EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin System Realization

Passive Circuit Elements Laplace transforms with zero-valued initial conditions Capacitor Inductor Resistor + – v(t)v(t) + – v(t)v(t) + – v(t)v(t)

First-Order RC Lowpass Filter x(t)x(t)y(t)y(t) ++ C R X(s)Y(s) ++ R Time domain Laplace domain i(t)i(t) I(s)

First-Order RC Highpass Filter x(t)x(t)y(t)y(t) ++ R C X(s)Y(s) ++ Time domain Laplace domain i(t)i(t) R I(s)I(s) Frequency response is also an example of a notch filter

Passive Circuit Elements Laplace transforms with non-zero initial conditions Capacitor Inductor

Operational Amplifier Ideal case: model this nonlinear circuit as linear and time-invariant Input impedance is extremely high (considered infinite) v x (t) is very small (considered zero) + _ y(t)y(t) + _ + _ vx(t)vx(t)

Operational Amplifier Circuit Assuming that V x (s) = 0, How to realize a gain of –1? How to realize a gain of 10? + _ Y(s)Y(s) + _ + _ Vx(s)Vx(s) Zf(s)Zf(s) Z(s)Z(s) +_ F(s)F(s) I(s)I(s)H(s)H(s)

Differentiator A differentiator amplifies high frequencies, e.g. high-frequency components of noise: H(s) = s for all values of s (see next slide) Frequency response is H(f) = j 2  f  | H( f ) |= 2  f | Noise has equal amounts of low and high frequencies up to a physical limit A differentiator may amplify noise to drown out a signal of interest In analog circuit design, one would generally use integrators instead of differentiators