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Lecture 25. Introduction to steady state sinusoidal analysis Overall idea Qualitative example and demonstration System response to complex inputs Complex arithmetic review Phasor representation of sinusoids Related educational materials: Chapter 10.1 - 10.3.

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Lecture 25

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Lecture 25

Introduction to steady state sinusoidal analysis

Overall idea

Qualitative example and demonstration

System response to complex inputs

Complex arithmetic review

Phasor representation of sinusoids

Related educational materials:

Chapter 10.1 - 10.3

Steady state sinusoidal response – overview

• We have examined the natural response and step response of electrical circuits

• We now consider the forced response of a circuit to sinusoidal inputs

• We will only consider the steady state response to the sinusoidal input

• Apply a sinusoidal input and let t  

• The steady state sinusoidal response

• Corresponds to the particular solution

Why is this important?

• Sinusoidal signals are very common

• Power signals commonly sinusoidal (AC signals)

• Carrier signals in communications often sinusoidal

• The mathematics is considerably simpler

• Differential equations become algebraic

• System behavior often specified in terms of the system’s steady-state sinusoidal response

• Example: Audio system specifications

• It’s “natural” – our senses often work this way

System response to sinusoidal input

• Apply a sinusoidal input, beginning at t = 0

• u(t) = Acos(t + ), t>0

Sinusoidal response of linear systems

• The steady state response of a linear system, to a sinusoidal input, will be a sinusoid of the same frequency (particular solution of same form as input)

• The amplitude and phase can change

• These changes are, in general, a function of the frequency

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• Demo system response

• Sinusoidal input to tower

• Indicate response: transient, steady-state, frequency dependence

RL circuit steady state sinusoidal response

• Apply a sinusoidal input to RL circuit:

u(t) = Acos(t + )

• Governing equation (t):

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• Note on previous slide that di/dt is NOT zero for steady-state sinusoidal response!

• Obtaining solutions in terms of sines and cosines is tedious!

• Try a “trick” involving complex exponentials:

Acos(t + ) = Re{Aej(t+)} = Re{Aejejt}

• Look at the response of the system to a complex exponential input, Aejejt

• Results in a complex exponential response, Bejejt

• The actual input is the real part of the complex input

• The actual output is the real part of the complex output

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• Note: Complex exponentials previously discussed in lecture 21.

• We’ll do a little review, but it may be worthwhile for you to review lecture 21, if you are insecure about complex exponentials – we’ll be using them a LOT now

• Point out that complex input not physically realizable!

RL circuit response – revisited

• Apply a complex exponential input:

u(t) = Aej ejt

• Governing equation (t)

• Assume form of solution:

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• Annotate last bullet of previous slide, to show di/dt and where terms go in governing differential equation

RL circuit response to complex input

• Substitute assumed solution into governing equation:

• We can cancel ejt :

• Since [Lej+R] is simply a complex number:

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• Note in previous slide:

• In equation 1, we no longer have a differential equation – it’s algebraic!

• In equation 2, the governing equation is no longer even a function of time! (The coefficients are, however, functions of frequency)

• The drawback: complex numbers are now involved. (Point out in equation 2)

• We will do a little complex arithmetic review in the next few slides.

Complex numbers – review

• Rectangular coordinates:

• Polar coordinates:

• Relationships:

Review of complex arithmetic

• Given two complex numbers:

• Subtraction:

Review of complex arithmetic – continued

• Same two complex numbers:

• Multiplication:

• Division:

Review of complex arithmetic

• Same two complex numbers, but in polar form:

• Multiplication:

• Division:

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• Annotate first “division” equation to note that 1/exp(phi) = exp(-phi)

Complex arithmetic – summary

• Addition, subtraction generally easiest in rectangular coordinates

• Add or subtract real and imaginary parts individually

• Multiplication, division generally easiest in polar coordinates

• Multiplication: multiply magnitudes, add phases

• Division: divide magnitudes, subtract phases

Phasors

• Recall:

• Complex exponentials can be used to represent sinusoids

• Complex exponentials can be written as a complex number multiplying a time-varying complex exponential

• The complex number Aejprovides the magnitude and phase of the original sinusoidal signal

Phasors – definition

• The complex number (in polar form) providing the magnitude and phase of a sinusoidal signal is called a phasor.

Example

• Use phasors to determine the current i(t) in the circuit below if Vs(t) = Vmcos(100t).

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• In previous slide, show:

• Change of input to complex exponential

• Derivation of governing differential equation.

Example – continued

• The governing differential equation is:

• Since the input has the form , where is a phasor representing the input magnitude and phase, the output must be of the form , where is a phasor representing the output magnitude and phase.

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• In previous slide, emphasize concepts. Do substitution and write algebraic equation.

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• In previous slide, do complex arithmetic to solve for current phasor; convert back to time domain.

Example – Time domain signals

• Input:

• Response:

• The response lags the input by 45