CENG4480 Lecture 03: Operational Amplifier 2 Bei Yu - - PowerPoint PPT Presentation

CENG4480 Lecture 03: Operational Amplifier – 2

Bei Yu

byu@cse.cuhk.edu.hk

(Latest update: September 28, 2019) Fall 2019

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Overview

Preliminaries Integrator & Differentiator Filters

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Overview

Preliminaries Integrator & Differentiator Filters

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Euler’s Identity

ejθ = cosθ + jsinθ ◮ real component ◮ imaginary component ◮ magnitude |ejθ| =

• cos2θ + sin2θ = 1

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Prove: | 1 1 + ja| = 1 √ 1 + a2

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Prove: | 1 1 + ja| = 1 √ 1 + a2

Tips:

1 1 + ja = 1 1 + ja · 1 − ja 1 − ja = 1 − ja 1 + a2 = 1 1 + a2 − ja 1 + a2

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Sinusoidal Signal

x(t) = Acos(ωt + φ) ◮ Periodic signals ◮ A: amplitude ◮ ω: radian frequency ◮ φ: phase

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Time Domain

◮ Voltage gain against time For sinusoidal signal: v(t) = Acos(ωt + φ)

Time (second) (usually linear scale) Voltage

+1V

• 1V

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Frequency Domain

◮ Voltage gain against frequency For sinusoidal signal: V(jω) = Aejφ = A∠φ = Acosφ + jAsinφ

Frequency (Hz) (can use log scale) Power Gain (dB)

0dB

• 3dB

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Impedance

A complex resistance or frequency-dependent resistance. That is, as resistors whose resistance is a function of the frequency of the sinusoidal excitation.

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Resistor Impedance

Assume source voltage v = A cos(ωt), then

◮ V(jω) = A∠0 ◮ I(jω) = A R∠0 Impedance of A Resistor ZR(jω) = V(jω) I(jω) = R∠0 = R

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Capacitor ABC

Capacitance C

A measure of how much charge a capacitor can hold.

◮ Amount of charge Q = C · V ◮ current is the rate of movement of charge: I = dQ dt = C · dV dt

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Capacitor Impedance

V(jω) = A∠0 I(jω) = ωCA∠π/2 Impedance of A Capacitor ZC(jω) = V(jω) I(jω)

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Capacitor Impedance

V(jω) = A∠0 I(jω) = ωCA∠π/2 Impedance of A Capacitor ZC(jω) = V(jω) I(jω) = 1 ωC∠ − π/2 = 1 ωC[cos(−π/2) + jsin(−π/2)] = − j ωC = 1 jωC

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ZC(jω) = 1 jωC Capacitor Rule 1

Low Frequency ⇒ Open circuit

Capacitor Rule 2

High Frequency ⇒ Short circuit

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Overview

Preliminaries Integrator & Differentiator Filters

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Frequency Response of An Op-Amp

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Frequency Response of An Op-Amp

◮ Inverting amplifier: Vout VS (jω) = −ZF ZS

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Frequency Response of An Op-Amp

◮ Inverting amplifier: Vout VS (jω) = −ZF ZS ◮ Non-Inverting amplifier: Vout VS (jω) = 1 + ZF ZS

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Integrator

iS(t) = −iF(t) iS(t) = vS(t) RS iF(t) = CF · dvout(t) dt

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Integrator

iS(t) = −iF(t) iS(t) = vS(t) RS iF(t) = CF · dvout(t) dt Therefore: vout(t) = − 1 RSCF t

−∞

vS(t′)dt′

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Differentiator

iS(t) = CS · dvS(t) dt iF(t) = vout(t) RF

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Differentiator

iS(t) = CS · dvS(t) dt iF(t) = vout(t) RF Therefore: vout(t) = −RFCS · dvS(t) dt

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Overview

Preliminaries Integrator & Differentiator Filters

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Low-Pass Filter

A(jω) = −ZF ZS ZF = RF|| 1 jωCF = RF 1 + jωCFRF ZS = RS

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Low-Pass Filter

A(jω) = −ZF ZS ZF = RF|| 1 jωCF = RF 1 + jωCFRF ZS = RS

⇒

A(jω) = −ZF ZS = − RF/RS 1 + jωCFRF

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Given: A(jω) = −ZF ZS = − RF/RS 1 + jωCFRF wc = 1 RFCF Prove: |A| = RF RS · 1

• 1 + ω2/w2

c

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Low-Pass Filter

|A| = RF RS · 1

• 1 + ω2/w2

c

◮ wc = 1 RFCF ◮ 3-dB frequency ◮ or cutoff frequency

BTW, limω→0 |A| = RF

RS

, limω→∞ |A| = 0

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High-Pass Filter

A(jω) = −ZF ZS ZS = RS + 1 jωCS ZF = RF

⇒

A(jω) = −ZF ZS = − jωCSRF 1 + jωCSRS

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High-Pass Filter

A(jω) = −ZF ZS = − jωCSRF 1 + jωCSRS lim

ω→0 |A| = 0

lim

ω→∞ |A| = RF

RC

High freq. cutoff unintentionally created by Op-amp

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Band-Pass Filter

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