1. Operational amplifiers Building blocks of servos

2. Operational amplifiers (Op-amps) Want perfect amplifier Infinite gain Infinite input impedance –will not load down source Zero output impedance –will drive anything Have operational amplifiers Gain ~ 10 6 Input impedance ~ 100 M  Output impedance ~ 100  Problems Gain too high –slightest input noise causes max output Other problems to be discussed later Solutions Use feedback Gain depends only on resistance: R f / R in –can control precisely Op amp with feedback V 1 = (V out - V in ) R in /(R in +R f ) + V in V out = -A V 1 V out = -V in (R f / R in ) / [1 + (1+ R f / R in )/A] Small gain: V out = -A V in R f / (R f + R in ) ~ divider Large gain: V out = -V in (R f / R in ) Note: V 1 = -V out / A ~ 0 R in - + Op amp V in V out RfRf Summing junction inverting non-inverting V1V1 IfIf I in

3. Differential amplifier Op amp output actually depends on voltage difference at two inputs V out = - (V in1 - V in2 ) (R f / R in ) Insensitivity to common voltage at both inputs = CMRR Real op amps have problems with unbalanced input impedance Solution: Add input resistor and pot. Op amp with built-in resistors = instrumentation amp. R in - + V in1 V out RfRf R in V in2 Non-ideal op-amp ~ R f Resistors to compensate for non-ideal op-amp Ideal op-amp circuit R in - + V in1 V out RfRf inverting non-inverting V1V1 V in2

4. Integrators Put capacitor in op amp feedback path V out = - V in (Z f / R in ) = - V in / (2  j  f C R in ) Similar to low pass filter in high frequency limit –except applies to low frequencies also –can show large gain near dc Recall V 1 ~ 0 forces I in = -I feedback –charge on capacitor is integral of I f –since Vout = Q/C f, V out is integral of I in Result is integrator –integration speed ~ 1 / R in C f Integrator R in - + V in V out CfCf V1V1 I in IfIf log(V out /V in ) log(  f ) Unity gain at f = 1 / 2  RC Gain response Phase shift log(  f ) Phase response -90 degrees 0 degrees Single-pole rolloff 6 dB/octave = 10 dB/decade RC 0 >60dB

5. Shunted integrator Limit dc gain Advantages: –dc input voltage no longer saturates op amp output –prevents servo runaway Dis-advantages –long term errors not well corrected by servo log(V out /V in ) log(  f ) Unity gain at f = 1 / 2  R in C f Gain response Phase shift log(  f ) Phase response -90 degrees 0 degrees Max gain = R f /R in at f < 1 / 2  R f C f 0 Shunted integrator R in - + V in V out CfCf V1V1 I in IfIf RfRf

6. Real op amp Op amp without feedback Acts like shunted integrator Stability condition: unity gain freq. before second pole otherwise feedback becomes positive –oscillation Real op amp - + V in V out log(V out /V in ) log(  f ) Unity gain Gain response Phase shift log(  f ) Phase response -90 degrees 0 degrees Max gain 0 -180 degrees Single pole 6 dB Double pole 12 dB

7. Integrator with lead High frequency gain has minimum value Purpose: Provides phase lead Can compensate for pole in servo Alternate circuits for lead Capacitor at input, inductor in feedback Overall positive phase –analog to faster than light propagation –output anticipates input Integrator with lead R in - + V in V out CfCf V1V1 I in IfIf RfRf log(V out /V in ) log(  f ) Gain response Phase shift log(  f ) Phase response -90 degrees 0 degrees Min gain 0 Single pole 6 dB

8. Summing amplifier High gain forces V 1 ~ 0 Feedback current must cancel input current Generalize to multiple inputs –V out = -R feedback  I in = -(R f / R in )  V in –Works because V 1 ~ 0 Summing amplifier –Op amp input is summing junction –Useful for combining multiple inputs - + R in1 V in1 V out RfRf Summing junction IfIf I in R in2 V in2 R in3 V in3 Summing amplifier V1V1 Op amp - + V out RfRf inverting non-inverting IfIf I in V1V1 R in V in

9. Drift-compensated integrator Real op amps have leakage current Can saturate integrator Compensate with dc current to summing junction Drift-compensated integrator R in - + V in V out CfCf V1V1 I in IfIf RcRc IcIc +V - V Integrator leakage compensation

10. Trans-impedance amplifiers Input is current source –model as voltage source with high impedance –I in = V source / Z source –V out = -Z feedback V source / Z source = -I in Z feedback Trans-impedance amplifier –current in, voltage out –gain expressed in Ohms V1V1 Trans-impedance amplifier - + V out RfRf inverting non-inverting IfIf I in Current source V1V1 I in V source Z source

11. Photodiode amplifier Photodiode like current source but with capacitor Input capacitor causes op amp gain to diverge at high freq. –Amplifies high freq noise –Oscillation Solution: –Shunt capacitor in feedback Photo-diode amplifier - + V out RfRf IfIf I pd V bias light Photo-diode equiv. circuit I pd C pd R pd Shunt capacitor CsCs log(V out /I in ) log(  f ) Gain response Unshunted Shunted

12. Integrator design tips Shunting switch with small value resistor Discharges capacitor –initialize integrator/ servo –simplify servo testing allow rest of circuit Resistors on power supply rails Limits current to saturated op amp –prevents burn out Capacitors on supply rails Reduces noise Note on capacitors High values –electrolytic, polar –leakage resistance Use low value in parallel Shunt switch on integrator R in - + V in V out CfCf V1V1 I in IfIf RsRs switch +15V -15V Filtered and limited supply rails