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1 Electronics in High Energy Physics Introduction to electronics in HEP Operational Amplifiers (based on the lecture of P.Farthoaut at Cern)

1 Electronics in High Energy Physics Introduction to electronics in HEP Operational Amplifiers (based on the lecture of P.Farthoaut at Cern)

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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)

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