Electronics in High Energy Physics Introduction to electronics in HEP

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. y x m s e s d b Feedback • Y is a source linked to X • Y = m x • Open loop • x = d e • y = m x • s = s y = s d m x • Closed loop • m is the open loop gain • bm is the loop gain

4. x m s e s d b Interest of the feedback • In electronics • m is an amplifier • b is the feedback loop • d and s are input and output impedances • If m is large enough the gain is independent of the amplifier

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

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

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

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

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

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

11. C R - Iin Vout + Current-to-Voltage converter (1) • Vout = - R Iin • For high gain and high bandwidth, one has to take into account the parasitic capacitance

12. R1 R2 - r Iin Vout + 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 • High resistor value with small ones

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

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

15. C - Vin -A e e Vout + Y X X Y Z Z2 Z1 Miller effect • Charge amplifier • Vin = e • Vout = -A e • The capacitor sees a voltage (A+1) e • It behaves as if a capacitor (A+1)C was seen by the input • Miller’s theorem • Av = Vy / Vx • Two circuits are equivalent • Z1 = Z / (1 - Av) • Z2 = Z / (1-Av-1)

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

18. R2 I R1 - Vd Vout + 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 mV and DVd = 0.1 mV / month • Otherwise a few mV

19. R2 Ib- R1 - + Vout Ib+ R3 Input bias current (1) • (Vout) = R2 Ib- • (Vout) = - R3 (1-G) Ib+ • Error null forR3 = (R1//R2) if Ib+ = Ib-

20. C Ib- - Vout R3 + Ib+ 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 mA bipolar

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

22. R2 I R1 - -A e Vin e + Vout Gain limitation • A is of the order of 105 • Error is very small

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

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

25. Maximum Output Swing R2 RL*Imax I R1 - RL + Vout Vin RL 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

26. f3db= fT/G fT 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+w0) • G = (R1+R2)/R1 40 dB • Asymptotic plot • G < A G(p) = G • G > A G(p) = A(p)

27. Slew Rate • Limit of the rate at which the output can change • Typical values : a few V/ms • A sine wave of amplitude A and frequency f requires a slew rate of 2pAf • S (V/ms) = 0.3 fT (MHz); fT = frequency at which gain = 1

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 t to 10 t with t = G / 2 p fT

29. Stability Unstable amplifier • 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 -12 dB/octave -12 dB/octave - Open loop gain A(p) - Ideal gain G - Loop gain A(p)/G

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

31. R2 R1 - C = 20 pF 10 C Load = 0.5 mF + Capacitive load • Buffering to drive lines • 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+w0) • 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

32. Examples of data sheets (1)

33. Examples of data sheets (2)

34. - -A e e + Current feedback amplifiers • Voltage feedback - Zt ie ie + • Current feedback • Zt = Vout/Ie is called the transimpedance gain of the amplifier

35. R2 I R1 - Zt ie ie Vout + Vin Applying Feedback • Non-inverting amplifier • Same equations as the voltage feedback

36. R2 I R1 - Zt ie ie Vout + Vin 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

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

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

42. Lossless transmission lines (3) • Characteristic impedance pure resistance • Pure delay • Capacitance and inductance per unit of length • Example 1: coaxial cable • Z = 50 W • t = 5 ns/m • L = 250 nH/m; C = 100 pF/m • Example 2: twisted pair • Z = 100 W • t = 6 ns/m • L = 600 nH/m ; C = 60 pF/m

43. Zs Zo Vs Is Z L V Reflection (1) • Source generator • V, Output impedance Zs • 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 • Line appears as Z0 • The reflected wave has an amplitude

44. ZS = 1/3 Z0 ZL = 3 Z0 ZS = 3 Z0 ZL = 3 Z0 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 • Zs < Z0 & ZL > Z0Dumped oscillation • Zs > Z0 & ZL > Z0Integration like

45. Zs Zo V 1 transit time 2 transit time Vs Reflection (3) • Adaptation is always better • At the destination: no reflection at all • At the source: 1 reflection dumped • Ex. ZL = 3 Z0 • Can be used to form signal • Clamping

46. Transmission lines on PCB • Microstrip • Stripline

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

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