Dr. Un-ki Yang Particle Physics Group

1. Amplifiers and Feedback 1 Dr. Un-ki Yang Particle Physics Group ukyang@hep.manchester.ac.uk or Shuster 5.15

2. Real Experiment • How can we catch cosmic particle and measure it’s energy?

3. Real Experiment Trigger cosmic ray scintillator coincidence integration Signal ADC X10 Amp.

4. 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: 3 lectures (2 hours per each) • Nov 10, Nov 17, and Nov 24 • Learning outcomes • To understand the behavior of an ideal amplifier under negative (positive) feedback • To be able to apply this to simple amplifier, summer, integrators, phase shifter, and oscillator • To understand the limitations of a real amplifier ( gain, bandwidth, and impedance) • To understand basic methods of analogue-to-digital conversion (ADC)

5. Lecture notes and references

6. Basic Circuit Theory • Kirchoff’s Laws • Conservation of energy: for a closed loop • Conservation of charge: net charge into a point (node) • 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

7. Dividers • Voltage Divider • Current Divider

8. AC Circuit • Z is a complex number  is a phase • Alternating current (AC) circuits: v(t), i(t) Consider v(t), i(t) with sinusoidal sources • Extension of Ohm’s law to AC circuits

9. 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, dv/dt = 1/C dq/dt = i/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)

10. RC Circuit with Sinusoidal Source • Resistive impedance: ZR=R, • same phase • Capacitive impedance: Zc = 1/jC, • -/2 phase • Inductive impedance: ZL = jL, • /2 phase

11. Capacitor i(t) V C -/2 phase Z() • Circuit with capacitor • In a DC circuit, inf it acts like an open circuit • The current leads the voltage by 90o

12. RC Low-Pass Filter R Vin Vout C

13. RC Low-pass filter R Vin Vout C • Low pas filter acts as an integrator at high frequency

14. RC High-pass filter • High pass filter acts as a differentiator at low frequency Vin Vout

15. RC circuits

16. Combined Impedance Vin Vout

17. Amplifiers • The amplification (gain) of a circuit • 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”

18. 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 )

19. OP Amplifier 741 +15V V+ Vout V- -15V Many interesting features about OP amplifier http://www.allaboutcircuits.com/vol_3/chpt_8/3.html

20. Negative Feedback • An overall gain G is independent of G0, but only depends on  • Stable gain

21. Non-inverting Amplifier • Golden rules: Infinite Gain Approximation (IGA) • Small v(=v+- v-): v+=v- • Small input currents: I+=I-=0 (large Zin)

22. Inverting Amplifier • Inverting Amplifier Golden rule: v+= v- (v- is at virtual ground) Calculate gain!

23. Differentiation • Differentiation circuit • Golden rule: v+= v- • (v- is at virtual ground) • Prove this is a differentiation circuit! • How would you configure to make an integration circuit?

24. Summer circuit • Summer Circuit v- is a virtual ground Prove that

25. Phase shifter • Golden rule: v+= v- • Calculate a phase shift