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System Realization

System Realization. Laplace transforms with zero-valued initial conditions Capacitor. Inductor Resistor. +. v ( t ). –. +. v ( t ). +. –. v ( t ). –. Passive Circuit Elements. Transfer Function. First-Order RC Lowpass Filter. R. +. +. x ( t ). C. y ( t ). i ( t ).

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System Realization

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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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