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Realizing Laplace Transform Systems with Passive Circuit Elements

Learn about implementing Laplace transforms with zero-valued initial conditions using capacitors, inductors, and resistors in passive circuits. Explore first-order RC lowpass and highpass filters in time and Laplace domains, and understand notch filters and operational amplifier applications. Discover strategies for achieving specific gains and the function of differentiators in signal processing.

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Realizing Laplace Transform Systems with Passive Circuit Elements

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  1. System Realization

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

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

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

  5. Laplace transforms with non-zeroinitial conditions Capacitor Inductor Passive Circuit Elements

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

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

  8. 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 p f  | H( f ) |= 2 p | 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

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