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Electrical Engineering BA (B), Analog
Electronics,
Lecture 2
ET065G 6 Credits
ET064G 7.5 Credits
Muhammad Amir Yousaf
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Frequency Response of R,L,C
 How varying frequency affects the opposition offered
by R,L and C
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Impedance Diagram
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Impedance Diagram
 The resistance appears on the positive real axis, the inductive reactance on the
positive imaginary axis, and the capacitive reactance on the negative imaginary
axis.
 Circuits combining different types of elements will have total impedances
that extend from 90° to -90°
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AC Circuit Analysis
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Complex Numbers
• A complex number represents a point in a two-dimensional plane
located with reference to two distinct axes.
• This point can also determine a radius vector drawn from the origin to
the point.
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Complex Numbers
 Rectangular and Polar forms
 Polar Form
 Rectangular Form
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Conversion between Forms
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MATHEMATICAL OPERATIONS
WITH COMPLEX NUMBERS
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MATHEMATICAL OPERATIONS
WITH COMPLEX NUMBERS
Complex Conjugate
simply changing the sign of the imaginary part
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MATHEMATICAL OPERATIONS
WITH COMPLEX NUMBERS
Reciprocal
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MATHEMATICAL OPERATIONS
WITH COMPLEX NUMBERS
Addition
Subtraction
Addition or subtraction cannot be performed in polar form unless the
complex numbers have the same angle u or unless they differ only by
multiples of 180°.
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MATHEMATICAL OPERATIONS
WITH COMPLEX NUMBERS
Multiplication
Division
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Phasors
• The radius vector, having a constant magnitude (length) with one end
fixed at the origin, is called a phasor when applied to electric
circuits.
It should be
pointed out
that in phasor
notation, the
sine wave is
always the
reference,
and the
frequency is
not
represented.
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Phasors
Phasor algebra for sinusoidal quantities is applicable only for waveforms having the same
frequency.
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R,L,C in series
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Voltage Divide Rule
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Frequency response of series R-C circuit
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Bode Diagram
• It is a technique for sketching the frequency response of systems (i.e.
filter, amplifiers etc) on dB scale . It provides an excellent way to
compare decibel levels at different frequencies.
• Absolute decibel value and phase of the transfer function is plotted
against a logarithmic frequency axis.
H  f  dB
  angleH  f 
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Decibel, dB
 decibel, dB is very useful measure to compare two levels
of power.
P
Out
APdB  10 log AP  10 log
PIn
V2
P V  I 
R
2
VOut
POut
APdB 10 log
10 log R 10 log
2
PIn
VIn
R
 VOut 
V 
 In 
2
VOut
 20 log
VIn
VOut
AVdB  20 log AV  20 log
VIn
 It is used for expressing amplification (and attenuation)
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Bode Plot for a RC Circuit
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Bode Plot for a RC Circuit
This gives an idealized bode plot.
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Bode Plot for a RC Circuit
Note that as the frequency of interest
approaches f , the dB gain becomes
less negative and approaches the
final normalized value of 0 dB.
c
The resulting plot is a straight line intersecting
the 0 dB line at fc . It increases to the right at a rate
of 6 dB per octave or 20 dB per decade.
At higher frequencies:
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Bode Plot for a RC Circuit
The phase response can also be
sketched using straight-line
asymptotes by considering a few
critical points in the frequency
spectrum.
An asymptote at theta = 90 for f << fc/10, an asymptote at theta = 0 for f >> 10fc and
an asymptote from fc/10 to 10fc that passes through theta = 45 at f= fc.
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Bode diagram for multiple stage filter
 According to logarithmic laws
Atot  A1  A2  A3
Atot
 A1
 A2
 A3
dB
dB
dB
dB
   angleA1   angleA2   angleA3 
angle Atot
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Bode diagram for multiple stage filter
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Bode diagram for multiple stage circuit
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Bode diagram
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Bode diagram
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Exercise
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Exercise
Draw a detailed asymptotic bode-diagram for a system’s gain. Both the
amplitude and phase should be clearly visualized.
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Exercise
 Derive to get Bode plot format equation for the system shown
in the figure
Z2
Z1
H  f  dB
  angleH  f 
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Gain = -Z2/Z1
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Thank You
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References
• Introductory Circuit Analysis By Boylestad
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