find the input impedance of the circuit. Assume that the circuit operates at ω = 50 rad/s.
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BEE1123[CH1]: Sinusoids and Phasor

28. Circuit Analysis Techniques
While Obeying Passive Sign Convention
Ohm’s Law; KCL; KVL
Voltage and Current Division
Series/Parallel Impedance combinations
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BEE1123[CH1]: Sinusoids and Phasor

29. Kirchoff’s Current Law (KCL)
i1(t)
i5(t)
i2(t)
i4(t)
i3(t)
The sum of currents entering the node is zero:
Analogy: mass flow at pipe junction
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BEE1123[CH1]: Sinusoids and Phasor

30. Kirchoff’s Voltage Law (KVL)+ -
v(t)1
v(t)2
v(t)3
The sum of voltages around a loop is zero:
Analogy: pressure drop thru pipe loop
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BEE1123[CH1]: Sinusoids and Phasor

31. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 9
A 90Ω, 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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BEE1123[CH1]: Sinusoids and Phasor

32. BEE1123[CH1]: Sinusoids and Phasor
VOLTAGE DIVIDER
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BEE1123[CH1]: Sinusoids and Phasor

33. BEE1123[CH1]: Sinusoids and Phasor
EXAMPLE 10
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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BEE1123[CH1]: Sinusoids and Phasor

34. BEE1123[CH1]: Sinusoids and Phasor
CURRENT DIVIDER
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BEE1123[CH1]: Sinusoids and Phasor

35. BEE1123[CH1]: Sinusoids and Phasor
Exercise
Determine vo(t) in the circuit using current divider
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BEE1123[CH1]: Sinusoids and Phasor

36. BEE1123[CH1]: Sinusoids and Phasor
Delta(Δ)-Wye(Y) a Z1 Zb Zc Z3 Z2 c b Za
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BEE1123[CH1]: Sinusoids and Phasor

37. BEE1123[CH1]: Sinusoids and Phasor
Wye(Y)-Delta(Δ) a b c Za Zc Zb Z1 Z2 Z3
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BEE1123[CH1]: Sinusoids and Phasor

38. BEE1123[CH1]: Sinusoids and Phasor
Exercise
Find current I in the circuit.
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BEE1123[CH1]: Sinusoids and Phasor

39. BEE1123[CH1]: Sinusoids and Phasor
Exercise
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BEE1123[CH1]: Sinusoids and Phasor