Transcript PowerPoint-presentation

Kent Bertilsson
Muhammad Amir Yousaf
DC and AC Circuit analysis
 Circuit analysis is the process of finding the voltages across, and the
currents through, every component in the circuit.
 For dc circuits the components are resistive as the capacitor and
inductor show their total characteristics only with varying voltage or
current.
 Sinusoidal waveform is one form of alternating waveform where the
amplitude alternates periodically between two peaks.
Kent Bertilsson
Muhammad Amir Yousaf
Sinusoidal Waveform
 Unit of measurement for horizontal axis can be time ,
degrees or radians.
Kent Bertilsson
Muhammad Amir Yousaf
Sinusoidal Waveform
 Unit of measurement for horizontal axis can be time ,
degrees or radians.
Vertical projection of radius
vector rotating in a uniform
circular motion about a fixed
point.
 Angular Velocity
 Time required to complete one
revolution is T
Kent Bertilsson
Muhammad Amir Yousaf
Sinusoidal Waveform
 Mathematically it is represented as:
Kent Bertilsson
Muhammad Amir Yousaf
Frequency of Sinusoidal
 Every signal can be described both in the time domain and the
frequency domain.
 Frequency representation of sinusoidal signal is:
Muhammad Amir Yousaf
A periodic signal in frequency domain
 Every signal can be described both in the time domain and the
frequency domain.
 A periodic signal is always a sine or cosine or the sum of sines
and cosines.
 Frequency representation of periodic signal is:
V
fs
Kent Bertilsson
Muhammad Amir Yousaf
2 fs
3 fs
4 fs
5 fs
f
A periodic signal in frequency domain
 A periodic signal (in the time domain) can in the frequency
domain be represented by:
 A peak at the fundamental frequency for the signal, fs=1/T
 And multiples of the fundamental f1,f2,f3,…=1xfs ,2xfs ,2xfs
V
V
T=1/fs
t
fs
Kent Bertilsson
Muhammad Amir Yousaf
2 fs
3 fs
4 fs
5 fs
f
Non periodic signal in frequency domain
 A non periodic (varying) signal time domain is spread in the
frequency domain.
 A completely random signal (white noise) have a uniform
frequency spectra
V
Noise
f
Kent Bertilsson
Muhammad Amir Yousaf
Why Frequency Representation?
 All frequencies are not treated in same way by nature and
man-made systems.
 In a rainbow, different parts of light spectrum are bent differently
as they pass through a drop of water or a prism.
 An electronic component or system also gives frequency
dependent response.
Kent Bertilsson
Muhammad Amir Yousaf
Phase Relation
 The maxima and the minima at pi/2,3pi/2 and 0,2pi
can be shifted to some other angle.
The expression in this case would be:
Kent Bertilsson
Muhammad Amir Yousaf
Derivative of sinusoidal
Kent Bertilsson
Muhammad Amir Yousaf
Response of R to Sinusoidal Voltage or
Current
 Resistor at a particular
frequency
Kent Bertilsson
Muhammad Amir Yousaf
Response of L to Sinusoidal Voltage or
Current
 Inductor at a particular frequency
Kent Bertilsson
Muhammad Amir Yousaf
Response of C to Sinusoidal Voltage or
Current
 Capacitor at a particular frequency
Kent Bertilsson
Muhammad Amir Yousaf
Frequency Response of R,L,C
 How varying frequency affects the opposition offered
by R,L and C
Kent Bertilsson
Muhammad Amir Yousaf
Complex Numbers
 Real and Imaginary axis on complex plane
 Polar Form
 Rectangular Form
Kent Bertilsson
Muhammad Amir Yousaf
Conversion between Forms
 Real and Imaginary axis on complex plane
Kent Bertilsson
Muhammad Amir Yousaf
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.
Kent Bertilsson
Muhammad Amir Yousaf
R,L,C and Phasors
 How to determine phase changes in voltage and
current in reactive circuits
Kent Bertilsson
Muhammad Amir Yousaf
R,L,C and Phasors
 How to determine phase changes in voltage and
current in reactive circuits
Kent Bertilsson
Muhammad Amir Yousaf
Impedance Diagram
 The resistance will always appear 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°
Kent Bertilsson
Muhammad Amir Yousaf
R,L,C in series
Kent Bertilsson
Muhammad Amir Yousaf
Voltage Divide Rule
Kent Bertilsson
Muhammad Amir Yousaf
Frequency response of R-C circuit
Kent Bertilsson
Muhammad Amir Yousaf
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 
Kent Bertilsson
Muhammad Amir Yousaf
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)
Kent Bertilsson
Muhammad Amir Yousaf
Bode Plot for High-Pass RC Filter
Kent Bertilsson
Muhammad Amir Yousaf
Sketching Bode Plot for High-Pass RC
Filter
 High-Pass R-C Filter
Voltage gain of the system is:
Av

1 / 1  j ( fc / f )
In magnitude and phase form
Av

Av

1 / 1 ( fc / f )^ 2 tan^1( fc / f )
AvdB  20 log 10Av
f
Avd B  20 log 1 0
fc
A vdB  0
Kent Bertilsson
Muhammad Amir Yousaf
For f << fc
For fc << f
Bode Plot Amplitude Response
 Must remember rules for sketching Bode Plots:
Two frequencies separated by a 2:1 ratio are said to be an octave
apart. For Bode plots, a change in frequency by one octave will result
in a 6dB change in gain.
Two frequencies separated by a 10:1 ratio are said to be a decade
apart. For Bode plots, a change in frequency by one decade will result
in a 20dB change in gain.
True only for f << fc
Kent Bertilsson
Muhammad Amir Yousaf
Asymptotic Bode Plot amplitude response
 Plotting eq below for higher frequencies:
Avd B  20 log 1 0




