dc and ac circuit analysis
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DC and AC Circuit analysis. Circuit analysis is the process of finding the voltages across, and the currents through, every component in the circuit. - PowerPoint PPT PresentationTRANSCRIPT
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Kent Bertilsson Muhammad Amir Yousaf
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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.
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Sinusoidal Waveform Unit of measurement for horizontal axis can be time ,
degrees or radians.
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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
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Sinusoidal Waveform Mathematically it is represented as:
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Frequency of Sinusoidal Every signal can be described both in the time domain and the
frequency domain.
Frequency representation of sinusoidal signal is:
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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 2 fs 3 fs 4 fs 5 fs f
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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
T=1/fs t
V
fs 2 fs 3 fs 4 fs 5 fs f
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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
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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.
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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:
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Derivative of sinusoidal
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Response of R to Sinusoidal Voltage or Current
Resistor at a particular frequency
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Response of L to Sinusoidal Voltage or Current
Inductor at a particular frequency
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Response of C to Sinusoidal Voltage or Current
Capacitor at a particular frequency
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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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Complex Numbers Real and Imaginary axis on complex plane
Rectangular Form
Polar Form
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Conversion between Forms Real and Imaginary axis on complex plane
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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.
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R,L,C and Phasors How to determine phase changes in voltage and
current in reactive circuits
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R,L,C and Phasors How to determine phase changes in voltage and
current in reactive circuits
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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°
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R,L,C in series
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Voltage Divide Rule
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Frequency response of 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.
fHangle
fHdB
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Decibel, dB decibel, dB is very useful measure to compare two levels
of power.
It is used for expressing amplification (and attenuation)InVOutV
VAVdBA
InVOutV
InVOutV
RInVROutV
InPOutP
PdBA
RVIVP
InPOutP
PAPdBA
log20log20
log20
2
log102
2
log10log10
2
log10log10
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Bode Plot for High-Pass RC Filter
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Sketching Bode Plot for High-Pass RC Filter
High-Pass R-C Filter
Voltage gain of the system is:
In magnitude and phase form)/(1/1 ffjA cv
)/(1tan^2)^/(1/1 ffcffcAA vv
vvdB AA 10log20
For f << fcc
vdB ff
A 10log20
0vdBA For fc << f
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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
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Asymptotic Bode Plot amplitude response Plotting eq below for higher frequencies:
For f= fc/10 AvdB = -20dB
For f= fc/4 AvdB = -12dB
For f= fc/2 AvdB = -6dB
For f= fc AvdB = 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.
cvdB f
fA 10log20
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Actual Bode Plot Amplitude Response For actual plot using equation
For f >> fc , fc / f = 0 AvdB = 0dB
For f = fc , fc / f = 01AvdB = -3dB
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.
)2)^/(1/1log(20 ffcAv
dBAv 3)2/1log(20)2)^0(1/1log(20
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Asymptotic Bode Plot Phase Response
Phase response can also be sketched using straight line asymptote by considering few critical points in frequency spectrum.
Plotting above equation
For f << fc , phase aproaches 90 For f >> fc , phase aproches 0 At f = fc tan^-1 (1) = 45
)/(1tan^ ffc
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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.
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Actual Bode Plot Phase Response At f = fc/10
90 – 84.29 = 5.7
At f = 10fc
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
29.84)10(1tan^
)10//(1tan^)/(1tan^ fcfcffc
7.5)10/1(1tan^
)10*/(1tan^)10/(1tan^ fcfcfcfc
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Bode Plot for RC low pass filter
Draw an asymptotic bode diagram for the RC filter.
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Bode Plot for RC low pass filter Draw an asymptotic bode diagram for the RC
filter.
cfjf
fRCj
jwRC
RjwCjwC
RZcZcZ
InVOutV
/11
211
11
/1/1
In terms of poles and Zeros:
cws
sRC
jwRC
1
1
11
jw S
11
Pole at wc
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Bode diagram for multiple stage filter
According to logarithmic laws
dBAdBAdBAdBtotA
AAAtotA
321
321
321 AangleAangleAangletotAangle
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Bode diagram for multiple stage filter
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Bode diagram for multiple stage filter
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Bode diagram
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Bode diagram
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Exercise
R
R2 CVIn R3 VOut
Draw an asymptotic bode diagram for the shown filter.
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Amplifier
• Voltage amplification
• Current amplification
• Power amplification
IIN IOut
PIN VIn VOut POut
In
OutV VVA
In
OutI IIA
In
OutP P
PA
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Amplifier model
• RIn – Input impedance• AV – Voltage gain• ROut – Output impedance
ROut
VIn RIn AVVIn VOut
• The amplifier model is often sufficient describing how an amplifier interacts with the environment
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Amplifier model
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H(f) AVmax
0.707AVmax
f1 f2 f
Bandwidth
The bandwidth is the frequency range where the transferred power are more than 50%.
12
maxmax
max
707.02
5.0
ffBAAA
AA
VVV
PP
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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
Distortion
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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
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RC Filters in Mindi
Design a RC filter in Mindi.
Simulate output for diffrent frequencies
Analyse the results.
dB
Bode Plots
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References
• Introductory Circuit Analysis By Boylestad
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