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ECE 3336 Introduction to Circuits & Electronics

Note Set #8Phasors

Spring 2013TUE&TH 5:30-7:00 pmDr. Wanda Wosik

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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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2

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Results: 4

Notice the phase shift

Static part

}

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