# 05-Chapter 5 - Sinusoidal

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```					Chapter 5
Sinusoidal Analysis
Where are we going?
HighPass Filter
Low Pass Filter

Band Pass Filter

Three phase Source
5-1 Sinusoidal Currents and Voltages

v(t) = Vmsin(wt + q)
Vm = peak value
w = angular frequency
q = phase angle
T = period
f = Frequency
f = 1/T
w = 2pf = 2p/T
Plot   v versus t
Label   T

Plot   v versus wt
Label   q
Chapter 5
Sinusoidal Analysis
Useful Trig Identities
sin(z) = cos(z - 90°)

Example: v(t) = 10sin(200t + 30°)
(a) Write v(t) as a cosine function.
(b) Find Vm, w, f, T and q.
Definitions
 Vrms = root mean square voltage
 Vpp = peak-to-peak voltage
 Vave = average voltage
 Pave = average power

Vm               Vave  0
Vrms 
2
V pp  2Vm            Pave  I rms Vrms
R=50W

 Example:
 Find  v(t) and i(t)
 Vrms = root mean square voltage
 Vpp = peak-to-peak voltage
 Vave = average voltage
 Pave = average power
What are the following for a wall outlet?

f
Vrms = VDCEquivalent
Vm
Vave
Vpp
Notation Summary
Sinusoidal
v1(t) = V1 cos(wt + q1)
Polar Phasor
V1 = V1 L q1
Complex Phasor
V1 = V1 cos(q1) + j V1 sin(q1)

j  1
Math
Imaginary
C=A + jB

q = phase angle
Real

What is the magnitude of C?

C  A B    2        2

What is the phase (or direction) of C?
B    1
q  tan
A
Example Problems
 Convert  the following voltages to phasors
in polar form and complex form.
v1(t) = 20 cos(wt - 45°)
v2(t) = 20 sin(wt + 60°)
Example Problems
 Convertthe following from complex
phasors to polar form.
V1 = 30 + j40
V2 = 4 - j20

V1 + V2
Phasor Math in Polar Form
 (C1q1)   (C2q2) = C1 C2 (q1 + q2)

 (C1q1)   /(C2q2) = C1/C2 (q1 - q2)
Ohm’s Law for AC Circuits

V=IZ

Impedance
Resistors
 Suppose that       v(t) = Vmcos(wt)

 What   is i(t) ?
 Hint:   v=iR

 What   is the phase relationship between i and v?
Capacitors
 Suppose that       v(t) = Vmcos(wt)

 What   is i(t) ?
 Hint:   v = q/C

 What   is the phase relationship between i and v?
Inductors
 Suppose that     i(t) = Imcos(wt)

 What   is v(t) ?
 Hint:   v(t) = L di/dt

 What   is the phase relationship between i and v?
Reactance, Impedance and Phasors

 Go   to notes…

 Phasor   Diagrams for V and I

 Impedance Diagrams    for R, L and C circuits
The R, L, and C Elements
   Ohm’s Law for Peak Values
Resistors: Vp=IpR
Capacitors: Vp=IpXC
Inductors: Vp=IpXL

   Go to notes...
Example Problems
 Example   3.15:

Convert the following from polar to rectangular
form.
1053.13
16-30
25120
Example Problems
 Example   3.15:

Convert the following from rectangular to polar
form.
30 + j40
4 - j20
-3 - 4j
Impedance Diagrams
Resistor
ZR = R0

Capacitor
ZC = XC-90

Inductor
ZL = XL90
RL Circuit Example
Connect at AC power supply in series with an
inductor and a resistor.
How does VR vary with the input frequency?
RC Circuit Example
Connect at AC power supply in series with an
capacitor and a resistor.
How does VR vary with the input frequency?
RLC Circuit Example
Connect at AC power supply in series with an
inductor, capacitor and a resistor.
How does VR vary with the input frequency?
3.18 Tuned Resonant Networks
 RLC  Series circuits are used in radios.
 Series RLC networks have a resonant
frequency that depends on C and L only.

1
fs 
2p LC
 Whatcapacitance do you need to listen to
107.7 MHz on a radio with a 1mH inductor?

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