1 ece 3336 introduction to circuits & electronics note set #8 phasors spring 2013 tue&th...
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ECE 3336 Introduction to Circuits & Electronics
Note Set #8Phasors
Spring 2013TUE&TH 5:30-7:00 pmDr. Wanda Wosik
2/22
AC Signals (continuous in time)
Voltages and currents v(t) and i(t) are functions of time now.
We will focus on periodic functionsf(t)
t
Periodiccos(x)
Even functions
y(x)=x2
Periodicsin(x)
Odd functions
y(x)=x3-x
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AC Circuit Analysis (Phasors)
AC signals in circuits are very important both for circuit analysis and for design of circuits
It can be very complicated to analyze circuits since we will have differential equations (derivatives & integrals from v-i dependences)
Techniques that we will use will rely on• complex numbers to solve these
equations, and on • Fourier’s Theorem to represent the
signals as sums of sinusoids.
Periodic signal waveforms
4http://dwb4.unl.edu/chem/chem869m/chem869mmats/sinusoidalfns.html
Sine waves Amplitude change
Frequency change
Phase shift
+Amplitude and DC shift
STEPPED FREQUENCIES
• C-major SCALE: successive sinusoids– Frequency is constant for each note
IDEAL
5© 2003, JH McClellan & RW Schafer
SPECTROGRAM EXAMPLE
• Two Constant Frequencies: Beats
))12(2sin())660(2cos( tt
6© 2003, JH McClellan & RW Schafer
Modulating frequency
Frequencies Fo±Fm:660Hz±12Hz
Periodic Signals
EXAMPLES: http://www.falstad.com/fourier/
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Fourier series is used to represent periodic functions as sums of cosine waves.
Fundamental frequency in Fourier series corresponds to signal frequency and added harmonics give the final shape of the signal.
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AC Circuit Analysis What are Phasors?
A phasor is a transformation of a sinusoidal voltage or current.
• Using phasors and their analysis makes circuit solving much easier.
• It allows for Ohm’s Law to be used for inductors and capacitors.
While they seem difficult at first our goal is to show that phasors make analysis so much easier.
Transformation – Complex Numbers
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ω0 means rotation frequency of the rotating phasor
Solving circuits:
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Results: 4
Notice the phase shift
Static part
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Drawing by Dr. Shattuck
Continuous time dependent periodic signals represented by complex numbers phasors
t=0
Corresponds to the time dependent voltage changes
Graphical Correlation Between CT Signals and Their Phasors
At t=0
Rotation of the phasor (voltage vector) V with the angular frequency
In general, the vector’s length is r (amplitude) so
V=a+jb •in the rectangular form:
•in the polar form:
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http://www.jhu.edu/~signals/phasorapplet2/phasorappletindex.htm
Current lagging voltage by 90° Current leading
voltage by 90°
Inductance Capacitance
For resistance R both vectors VR(jt) and IR (jt) are the same and there is no phase shift! 11
Phasors
Impedances Represented by Complex Numbers
Current leading voltage by 90°
Current lagging voltage by 90°
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Transformation of Signals from the Time Domain to Frequency Domain
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Euler identity
Euler identity
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Complex Numbers - Reminder
Equivalent representationsRectangular
Polar
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Complex Numbers – Reminder
Example:
Use complex conjugate
and multiply
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The Limitations
The phasor transform analysis combined with the implications of Fourier’s Theorem is significant.
Limitations.
•The number of sinusoidal components, or sinusoids, that one needs to add together to get a voltage or current waveform, is generally infinite.
•The phasor analysis technique only gives us part of the solution. It gives us the part of the solution that holds after a long time, also called the steady-state solution.
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Phasors Used to Represent Circuits
Steady state value of a solution the one that remains unchanged after a long time is obtained with the phasor transform technique.
• Sinusoidal source vs. • What is the current that results
for t > 0?
Kirchhoff’s Voltage Law in the loop:
This is a first order differential equation with constant coefficients and a sinusoidal forcing function. The current at t = 0 is zero.
The solution of i(t), for t > 0, can be shown to be
Will disappear=transient Steady State – use only that
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More on Transient and Steady State
The solution of i(t), for t > 0 is
Decaying exponential with Time constant = L/R.
It will die away and become relatively small after a few .
This part of the solution is the transient response.
This part of the solution varies with time as a sinusoid.
It is also a sinusoid with the same frequency as the source, but with different amplitude and phase.
This part of the solution is the steady-state response.
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“Steady State solution” for Phasors
• Frequency of iss is the same as the source’s
• Both the Amplitude and Phase depend on: , L and R
• Finding the phasor means to determine the Amplitude and Phase
Frequency dependence is very important in ac circuits.
Euler identity
Phasors
It was input voltage
Calculated current