electronics in high energy physics introduction to electronics in hep
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Electronics in High Energy Physics Introduction to electronics in HEP. Operational Amplifiers (based on the lecture of P.Farthoaut at Cern). Operational Amplifiers. Feedback Ideal op-amp Applications Voltage amplifier (inverting and non-inverting) Summation and differentiation - PowerPoint PPT PresentationTRANSCRIPT
1
Electronics in High Energy Physics Introduction to electronics in HEP
Operational Amplifiers(based on the lecture of P.Farthoaut at Cern)
2
Operational Amplifiers
Feedback Ideal op-amp Applications
– Voltage amplifier (inverting and non-inverting)– Summation and differentiation – Current amplifier– Charge amplifier
Non-ideal amplifier– Offset– Bias current– Bandwidth– Slew rate– Stability– Drive of capacitive load
Data sheets Current feedback amplifiers
3
Feedback
Y is a source linked to X– Y = x
Open loop– x = e
– y = x
– s = y = x Closed loop
ex y s
1e
s
1
eys
1
ey
yexy
yex
is the open loop gain is the loop gain
4
Interest of the feedback
In electronics– is an amplifier
– is the feedback loop
– and are input and output impedances If is large enough the gain is independent of the amplifier
ex s
1e
s
5
Operational amplifier
Gain A very large Input impedance very high
– I.e input current = 0 A(p) as shown
-40
-20
0
20
40
60
80
100
120
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Frequency (Hz)
A (
dB
)
-
+
-A
6
How does it work?
Direct gain calculation
-
+
-A
R1
R2
Vin
Vout
I
2R1R1R
A1
A
Vin
Vout
1;2R1R
1R;A
1e
s
2R1R1R
A1
A
Vin
Vout
I)2R1R(Vout;AVout
1RIVin
Feed-back equation
Ideal Op-Amp
1R
2R1R
Vin
Vout;A
7
Non-inverting amplifier
Input impedance
-
+
R1
Vin
I
R2
Vout
Gain
1R
2R1R
Vin
Vout
I)2R1R(Vout
1RIVin
Called a follower if R2 = 0
Zin
8
Inverting amplifier
Gain
-
+
R1
Vin
I
Vout
R2
1R
2R
Vin
Vout
I2RVout
1RIVin
Input impedance
Gain error
R
R2
G
G1R
2R
Vin
VoutG
1RZin
9
Summation
If Ri = R
-
+
R1
V1
I
Vout
R
Rn
Vn
I1
In
Ri
ViRIRVout
IiIRi
ViIi
Transfer function
ViVout
10
Differentiation
-
+
R1
V1
I1
Vout
R2
I1
R1
R2
V2I2
)1V2V(1R
2RVout
)1I2I(1R1V2V2V2I1R1I1R1V
)1I2I(2R2I2R1I2RVout
11
Current-to-Voltage converter (1)
Vout = - R Iin For high gain and high bandwidth, one has to take into
account the parasitic capacitance
-
+
Iin
Vout
R
C
12
Current-to-Voltage converter (2)
Equivalent feedback resistor = R1 + R2 + R2 * (R1/r) – ex. R1 = R2 = 100 k ; r = 1 k ; Req = 10.2 M
Allows the use of smaller resistor values with less problems of parasitic capacitance
r
R1 R2
-
+
Iin
Vout
High resistor value with small ones
13
Charge amplifier (1)
Requires a device to discharge the capacitor– Resistor in //
– Switch
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
Time
Inp
ut
and
Ou
tpu
t
Input current
Capacitor only
RC network
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12 14 16
Time
Inpu
t & O
utpu
t
Output
Input current
-
+
I
Vout
C
R
)t(C
1)t(Vout;
Cp
1)p(Vout
1)p(I;)t()t(I
)p(ICp
1)p(Vout
14
Charge amplifier (2)
-
+
I
V1
C
R
C1R1 V2
R2C2
Input ChargeIn a few ns
Output of the charge amplifierVery long time constant
Shapinga few 10’s of ns
15
Miller effect
Charge amplifier– Vin =
– Vout = -A
– The capacitor sees a voltage (A+1)
– It behaves as if a capacitor (A+1)C was seen by the input
-
+Vout
C
Vin
Miller’s theorem
–Av = Vy / Vx
–Two circuits are equivalent
»Z1 = Z / (1 - Av)
»Z2 = Z / (1-Av-1)
X YZ
X Y
Z2Z1
16
Common mode
The amplifier looks at the difference of the two inputs– Vout = G * (V2 - V1)
The common value is in theory ignored– V1 = V0 + v1
– V2 = V0 + v2 In practice there are limitations
– linked to the power supplies
– changes in behaviour Common mode rejection ratio CMRR
– Differential Gain / Common Gain (in dB)
17
Non-ideal amplifier
Input Offset voltage Vd
-
+
-A
Ib+
Ib-
Vd
Zd
Zc
Zc
Zout
Input bias currents Ib+ and Ib-
Limited gain
Input impedance
