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Amplifiers and Feedback 1
Dr. Un-ki Yang
Particle Physics Group
[email protected] or Shuster 5.15
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Real Experiment
 How can we catch
cosmic particles &
measure their energies?
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Real Experiment
Trigger
cosmic ray
scintillator
coincidence
integration
Signal
X10
Amp.
ADC
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Outline
 Aims: to understand how analogue signals are amplified, manipulated,
and how they can be interfaced to digital systems
 Prerequisites: 1st-year electronics, and vibration & waves
 Lectures: 4 hours lectures (2 hours per day)
• Oct. 5 & Oct. 12 (1st) , Oct 19 & Oct 26 (2nd)
 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)
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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
 iVi  0
• Conservation of charge: net charge into a point (node)
i 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( t  v ), i(t)  I 0 cos( t   I )
v(t)  V0 e j ( t v ) , i(t)  I 0 e j ( t I )
 Extension of Ohm’s law to AC circuits
v( ,t)  Z( )i( ,t),
Z is a generalized resistance: "impedance"
 Z is a complex number
Z  Z ei
 is a phase
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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
j t
v(t)  V0 e , i(t)  I 0 e
j t
v(t)  Ri(t)  0
 Resistive impedance: ZR=R,
• same phase
 Capacitive impedance: Zc = 1/jC,
• -/2 phase
 Inductive impedance: ZL = jL,
• /2 phase
v(t)  q(t) / C  0
v(t)  L di(t) / dt  0
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Capacitor
 Circuit with capacitor
v  V0 cos  t  V0 e j t
v q/C
V
C
v(t )  i(t ) / j C
Z   j / C
 In a DC circuit, inf
it acts like an open circuit
 The current leads the voltage
by 90o
i(t)
Z()
-/2 phase
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RC Low-Pass Filter
R
Vin
C
G
Vout
Vout
1

Vin 1  j RC
  0  G( )  Glow  1
    G( )  Ghigh
Ghigh
1

j RC
1 1


RC
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RC Low-pass filter
 Low pas filter acts as an integrator at high frequency
R
Vin
C
VIN (t)  Ve j t
1
Ghigh 
jwRC
Vout
I R  IC
Vin  Vout
dVout
IR 
, IC  C
R
dt
Vin  Vout
dV
 C out
R
dt
if Vin ? Vout (low gain: high  )
Vin
dVout
C
R
dt
1
Vout 
Vin dt

RC
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RC High-pass filter
 High pass filter acts as a differentiator at low frequency
Vin
R
Vout 
Vin
R  1 / j C
j RC
Vout 
Vin
1  j RC
Vout
j RC
G

Vin 1  j RC
Vout
  0  G( )  Glow  j RC
    G( )  Ghigh  1
Vout
d
 RC VIN at low frequency
dt
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RC circuits
  0  
Low-pass
filter
High-pass
filter
1
1
jwRC
high 
Vout
1

Vin dt

RC
low 
jwRC
1
Vout
d
 RC VIN
dt
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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
V
 G v, VV=V
 IN
VINV
Vout
VOUT
OUT
out  G00V, OUT
VOUT 
G0
VIN
1   GO
1
G   , if G 0

 An overall gain G is independent of
G0, but only depends on 
 Stable gain
1
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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
+15V
V+
V-
Vout
-15V
Many interesting features about OP amplifier
http://www.allaboutcircuits.com/vol_3/chpt_8/3.html
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Non-inverting Amplifier
G
VOUT
1 R1  R2


,
VIN

R1
if G 0 ? 1
R1
V  VOUT
R1  R2
VOUT  G0 (V  V ), VIN  V
VOUT
G0
G

VIN 

R1
 1 R  R G0 
1
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Golden rules
Infinite Gain Approx. (IGA)
 Small v(=V+- V-): V+=V Small input currents:
I+=I-=0 (large Zin)
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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 V  Vout

ZC
R
Vout
 jwCVin 
R
where V  0 & V  V ,
thus V =0
VOUT
R
G

  jwCR
VIN
ZC
Vout
d
 RC VIN
dt
Not
necessary
to assume
Vin>>V22
Realistic OP Amplifier
 Gain is NOT infinite
Gain drops at high frequency
 Gain is NOT constant against frequency
Bandwidth: a stable range, -3dB
 Output response is NOT instantaneous
Slew rate: response rate
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Gain
Go
 Open gain, Go ~ 105:
const. for a small range
 Closed gain, G(R,C):
const. for a wide range
G
Bandwidth
-3dB
VOUT
POUT
dB  20log
 10log
VIN
PIN
20log(G / 2)  20logG  3
 Bandwidth: the range of frequencies
for gain to be within 3dB
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Slew Rate
 Output response is not instantaneous
t
 Slew rate: the rate at which the output voltage
can change: V/t
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Output Impedance
VOUT
r

VOUT
rR
 Vout will drop by r/(r+R), thus output impedance
can be measured using an external register, r
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