Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
1
Learning Objectives
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
2
Introduction
To perform AC analyses using phasors and impedances
We utilize all our circuit analysis tools to accomplish this
– KCL, KVL, Ohm’s Law
– Series and Parallel circuits.
– Voltage and current division
– Superposition, source transformation
– Nodal analysis.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
3
Sinusoids
the the sinusoid repeats itself every T seconds,
T is called the period of the sinusoid.
we observe that
T=2 ,
the sinusoidal voltage
Vm = the amplitude of the sinusoid
= the angular frequency in radians/s
t = the argument of the sinusoid
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
4
The fact that v(t) repeats itself every T seconds.
by replacing t by t + T
v has the same value
at t + T as it does at t
v(t) is said to be
periodic.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
5
A periodic function is one that satisfies
f (t) = f (t + nT),
for all t and for all integers n.
the number of cycles per second, known as the cyclic frequency f of
the sinusoid.
is in radians per second (rad/s),
f is the unit of frequency in hertz (Hz).
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
6
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
7
For
v1(t) = Vm sin
t and v2(t) = Vm sin( t + φ )
• The starting point of v2 occurs first in time.
• we say that v2 leads v1 by or that v1 lags v2 by φ .
• If φ = 0, then v1 and v2 are said to be in phase; they reach their minima
and maxima at exactly the same time.
A sinusoid can be expressed in either sine or cosine form.
by using the following trigonometric identities:
sin(A ± B) = sinAcosB ± cosAsinB
cos(A ± B) = cosAcosB sinAsinB
sin( t ± 180 ) = −sin t
cos( t ± 180 ) = −cos t
sin( t ± 90 ) = ±cos t
cos( t ± 90 ) = ± sin t
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
8
adding 180 to
sin t gives
−sin t, or
sin( t − 180 ) =
−sin t,
subtracting 90 from
cos t gives
sin t, or
cos( t−90 ) = sin t
To add Acos t and B sin t, we note
Thus,
Acos t + B sin t = C cos( t − )
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
9
Example
Add 3 cos
t and −4 sin
t
B
C
Solution
A
c=
(3) + (− 4)
2
2
=5
A
θ = tan
= 53.1
B
−1
3 cos
t − 4 sin
t = 5 cos( t + 53.1 )
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
10
Example
Find the amplitude, phase, period, and frequency of the sinusoid
v(t) = 12 cos(50t + 10 )
Solution
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
11
Phasors
A phasor is a complex number that represents the
amplitude and phase of a sinusoid.
Phasors provide a simple means of analyzing linear circuits excited by
sinusoidal sources
A complex number z can be written in rectangular form
z = x + jy
A complex number z can be written in polar form
A complex number z can be written in exponential form
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
z = re jφ
ELCT708: Electronics for Biotechnology
12
If x and y are given
z = x + jy
the x axis represents the real part and
the y axis represents the imaginary
If r and φ are given
x = r cos φ
y = r sin φ
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
13
j x j = -1
-j x j = 1
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
14
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
15
Example
Given a sinusoid v(t) = Vm cos( t + φ ),
Represent v(t) in terms of polar and exponential form
Solution
V is thus the phasor representation of the sinusoid v(t),
it is a complex representation of the magnitude and phase of a sinusoid.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
16
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
17
Example
Plot the graphical presentation of the following
Solution
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
18
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
19