For f= fc/10 AvdB
For f= fc/4 AvdB
For f= fc/2 AvdB
For f= fc
AvdB
f
fc
= -20dB
= -12dB
= -6dB
= 0dB
 This gives an idealized bode plot.
 Through the use of straight-line segments called idealized Bode plots,
the frequency response of a system can be found efficiently and
accurately.
Kent Bertilsson
Muhammad Amir Yousaf
Actual Bode Plot Amplitude Response
 For actual plot using equation
Av

20 log(1 / 1 ( fc / f )^ 2 )
 For f >> fc , fc / f = 0 AvdB = 0dB
 For f = fc , fc / f = 01AvdB = -3dB
Av

20 log(1 / 1 ( 0 )^ 2 )  20 log(1 /
2 )  3 dB
 For f = 2fc AvdB = -1dB
 For f = 1/2fc AvdB = -7dB
 At f = fc the actual response curve is 3dB down from the idealized Bode
plot, whereas at f=2fc and f = fc/2 the acutual response is 1dB down
from the asymptotic response.
Kent Bertilsson
Muhammad Amir Yousaf
Asymptotic Bode Plot Phase Response
 Phase response can also be sketched using straight line asymptote
by considering few critical points in frequency spectrum.
  tan^1( fc / f )
 Plotting above equation
 For f << fc , phase aproaches 90
 For f >> fc , phase aproches 0
 At f = fc tan^-1 (1) = 45
Kent Bertilsson
Muhammad Amir Yousaf
Asymptotic Bode Plot Phase Response
 Must remember rules for sketching Bode Plots:
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.
Kent Bertilsson
Muhammad Amir Yousaf
Actual Bode Plot Phase Response
 At f = fc/10
  tan^ 1( fc / f )  tan^ 1( fc / fc / 10 )
 tan^ 1(10 )  84.29
 90 – 84.29 = 5.7
 At f = 10fc
  tan^ 1( fc / 10 fc )  tan^ 1( fc / fc * 10 )
 tan^ 1(1 / 10 )  5.7
 At f= fc theta = 45 whereas at f=fc/10 and f=10fc, the difference the
actual and asymptotic phase response is 5.7 degrees
Kent Bertilsson
Muhammad Amir Yousaf
Bode Plot for RC low pass filter

Kent Bertilsson
Muhammad Amir Yousaf
Draw an asymptotic bode
diagram for the RC filter.
Bode Plot for RC low pass filter
 Draw an asymptotic bode diagram for the RC
filter.
Zc
VOut

Zc  Z R
VIn

1 / jwC
1 / jwC  R
1

1 jwRC
1
1 j 2  fRC
1
1 jf / f c
Kent Bertilsson
Muhammad Amir Yousaf
In terms of poles and Zeros:

1
1 jwRC
S  jw

1
1 sRC

1
1
s
wc
Pole at wc
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
Kent Bertilsson
Muhammad Amir Yousaf
Bode diagram for multiple stage filter
Kent Bertilsson
Muhammad Amir Yousaf
Bode diagram for multiple stage filter
Kent Bertilsson
Muhammad Amir Yousaf
Bode diagram
Kent Bertilsson
Muhammad Amir Yousaf
Bode diagram
Kent Bertilsson
Muhammad Amir Yousaf
Exercise
 Draw an asymptotic bode
diagram for the shown filter.
R
VIn
Kent Bertilsson
Muhammad Amir Yousaf
R2
C
R3
VOut
Amplifier
• Voltage amplification
PIN
IIN
IOut
VIn
VOut
POut
VOut
AV 
VIn
• Current amplification
I Out
AI 
I In
• Power amplification
POut
AP 
PIn
Kent Bertilsson
Muhammad Amir Yousaf
Amplifier model
• The amplifier model is often sufficient describing
how an amplifier interacts with the environment
ROut
VIn
RIn
AVVIn
VOut
• RIn – Input impedance
• AV – Voltage gain
• ROut – Output impedance
Kent Bertilsson
Muhammad Amir Yousaf
Amplifier model
Kent Bertilsson
Muhammad Amir Yousaf
Bandwidth
 The bandwidth is the frequency range where the
transferred power are more than 50%.
AP  0.5 AP max
H(f)
AVmax
0.707AVmax
AV  2 AV max  0.707AV max
B  f 2  f1
f1
Kent Bertilsson
Muhammad Amir Yousaf
f2
f
Distortion
 A nonlinear function between UIn and UOut distorts the
signal
 An amplifier that saturates at high voltages
 A diode that conducts only in the forward direction
Kent Bertilsson
Muhammad Amir Yousaf
Noise
• Random fluctuation in the signal
• Theoretically random noise contains all possible
frequencies from DC to infinity
• Practical noise is often frequency limited to an upper
bandwidth by some filter
• A limited bandwidth from the noisy reduce the noise
power
Kent Bertilsson
Muhammad Amir Yousaf
RC Filters in Mindi
 Design a RC filter in Mindi.
 Simulate output for diffrent frequencies
 Analyse the results.
 dB
 Bode Plots
Kent Bertilsson
Muhammad Amir Yousaf
References
• Introductory Circuit Analysis By Boylestad
Kent Bertilsson
Muhammad Amir Yousaf