Output impedance
Common mode rejection Noise Bandwidth limitation & Stability
18
Input Offset Voltage
“Zero” at the input does not give “Zero” at the output
In the inverting amplifier it acts as if an input Vd was applied (Vout) = G Vd
Notes:– Sign unknown
– Vd changes with temperature and time (aging)
– Low offset = a few V and Vd = 0.1 V / month
– Otherwise a few mV
-
+
R1I
Vout
R2
Vd
19
Input bias current (1)
(Vout) = R2 Ib- (Vout) = - R3 (1-G) Ib+ Error null for
R3 = (R1//R2) if Ib+ = Ib-
Ib+
Ib-
-
+
R1
R2
R3Vout
20
Input bias current (2)
In the case of the charge amplifier it has to be compensated
Switch closed before the measurement and to discharge the capacitor
Values– less than 1.0 pA for JFET
inputs
– 10’s of nA to A bipolar
-
+Ib+
Ib-
R3Vout
C
21
Common mode rejection
Input voltage Vc/Fr (Vc common mode voltage) Same effect as the offset voltage
-
+
R1I
R2
VoutVc/Fr
Non-inverting amplifier
22
Gain limitation
-A -
+
R1
R2
Vin
Vout
I
AGiifA
Gi1GiGiGG
1R
2RGi
Gi1A
AGi
2R1RA1R
A1R
1R
2RG
A is of the order of 105 – Error is very small
23
Input Impedance
Zin = Zc+ // (Zd A / G) ~ Zc+ G= (R1+R2)/R1
Zd
Zc-
Zc+
-
+
R1
Vin
Vout
R2
Non-inverting amplifier
24
Output impedance
Non-inverting amplifierR2
-
+
R1
Vout
I0 + Iout
Iout-A
Z0
I0
R1
R2R1 G
A
G Zo Zout
Iout) (Io R2) (R1 Vout
Io; Zo - Ae- G Vout
0 Vin when Iout
Vout Zout
25
Current drive limitation
Vout = R I = RL IL
The op-amp must deliver I + IL = Vout (1/R + 1/RL)
Limitation in current drive limits output swing
-
+
R1
Vin
I
R2
Vout RL
RL
MaximumOutputSwing
RL*Imax
26
Bandwidth
Gain amplifier of non-inverting G(p) = G A(p) / (G + A(p))– A(p) with one pole at low frequency and -6dB/octave
» A(p) = A0 / (p+0)
– G = (R1+R2)/R1 40 dB – Asymptotic plot
» G < A G(p) = G» G > A G(p) = A(p)
-40
-20
0
20
40
60
80
100
120
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Frequency
Gai
n [
dB
]
f3db= fT/G
fT
27
Slew Rate
Limit of the rate at which the output can change Typical values : a few V/s A sine wave of amplitude A and frequency f requires a slew rate of
2Af S (V/s) = 0.3 fT (MHz); fT = frequency at which gain = 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
28
Settling Time
Time necessary to have the output signal within accuracy– ±x%
Depends on the bandwidth of the closed loop amplifier – f3dB = fT / G
Rough estimate – 5 to 10 with = G / 2 fT
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20
Time
Am
pli
tud
e
29
Stability
G(p) = A(p) G / (G + A(p))– A(p) has several poles
If G = A(p) when the phase shift is 180o then the denominator is null and the circuit is unstable
Simple criteria– On the Bode diagram G should cut A(p) with
a slope difference smaller than -12dB / octave – The loop gain A(p)/G should cut the 0dB axe
with a slope smaller than -12dB / octave Phase margin
– (1800 - Phase at the two previous points) The lower G the more problems
Unstable amplifier
- Open loop gain A(p)- Ideal gain G- Loop gain A(p)/G
-80
-60
-40
-20
0
20
40
60
80
100
120
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06
Frequency
Gai
n [
dB
]
-12 dB/octave
-12 dB/octave
30
Stability improvement
Move the first pole of the amplifier – Compensation
-80
-60
-40
-20
0
20
40
60
80
100
120
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06
Frequency
Ga
in [
dB
]
Compensation
-60
-40
-20
0
20
40
60
80
100
120
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06
Frequency
Ga
in [
dB
]
Pole in the loop
-6 dB/octave-6 dB/octave
Add a pole in the feed-back
These actions reduce the bandwidth
31
Capacitive load
The output impedance of the amplifier and the capacitive contribute to the formation of a second pole at low frequency– A’(p) = k A(p) 1/(1+r C p) with r = R0//R2//R
– A(p) = A0 / (p+0)
10
C = 20 pF
Buffering to drive lines
Capacitance in the feedback to compensate – Feedback at high frequency from the op-amp
– Feedback at low frequency from the load
