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Amplifiers and Feedback 1 Dr. Un-ki Yang Particle Physics Group [email protected] or Shuster 5.15 1 Real Experiment How can we catch cosmic particle and measure it’s energy? 2 Real Experiment Trigger cosmic ray scintillator coincidence integration Signal X10 Amp. ADC 3 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) 4 Lecture notes and references 5 Basic Circuit Theory 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 Kirchoff’s Laws • Conservation of energy: for a closed loop iVi 0 • Conservation of charge: net charge into a point (node) i Ii 0 6 Dividers Voltage Divider Current Divider 7 AC Circuit Alternating current (AC) circuits: v(t), i(t) Consider v(t), i(t) with sinusoidal sources v(t) V0 cos( t v ), i(t) I 0 cos( t I ) v(t) V0 e j ( t v ) , i(t) I 0 e j ( t I ) Extension of Ohm’s law to AC circuits v( ,t) Z( )i( ,t), Z is a generalized resistance: "impedance" Z is a complex number Z Z ei is a phase 8 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) 9 RC Circuit with Sinusoidal Source j t v(t) V0 e , i(t) I 0 e j t v(t) Ri(t) 0 Resistive impedance: ZR=R, • same phase Capacitive impedance: Zc = 1/jC, • -/2 phase Inductive impedance: ZL = jL, • /2 phase v(t) q(t) / C 0 v(t) L di(t) / dt 0 10 Capacitor Circuit with capacitor v V0 cos t V0 e j t v q/C V C v(t ) i(t ) / j C Z j / C In a DC circuit, inf it acts like an open circuit The current leads the voltage by 90o i(t) Z() -/2 phase 11 RC Low-Pass Filter R Vin C G Vout Vout 1 Vin 1 j RC 0 G( ) Glow 1 G( ) Ghigh Ghigh 1 j RC 1 1 RC 12 RC Low-pass filter Low pas filter acts as an integrator at high frequency R Vin C VIN (t) Ve j t 1 Ghigh jwRC Vout I R IC Vin Vout dVout IR , IC C R dt Vin Vout dV C out R dt if Vin ? Vout (low gain: high ) Vin dVout C R dt 1 Vout Vin dt RC 13 RC High-pass filter High pass filter acts as a differentiator at low frequency Vin R Vout Vin R 1 / j C j RC Vout Vin 1 j RC Vout j RC G Vin 1 j RC Vout 0 G( ) Glow j RC G( ) Ghigh 1 Vout d RC VIN at low frequency dt 14 RC circuits 0 Low-pass filter High-pass filter 1 1 jwRC high Vout 1 Vin dt RC low jwRC 1 Vout d RC VIN dt 15 Combined Impedance ZR Vin Z R Z L ZC R Vin R j( L 1 / C) R e j G R 2 ( L 1 / C)2 Vout Vin Vout 1 / C L R tan 1 1 / LC : same phase 16 Amplifiers The amplification (gain) of a circuit G = VOUT / VIN 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” 17 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 ) 18 OP Amplifier 741 +15V V+ V- Vout -15V Many interesting features about OP amplifier http://www.allaboutcircuits.com/vol_3/chpt_8/3.html 19 Negative Feedback V G v, VV=V IN VINV Vout VOUT OUT out G00V, OUT VOUT G0 VIN 1 GO 1 G , if G 0 An overall gain G is independent of G0, but only depends on Stable gain 1 20 Non-inverting Amplifier G VOUT 1 R1 R2 , VIN R1 if G 0 ? 1 R1 v VOUT R1 R2 VOUT G0 (v v ), VIN v VOUT G0 G VIN R1 1 R R G0 1 2 Golden rules: Infinite Gain Approximation (IGA) • Small v(=v+- v-): v+=v• Small input currents: I+=I-=0 (large Zin) 21 Inverting Amplifier Inverting Amplifier Golden rule: v+= v(v- is at virtual ground) Calculate gain! 22 Differentiation Differentiation circuit VIN (t) Ve j t Vin Vout Vin ZC R 1 / jwC Vout jwRCVin d Vout RC VIN dt Golden rule: v+= v(v- is at virtual ground) Prove this is a differentiation circuit! How would you configure to make an integration circuit? 23 Summer circuit Summer Circuit v- is a virtual ground Prove that V (V V OUT 1 2 ) 24 Phase shifter Golden rule: v+= v Calculate a phase shift 25