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Systems Concepts. Dr. Holbert March 19, 2008. Introduction. Several important topics today, including: Transfer function Impulse response Step response Linearity and time invariance. System. X ( s ) ↔ x( t ). Y ( s ) ↔ y( t ). H ( s ) ↔ h( t ). Input. Output. Transfer Function.

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systems concepts

Systems Concepts

Dr. Holbert

March 19, 2008

EEE 202

introduction
Introduction
  • Several important topics today, including:
    • Transfer function
    • Impulse response
    • Step response
    • Linearity and time invariance

EEE 202

transfer function

System

X(s) ↔ x(t)

Y(s) ↔ y(t)

H(s) ↔ h(t)

Input

Output

Transfer Function
  • The transfer function, H(s), is the ratio of some output variable (y) to some input variable (x)
  • The transfer function is portrayed in block diagram form as

EEE 202

common transfer functions
Common Transfer Functions
  • The transfer function, H(s), is bolded because it is a complex quantity (and it’s a function of frequency, s = jω)
  • Since the transfer function, H(s), is the ratio of some output variable to some input variable, we may define any number of transfer functions
    • ratio of output voltage to input voltage (i.e., voltage gain)
    • ratio of output current to input current (i.e., current gain)
    • ratio of output voltage to input current (i.e., transimpedance)
    • ratio of output current to input voltage (i.e., transadmittance)

EEE 202

finding a transfer function
Finding a Transfer Function
  • Laplace transform the circuit (elements)
    • When finding H(s), all initial conditions are zero (makes transformation step easy)
  • Use appropriate circuit analysis methods to form a ratio of the desired output to the input (which is typically an independent source); for example:

EEE 202

transfer function example

+

+

R

R

vin(t)

+

Vin(s)

+

vout(t)

Vout(s)

C

1/sC

Transfer Function Example

Time Domain

Frequency Domain

Using voltage division, we find the transfer function

EEE 202

transfer function use
Transfer Function Use
  • We can use the transfer function to find the system output to an arbitrary input using simple multiplication in the s domain

Y(s) = H(s) X(s)

  • In the time domain, such an operation would require use of the convolution integral:

EEE 202

impulse response
Impulse Response
  • Let the system input be the impulse function: x(t) = δ(t); recall that X(s) = L [δ(t)] = 1
  • Therefore: Y(s) = H(s) X(s) = H(s)
  • The impulse response, designated h(t), is the inverse Laplace transform of transfer function

y(t) = h(t) = L -1[H(s)]

  • With knowledge of the transfer function or impulse response, we can find the response of a circuit to any input

EEE 202

unit step response
(Unit) Step Response
  • Now, let the system input be the unit step function: x(t) = u(t)
  • We recall that X(s) = 1/s
  • Therefore:
  • Using inverse Laplace transform skills, and a specific H(s), we can find the step response, y(t)

EEE 202

step response from convolution
Step Response from Convolution
  • We could also use the convolution integral in combination with the impulse response, h(t), to find the system response to any other input
  • Either form of the convolution integral above can be used, but generally one expression leads to a simpler, or more interpretable, result
  • We shall use the first formulation here

EEE 202

impulse step response relation
Impulse – Step Response Relation
  • The step input function is
  • The convolution integral becomes
  • We observe that the step response is the time integral of the impulse response

EEE 202

unit ramp response
(Unit) Ramp Response
  • Besides the impulse and step responses, another common benchmark is the ramp response of a system (because some physical inputs are difficult to create as impulse and step functions over small t)
  • The unit ramp function is t·u(t)

which has a Laplace transform of 1/s2

  • The ramp response is the time integral of the unit step response

EEE 202

pole zero plot
For a pole-zero plot place "X" for poles and "0" for zeros using real-imaginary axes

Poles directly indicate the system transient response features

Poles in the right half plane signify an unstable system

Consider the following transfer function

Im

Re

Pole-Zero Plot

EEE 202

linearity
Linearity
  • Linearity is a property of superposition

αx1(t) + βx2(t) → αy1(t) + βy2(t)

  • A system with a constant (additive) term is nonlinear; this aspect results from another property of linear systems, that is, a zero input to a linear system results in an output of zero
  • Circuits that have non-zero initial conditions are nonlinear
  • An RLC circuit initially at rest is a linear system

EEE 202

time invariant systems
Time-Invariant Systems
  • In broad terms, a system that does not change with time is a time-invariant system; that is, the rule used to compute the system output does not depend on the time at which the input is applied
  • The coefficients to any algebraic or differential equations must be constant for the system to be time-invariant
  • An RLC circuit initially at rest is a time-invariant system

EEE 202

class examples
Class Examples
  • Drill Problems P7-1, P7-2, P7-4

EEE 202