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Systems Concepts

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

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  1. Systems Concepts Dr. Holbert March 19, 2008 EEE 202

  2. Introduction • Several important topics today, including: • Transfer function • Impulse response • Step response • Linearity and time invariance EEE 202

  3. 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

  4. 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

  5. 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

  6. + + 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

  7. 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

  8. 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

  9. (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

  10. 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

  11. 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

  12. (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

  13. 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

  14. 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

  15. 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

  16. Class Examples • Drill Problems P7-1, P7-2, P7-4 EEE 202

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