Section 4 OpAmps2/24/20 Laplace adds a real component σto the phasor 256 •We use a new variable,...
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2/24/20 Sinusoidal Oscillator Op Amps can model any linear differential equation. Basis of “Analog Computers” used before powerful digital computers. 252 253 R 1 R 1 R 2 R 2 R 3 R 4 C 1 C 1 − integrator multiply by − 1 − integrator Vt () = − 1 R 1 C 1 1 R 1 C 1 Vt () dt ∫ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∫ dt Vt () is sinusoid ω = 1 R 1 C 1 Vt () d 2 Vt () dt 2 = − 1 R 1 C 1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 Vt () Amplitude of sinusoid determined by e − t R 3 C 1 e + t R 4 C 1 and
Transcript of Section 4 OpAmps2/24/20 Laplace adds a real component σto the phasor 256 •We use a new variable,...
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Sinusoidal OscillatorOp Amps can model any linear differential equation.
Basis of “Analog Computers” used before powerful digital computers.
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R1 R1 R2
R2
R3
R4
C1 C1
− integratormultiply by − 1
− integrator
V t( ) = − 1R1C1
1R1C1
V t( )dt∫⎛⎝⎜
⎞⎠⎟∫ dt
V t( ) is sinusoid ω = 1R1C1
V t( )
d 2V t( )dt2
= − 1R1C1
⎛⎝⎜
⎞⎠⎟
2
V t( )
Amplitude of sinusoid determined by
e− tR3C1 e
+ tR4C1and
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Why doesn’t oscillator keep expanding?
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see movie
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Laplace adds a real component σ to the phasor
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• We use a new variable, “s”• ! = # + %&• Basis function becomes '()
'*'+, = ' *-+,
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Inverse Fourier Transform
x t( ) = 12π
X ω( )e jωt dω−∞
+∞
∫
Recall the Fourier TransformApplies to any finite signal (not just periodic)
Fourier Transform
X ω( ) = x t( )e− jωt dt−∞
+∞
∫
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Now becomes Laplace TransformApplies to any signal (not just finite),
any linear differential equation.
Laplace Transform
! " = $%&
'&( ) *%+,-)
Inverse Laplace Transform
( ) = 1201$2%3&
2'3&! " *'+,-"
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Exponential Amp Log Amp
• Because current is exponential of voltage in diode.
• Now can multiply signals by taking log of each, then add them and take exponential.
261https://en.wikibooks.org/wiki/Electronics/Electronics_Formulas/Op_Amp_Configurations
Non-linear Op Amp circuits with diodes
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