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

2. 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. 3 Feedback  Y is a source linked to X –Y =  x  Open loop –x =  e –y =  x –s =  y =  x  Closed loop e x y s       is the open loop gain   is the loop gain

4. 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 e x s    

5. 5 Operational amplifier  Gain A very large  Input impedance very high –I.e input current = 0  A(p) as shown - + -A  

6. 6 How does it work?  Direct gain calculation - + -A   R1 R2 Vin Vout I  Feed-back equation  Ideal Op-Amp

7. 7 Non-inverting amplifier  Input impedance - + R1 Vin I R2 Vout  Gain  Called a follower if R2 = 0

8. 8 Inverting amplifier  Gain - + R1 Vin I Vout R2  Input impedance  Gain error

9. 9 Summation  If Ri = R - + R1 V1 I Vout R Rn Vn I1 In  Transfer function

10. 10 Differentiation - + R1 V1 I1 Vout R2 I1 R1 R2 V2 I2

11. 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. 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 R1R2 - + Iin Vout  High resistor value with small ones

13. 13 Charge amplifier (1)  Requires a device to discharge the capacitor –Resistor in // –Switch - + I Vout C R

14. 14 Charge amplifier (2) - + I V1 C R C1 R1 V2 R2 C2 Input Charge In a few ns Output of the charge amplifier Very long time constant Shaping a few 10’s of ns

15. 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 Y Z X Y Z2 Z1

16. 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. 17 Non-ideal amplifier  Input Offset voltage Vd - + -A   Ib+ Ib- Vd Zd Zc Zout  Input bias currents Ib+ and Ib-  Limited gain  Input impedance  Output impedance  Common mode rejection  Noise  Bandwidth limitation & Stability

18. 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 - + R1 I Vout R2 Vd

19. 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 R3 Vout

20. 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- R3 Vout C

21. 21 Common mode rejection  Input voltage Vc/Fr (Vc common mode voltage)  Same effect as the offset voltage - + R1 I R2 Vout Vc/Fr  Non-inverting amplifier

22. 22 Gain limitation -A  - +  R1 R2 Vin Vout I  A is of the order of 10 5 –Error is very small

23. 23 Input Impedance  Zin = Zc+ // (Zd A / G) ~ Zc+ G= (R1+R2)/R1 Zd Zc- Zc+ - + R1 Vin Vout R2  Non-inverting amplifier

24. 24 Output impedance  Non-inverting amplifier R2 - + R1 Vout I0 + Iout Iout -A   Z0 I0

25. 25 Current drive limitation  Vout = R I = R L I L  The op-amp must deliver I + I L = Vout (1/R + 1/R L )  Limitation in current drive limits output swing - + R1 Vin I R2 Vout RL Maximum Output Swing R L *I max

26. 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) f 3db = f T /G fTfT

27. 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 2  Af  S (V/  s) = 0.3 f T (MHz); f T = frequency at which gain = 1

28. 28 Settling Time  Time necessary to have the output signal within accuracy –±x%  Depends on the bandwidth of the closed loop amplifier –f 3dB = f T / G  Rough estimate –5  to 10  with  = G / 2  f T

29. 29 Stability  G(p) = A(p) G / (G + A(p)) –A(p) has several poles  If G = A(p) when the phase shift is 180 o 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 –(180 0 - 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 -12 dB/octave

30. 30 Stability improvement  Move the first pole of the amplifier –Compensation Compensation Pole in the loop -6 dB/octave  Add a pole in the feed-back  These actions reduce the bandwidth

31. 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. 32 Examples of data sheets (1)

33. 33 Examples of data sheets (2)

34. 34 Current feedback amplifiers  Voltage feedback - + -A   - + Z t i e ieie  Current feedback  Z t = V out /I e is called the transimpedance gain of the amplifier

35. 35 Applying Feedback  Non-inverting amplifier  Same equations as the voltage feedback - Z t i e ieie R1 Vin I R2 Vout +

36. 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 –f 3dB = f T / G - Z t i e ieie R1 Vin I R2 Vout +

37. 37 Data sheet of a current feedback amplifier

38. 38 Data sheet of a current feedback amplifier (cont’)  Very small change of bandwidth with gain

39. 39 Transmission Lines  Lossless Transmission Lines  Adaptation  Reflection  Transmission lines on PCB  Lossy Transmission Lines

40. 40 Lossless transmission lines (1)  L,C per unit length x  Impedance of the line Z Z Lx Cx Lx Cx  Pure resistance

41. 41 Lossless transmission lines (2)  Propagation delay Lx Cx I Z V2 V1  Pure delay

42. 42 Lossless transmission lines (3)  Characteristic impedance pure resistance  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. 43 Reflection (1)  All along the line V s = Z 0 I s  If the termination resistance is Z L a reflection wave is generated to compensate the excess or lack of current in Z L V Zs Zo Z L  Source generator –V, Output impedance Z s  Line appears as Z 0  The reflected wave has an amplitude IsIs VsVs

44. 44 Reflection (2)  The reflected wave travels back to source and will also generate a reflected wave if the source impedance is different from Z 0 –During each travel some amplitude is lost  The reflection process stops when equilibrium is reached –V S = V L Z S = 1/3 Z 0 Z L = 3 Z 0 Z S = 3 Z 0 Z L = 3 Z 0  Z s Z 0 Dumped oscillation  Z s > Z 0 & Z L > Z 0 Integration like

45. 45 Reflection (3)  Adaptation is always better –At the destination: no reflection at all –At the source: 1 reflection dumped »Ex. Z L = 3 Z 0 2 transit time 1 transit time  Can be used to form signal –Clamping V Zs Zo VsVs

46. 46 Transmission lines on PCB  Microstrip  Stripline

47. 47 Lossy transmission lines  Idem with R s L instead of L, R p //C instead of C L C Rp Rs  Characteristic impedance depends on  –Even R s is a function of  because of the skin effect  Signal is distorted  Termination more complex to compensate cable characteristic

48. 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)http://www.national.com/apnotes/apnotes_all_1.html  For feedback systems and their stability –FEED-2002 from CERN Technical Training