Systems ConceptsIntroductionTransfer FunctionCommon Transfer FunctionsFinding a Transfer FunctionTransfer Function ExampleTransfer Function UseImpulse Response(Unit) Step ResponseStep Response from ConvolutionImpulse – Step Response Relation(Unit) Ramp ResponsePole-Zero PlotLinearityTime-Invariant SystemsClass ExamplesLect15 EEE 202 1Systems ConceptsDr. HolbertMarch 19, 2008Lect15 EEE 202 2Introduction•Several important topics today, including:–Transfer function–Impulse response–Step response–Linearity and time invarianceLect15 EEE 202 3Transfer 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 asInputOutput(s)(s)(s) XYHH(s) ↔ h(t)X(s) ↔ x(t) Y(s) ↔ y(t)Input OutputSystemLect15 EEE 202 4Common 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)Lect15 EEE 202 5Finding 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:(s)(s)InputOutput(s)inoutVVH Lect15 EEE 202 6Transfer Function ExampleC+–vin(t)Rvout(t)+–1/sC+–Vin(s)RVout(s)+–Time Domain Frequency DomainUsing voltage division, we find the transfer function/1/1)/(1)/(1)()()(11)()/(1)/(1)()(sRCsRCssssRCssCRsCssinoutininoutVVHVVVLect15 EEE 202 7Transfer Function Use•We can use the transfer function to find the system output to an arbitrary input using simple multiplication in the s domainY(s) = H(s) X(s)•In the time domain, such an operation would require use of the convolution integral:d)x()-h(td)-x(t)h(y(t)Lect15 EEE 202 8Impulse 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 functiony(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 inputLect15 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)s1(s)(s)(s)(s) HXHY s(s)(s)](s)[(s)][y(t)HXHY111LLLLect15 EEE 202 10Step 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d)x()-h(td)-x(t)h(y(t)Lect15 EEE 202 11Impulse – 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td)h(d)h()-u(ty(t)tt)-u(t)-x(t01Lect15 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 responseLect15 EEE 202 13Pole-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)5.1)(4)(5()54)(5.3)(3()(22ssssssssHReImLect15 EEE 202 14Linearity•Linearity is a property of superpositionαx1(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 systemLect15 EEE 202 15Time-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 systemLect15 EEE 202 16Class Examples•Drill Problems P7-1, P7-2,