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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

- 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

- 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

- Apply a sinusoidal input, beginning at t = 0
- u(t) = Acos(t + ), t>0

- 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

- Demo system response
- Sinusoidal input to tower
- Indicate response: transient, steady-state, frequency dependence

- Apply a sinusoidal input to RL circuit:
u(t) = Acos(t + )

- Governing equation (t):

- 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{Aejejt}

- Look at the response of the system to a complex exponential input, Aejejt
- Results in a complex exponential response, Bejejt
- The actual input is the real part of the complex input
- The actual output is the real part of the complex output

- 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!

- Apply a complex exponential input:
u(t) = Aej ejt

- Governing equation (t)
- Assume form of solution:

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

- Substitute assumed solution into governing equation:
- We can cancel ejt :
- Since [Lej+R] is simply a complex number:

- 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.

- Rectangular coordinates:
- Polar coordinates:
- Relationships:

- Given two complex numbers:

- Addition:
- Subtraction:

- Same two complex numbers:

- Multiplication:
- Division:

- Same two complex numbers, but in polar form:

- Multiplication:
- Division:

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

- 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

- 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 Aejprovides the magnitude and phase of the original sinusoidal signal

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

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

- In previous slide, show:
- Change of input to complex exponential
- Derivation of governing differential equation.

- 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.

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

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

- Input:
- Response:
- The response lags the input by 45