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Dr. Un-ki Yang

Particle Physics Group

ukyang@hep.manchester.ac.uk or Shuster 5.15

Amplifiers and Feedback 1Amplifiers and Feedback 1

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Real Experiment

How can we catch cosmic particles & measure their energies?

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Real Experiment

Trigger

coincidence

cosmic ray

scintillator

Signal

X10Amp.

integration

ADC

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OutlineOutline

Prerequisites: 1st-year electronics, and vibration & waves

Aims: to understand how analogue signals are amplified, manipulated, and how they can be interfaced to digital systems

Learning outcomes• To understand the behavior of an ideal amplifier under negative (positive) feedback• To be able to apply this to simple amplifier, summer, integrator,

phase shifter, and oscillator• To understand the limitations of a real amplifier • To understand basic methods of analogue-to-digital conversion

(ADC)

Lectures: 4 hours lectures (2 hours per day)• Oct. 5 & Oct. 12 (1st) , Oct 19 & Oct 26 (2nd)

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Lecture notes and references

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Basic Circuit Theory

Ohm’s Law: V = IR • V is the potential difference across the resister• R is the resister (): typically k• I is the current (A): typically mA

Kirchoff’s Laws• Conservation of energy: for a closed loop

• Conservation of charge: net charge into a point (node)

ΣiVi = 0

Σi Ii = 0

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Dividers

Voltage Divider

Current Divider

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AC Circuit

Alternating current (AC) circuits: v(t), i(t)

Consider v(t), i(t) with sinusoidal sources

v(t) =V0 cos(ωt+φv), i(t) =I 0 cos(ωt+φI )

v(t) =V0ej (ωt+φv ), i(t) =I 0e

j (ωt+φI )

Extension of Ohm’s law to AC circuits

v(ω,t) =Z(ω)i(ω,t),Zisageneralizedresistance:"impedance"

Z is a complex number

is a phase

Z = Z eiφ

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AC Circuit with Capacitor & Inductance

In AC circuit, capacitance (C) and inductance (L) are used to store energy in electric and magnetic fields

Capacitance : v = q/C• Source of i and v• To smooth a sudden change in voltage• Typically F or pF (farad)

Inductance : v = L di/dt • To smooth sudden change in current• Typically H or mH (henry)

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RC Circuit with Sinusoidal Source

Resistive impedance: ZR=R,

• same phase

Capacitive impedance: Zc = 1/jωC, • -/2 phase

Inductive impedance: ZL = jωL,

• /2 phase

v(t)−Ri(t) =0

v(t)−q(t) /C =0

v(t)−Ldi(t) / dt=0

v(t) =V0ejωt, i(t) =I 0e

jωt

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Capacitor

Circuit with capacitor

v =V0 cosωt⇔ V0ejωt

v=q /C V C

i(t)

Z(ω) -/2 phase

In a DC circuit, ω0inf

it acts like an open circuit The current leads the voltage

by 90o

v(t) =i(t) / jωCZ=−j /ωC

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RC Low-Pass Filter

G ≡Vout

Vin

=1

1+ jωRC

R

CVin Vout

ω → 0 ⇒ G(ω) →Glow = 1

ω → ∞ ⇒ G(ω) →Ghigh =1

jωRC

Ghigh =1

RCω −1

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RC Low-pass filter

Low pas filter acts as an integrator at high frequency

IR =IC

IR =Vin −Vout

R, IC =C

dVout

dtVin −Vout

R=C

dVout

dtifVin ? Vout(lowgain:highω)Vin

R≈C

dVout

dt

Vout =1

RCVindt∫

R

CVin Vout

VIN (t) Vejωt

Ghigh 1

jwRC

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RC High-pass filter

High pass filter acts as a differentiator at low frequency

Vin Vout

Vout =R

R+1 / jωCVin

Vout =jωRC

1+ jωRCVin

G≡Vout

Vin

=jωRC

1+ jωRC

ω → 0 ⇒ G(ω ) →Glow = jωRC

ω → ∞ ⇒ G(ω ) →Ghigh = 1

Vout = RCd

dtVIN at low frequency

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RC circuits

Low-pass

filter

high ω

High-pass

filter

low ω

1

jwRC

ω → 0 ω → ∞

jwRC

1

1

Vout =1

RCVindt∫

Vout =RCddt

VIN

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Amplifiers

The amplification (gain) of a circuit

G = VOUT / VIN

Ideal amplifier• Large but stable gain• Gain is independent of frequency• Large input impedance (not to draw too much current) • Small output impedance

Obtained by “negative feedback”

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Negative Feedback

An overall gain G is independent of G0, but only depends on

Stable gain

VOUT =G0v, VOUT =VIN + VOUT

VOUT =G0

1−GO

VIN

G ≈−

1,ifG0 ? 1

Vout =G0V, V=VIN + Vout

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Operational Amplifier

Vout =G0 (V+ - V-) (called as differential amp.)• Vout = - G0 V- , if V+ =0 : inverting amplifier• Vout = G0 V+ , if V- =0 : non-inverting amplifier

Amplifier with a large voltage gain (~105) High Zin (~106 ) Low Zout(<100 )

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OP Amplifier 741

Many interesting features about OP amplifier http://www.allaboutcircuits.com/vol_3/chpt_8/3.html

+15V

-15V

VoutV-

V+

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Non-inverting Amplifier

Golden rules Infinite Gain Approx. (IGA)Small v(=V+- V-): V+=V-

Small input currents:

I+=I-=0 (large Zin)

G =VOUT

VIN

≈1=

R1 + R2

R1

,

ifG0 ? 1

V−=VOUT

R1

R1 + R2

VOUT =G0 (V+ −V−), VIN =V+

G =VOUT

VIN

=G0

1+R1

R1 + R2

G0

⎛

⎝⎜⎞

⎠⎟

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Inverting Amplifier

Inverting Amplifier

Golden rule: V+= V- (v- is at virtual ground)

Calculate gain!

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Differentiator

Vin −V−

ZC

=V−−Vout

R

−jwCVin =Vout

RwhereV+ =0&V+ =V−, thusV−=0

G =VOUT

VIN

=−RZC

=−jwCR

Vout =−RCddt

VIN

Not necessaryto assume

Vin>>V-

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Realistic OP Amplifier

Gain is NOT infinite

Gain is NOT constant against frequency

Output response is NOT instantaneous

Gain drops at high frequency

Bandwidth: a stable range, -3dB

Slew rate: response rate

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Gain

Open gain, Go ~ 105: const. for a small range

Go

G

Closed gain, G(R,C): const. for a wide range

-3dB dB =20 logVOUT

VIN

=10 logPOUT

PIN

20 log(G / 2 ) =20 logG−3Bandwidth

Bandwidth: the range of frequencies for gain to be within 3dB

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Slew Rate

Output response is not instantaneous

Slew rate: the rate at which the output voltage can change: V/t

t

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Output Impedance

Vout will drop by r/(r+R), thus output impedance can be measured using an external register, r

VOUT ⇒r

r + RVOUT