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ECE 3336 Introduction to Circuits & Electronics

Note Set #8

Phasors

Fall 2012

TUE&TH 5:30-7:00 pm

Dr. Wanda Wosik

AC Signals (continuous in time)

Voltages and currents v(t) and i(t) are functions of time now.

We will focus on periodic functions

f(t)

y(x)=x3-x

t

Even functions

Odd functions

y(x)=x2

Periodic

cos(x)

Periodic

sin(x)

Periodic signal waveformsAC 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.

Sine waves

Amplitude change

Phase shift

Frequency change

+Amplitude and DC shift

http://dwb4.unl.edu/chem/chem869m/chem869mmats/sinusoidalfns.html

STEPPED FREQUENCIES

- C-major SCALE: successive sinusoids
- Frequency is constant for each note

IDEAL

© 2003, JH McClellan & RW Schafer

Periodic Signals

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.

EXAMPLES: http://www.falstad.com/fourier/

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

Notice the phase shift f

Solving circuits:

Continuous time dependent periodic signals represented by complex numbers phasors

Results: 4

1

ω0 means rotation frequency of the rotating phasor

3

2

}

Static part

Drawing by Dr. Shattuck

Graphical Correlation Between CT Signals and Their Phasors

Rotation of the phasor (voltage vector) Vwith the angular frequency

- In general, the vector’s length is r (amplitude) so
- V=a+jb
- in the rectangular form:
- in the polar form:

At t=0

t=0

http://www.ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html

Corresponds to the time dependent voltage changes

Phasors

Inductance

Capacitance

Current lagging voltage by 90°

Current leading voltage by 90°

For resistance R both vectors VR(jt) and IR (jt) are the same and there is

no phase shift!

Impedances Represented by Complex Numbers

Current lagging voltage by 90°

Current leading voltage by 90°

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-statesolution.

Phasors Used to Represent Circuits

- Sinusoidalsource vs.
- What is the current that results for t > 0?

Steady state value of a solution the one that remains unchanged after a long time is obtained with the phasor transform technique.

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

Steady State – use only that

Will disappear=transient

More on Transient and Steady State

The solution of i(t), for t > 0 is

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.

Decaying exponential with Time constant t = L/R.

It will die away and become relatively small after a few t.

This part of the solution is the transient response.

“Steady State solution” for Phasors

It was input voltage

Calculated current

- 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.

Phasors

Euler identity

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