dc and ac circuit analysis

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.

Sinusoidal Waveform Unit of measurement for horizontal axis can be time ,

degrees or radians.

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

Sinusoidal Waveform Mathematically it is represented as:

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:

fs 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

T=1/fs t

fs 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

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.

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:

Derivative of sinusoidal

Response of R to Sinusoidal Voltage or Current

Resistor at a particular frequency

Response of L to Sinusoidal Voltage or Current

Inductor at a particular frequency

Response of C to Sinusoidal Voltage or Current

Capacitor at a particular frequency

Frequency Response of R,L,C How varying frequency affects the opposition offered

by R,L and C

Complex Numbers Real and Imaginary axis on complex plane

Rectangular Form

Polar Form

Conversion between Forms Real and Imaginary axis on complex plane

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.

R,L,C and Phasors How to determine phase changes in voltage and

current in reactive circuits

R,L,C and Phasors How to determine phase changes in voltage and

current in reactive circuits

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°

R,L,C in series

Voltage Divide Rule

Frequency response of R-C circuit

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

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

RInVROutV

InPOutP

PAPdBA

log20log20

log102

log10log10

Bode Plot for High-Pass RC Filter

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

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

Asymptotic Bode Plot amplitude response Plotting eq below for higher frequencies:

For f= fc/10 AvdB = -20dB

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

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

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

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.

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

Bode Plot for RC low pass filter

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.

RjwCjwC

RZcZcZ

InVOutV

In terms of poles and Zeros:

Pole at wc

Bode diagram for multiple stage filter

According to logarithmic laws

dBAdBAdBAdBtotA

AAAtotA

321 AangleAangleAangletotAangle

Bode diagram

Exercise

R2 CVIn R3 VOut

Draw an asymptotic bode diagram for the shown filter.

Amplifier

• Voltage amplification

• Current amplification

• Power amplification

IIN IOut

PIN VIn VOut POut

OutV VVA

OutI IIA

OutP P

Amplifier model

• RIn – Input impedance• AV – Voltage gain• ROut – Output impedance

VIn RIn AVVIn VOut

• The amplifier model is often sufficient describing how an amplifier interacts with the environment

Amplifier model

H(f) AVmax

0.707AVmax

f1 f2 f

Bandwidth

The bandwidth is the frequency range where the transferred power are more than 50%.

maxmax

707.02

ffBAAA

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

• 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

RC Filters in Mindi

Design a RC filter in Mindi.

Simulate output for diffrent frequencies

Analyse the results.

Bode Plots

References

• Introductory Circuit Analysis By Boylestad

dc and ac circuit analysis

frequency domainevery

fundamental frequency

fs tv fs

frequency dependent

sinusoidal voltage

random signal white

varying voltage

angular velocity time

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