– Typical values a few pF and a few Ohms series resistor
-
+
R1
R2
C Load = 0.5 F
32
Examples of data sheets (1)
33
Examples of data sheets (2)
34
Current feedback amplifiers
Voltage feedback
-
+
-A
-
+
Zt ie
ie
Current feedback Zt = Vout/Ie is called the transimpedance
gain of the amplifier
35
Applying Feedback
Non-inverting amplifier Same equations as the voltage feedback
Zt if1R
2R1R
Zt2R
1
11R
2R1R
VinVout
Ie Zt Vout
Ie 1RI)2R1R( Vout
1R)IeI(Vin-
Zt ie
ie
R1
Vin
I
R2
Vout
+
36
Frequency response
The bandwidth is not affected by the gain but only by R2– Gain and bandwidth can be defined independently
Different from the voltage feedback – f3dB = fT / G
-Zt ie
ie
R1
Vin
I
R2
Vout
+
0Z)p(2R
1
11R
2R1R
VinVout
pZ0
Zt
Zt2R
1
11R
2R1R
VinVout
37
Data sheet of a current feedback amplifier
38
Data sheet of a current feedback amplifier (cont’)
Very small change of bandwidth with gain
39
Transmission Lines
Lossless Transmission Lines Adaptation Reflection Transmission lines on PCB Lossy Transmission Lines
40
Lossless transmission lines (1)
L,C per unit length x Impedance of the line Z
Z
Lx
Cx
Lx
Cx
C
LZ
C
LZ;0x
0C
LpZLxZ
1pZCx
ZpLxZ
2
2
Pure resistance
41
Lossless transmission lines (2)
Propagation delay
Lx
Cx
I
ZV2V1
LC;)t()t(V)t(V
eVV;0x
)pxLC1(VV)cellsx
1(lengthunityAfter
)xpLC1(VZ
1VpLxVIpLxVV
12
pLC12
x
1
12
1112
Pure delay
42
Lossless transmission lines (3)
Characteristic impedance pure resistance C
LZ
LC
ZC
ZL
Example 1: coaxial cable– Z = 50
– = 5 ns/m
– L = 250 nH/m; C = 100 pF/m Example 2: twisted pair
– Z = 100
– = 6 ns/m
– L = 600 nH/m ; C = 60 pF/m
Pure delay
Capacitance and inductance per unit of length
43
Reflection (1)
All along the line Vs = Z0 Is
If the termination resistance is ZL a reflection wave is generated to compensate the excess or lack of current in ZL
V
Zs Zo
Z L
Source generator – V, Output impedance Zs
Line appears as Z0
The reflected wave has an amplitude
0S
0s
0S
s ZZ
ZVV;
ZZ
1VI
RsL
RsL
R0R
LLL
III
VVV
IZV
IZV
0L
0LsR ZZ
ZZVV
sRL
sRL
VV;0Z
VV;Z
IsVs
44
Reflection (2)
The reflected wave travels back to source and will also generate a reflected wave if the source impedance is different from Z0
– During each travel some amplitude is lost
The reflection process stops when equilibrium is reached– VS = VL
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25
Time
Vo
lt
V
Vs
VL
ZS = 1/3 Z0
ZL = 3 Z0
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Time
Vo
lt
V
Vs
VL
ZS = 3 Z0
ZL = 3 Z0
Zs < Z0 & ZL > Z0
Dumped oscillation
Zs > Z0 & ZL > Z0 Integration like
45
Reflection (3)
Adaptation is always better – At the destination: no
reflection at all
– At the source: 1 reflection dumped
» Ex. ZL = 3 Z0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Time
Vo
ltV
VS
VR
2 transit time
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Time
Vo
lts
V
VS
VL
1 transit time
Can be used to form signal– Clamping
V
Zs Zo
Vs
46
Transmission lines on PCB
Microstrip
Stripline
m/ns45.7feet/ns27.2t;inch/pf5C;53Z
35T;mm5.0W;mm8.0H;5:Example
feet/ns016.1tpd
inch/pF
TW8.0
H81.3ln
41.1C
TW8.0
TH29.1ln
60Z
pd00
r
r
r0
r0
m/ns80.5feet/ns77.1t;inch/pf4.1C;106Z
35T;mm5.0W;mm6.0H;5:Example
feet/ns67.0475.0016.1tpd
inch/pF
TW8.0
H98.5ln
41.167.0C
TW8.0
H98.5ln
41.1
87Z
pd00
r
r
r0
r0
47
Lossy transmission lines
Idem with RsL instead of L, Rp//C instead of CL
C Rp
Rs
Characteristic impedance depends on – Even Rs is a function of because of the skin effect
Signal is distorted Termination more complex to compensate cable characteristic
p
s
R
1Cp
LpRZ
48
Bibliography
The Art of Electronics, Horowitz and Hill, Cambridge – Very large covering
An Analog Electronics Companion, S. Hamilton, Cambridge– Includes a lot of Spice simulation exercises
Electronics manufacturers application notes– Available on the web
» (e.g. http://www.national.com/apnotes/apnotes_all_1.html)
For feedback systems and their stability– FEED-2002 from CERN Technical Training