Prof. Anyes Taffard. Physics 120/220. Operational Amplifier

Transcription

1 Prof. Anyes Taffard Physics 120/220 Operational Amplifier

2 Introduction Operational Amplifier are the most useful and important component of analog electronics. In many electronics circuits, the signal is very small. Thus amplification is essential (eg audio amp. from DVDà speakers). 2 An Op-Amp takes a low level voltage signal as inputs and produces large output voltage that vary linearly with input voltage. Primary limitation: not fast. Performance degrades for frequencies >1MHz Primarily use of Op-Amp in amplification circuits is related to the concepts of feedback ( + or - ): some of the output signal is feedback to the input.

3 Feedback concept 3 Positive feedback Feedback causes input to be enhanced Eg: mic speaker Waveform travels around the loop, becomes bigger each time. However, whistling noise, ie oscillation at the frequency where the gain is highest. Positive feedback - unstable Used for oscillators

4 Feedback concept (cont.) 4 Negative feedback Feedback causes input to diminish Results in: Less overall gain Better linearity Avoid oscillation Amplification nearly independent of f Negative feedback - stable Desirable for amplifier circuits

5 Operational-Amplifier: Op-Amp Op-Amp: Integrated Circuit (IC) eg: 411 Op-Amp contains 24 transistors on a single chip Note: PS to Op-Amp often omitted on circuit diagrams Count pins counter clockwise +V CC / V CC inverting input non-inverting input -V EE -V EE 8 pins DIP (Dual Inline Package) Some pins on chip are not used Op-Amp should always be powered The output goes more positive when the non-inverting input (+) goes more positive than the inverted input (-) & vice-versa. 411 model uses JFET transistors as input to achieve very large impedance (R in ~10 9 ) 741 is an all BJT design (R in ~10 6 ). Open loop gain is huge

6 Op-Amp Golden Rules Idealization Useful to describe overall performance. Applicable whenever Op-amp are configured with negative feedback 6 1 The output attempts to do whatever is necessary to make the voltage difference between the two inputs zero. ü Consequence of very high voltage gain. ü Note the output voltage cannot exceed the PS voltages. Clipping of the output voltage will occur. 2 The inputs draw no current. ü Consequence of the Op-Amp having a very high input impedance. Open loop behavior (ie no feedback) will swing to one of the PS rails ü If V + >V -, =+V CC ü If V + <V -, =-V EE like a comparator

7 Inverting amplifier Inverting: signal input comes to - input, thus has the opposite sign to output. 7 Negative feedback provided by R 2. V + =0 (connected to GND). According to rule #1, V - =0. For this reason the - input is said to be at a virtual GND ) The voltage across is V in -V - =V in & R 2 is -V - = According to rule #2 i - =0 (inputs draw no current) I R1 = I R2 I in = I out V A V in or = V A R 2 but V A =V - =0 V in = R 2 A v = V in = R 2 & R 2 define the gain The input impedance of the Op-Amp circuit is the impedance to GND from an input signal. Since - is a virtual GND, so Z in = Output impedance of Op-Amp is very small ( < 1 Ohm)

8 Non-inverting amplifier 8 Non-inverting: signal input connect to + Negative feedback via R 2. V + =V - thus V in =V A (rule #1) All current going thru R 2 also flows thru. Redrawing the /R 2 network, exploiting the fact that no current goes thru A to -, we get the voltage divider with: V A = + R 2 V in = 1+ R 2 Positive gain thus non-inverted Always > 1 Z in ~ as no current flows into + input (driven by Op-Amp input impedance)

9 Op-Amp gain (non-inverting case) 9 The wheel symbol indicates that negative feedback is being added to the input A: Open loop gain. It is the intrinsic gain, ie the gain in the absence of feedback to input. A is huge 10 5 to A varies with frequency. B: fraction of output feedback to input G: closed loop gain (ie gain of the device) AB: loop gain R B = R + R F G = V in Abstract schematic of negative feedback Let s derive the gain as a function of A & B, for the non-inverting case. The voltage at a : v a = v in Bv out The Op-Amp applies an open-loop gain A to produce v out : v out = Av a = Av in ABv out v out A = G = v in 1+ AB In the limit that A>>1 G = 1 B For the inverting-case: v out v in = G = ( ) 1+ AB A 1 B In the limit that A>>1 G = 1 1 B

10 Input & output impedance (non-inverting case) 10 Now we can calculate the effect that the closed-loop gain configuration has on the input & output impedance. Input impedance: since v out A = v in 1+ AB Z in = v in i in i in = v in v b R in = v in Bv out R in i in = v in 1 R in (1+ AB) Z in = R in (1+ AB) Output impedance: v open =v out (no external load) Z out = v open i short ie Z out =R TH Z in = R + R F 1+ A inverting case Schematic illustrating the input/ output impedance of negative feedback i short : what we get when load has zero input impedance. This means all the current from amplifier goes into the load, leaving none for the feedback loop. Thus B=0 i short = Av d v d = v in Bv out = v in i short = Av in R R 0 0 Z out = R 0 1+ AB valid also for the inverting case v out A = v in 1+ AB i short = 1+ AB R 0 Thus, the effect of the closed-loop circuit is to improve both the input and output impedances by the identical loop gain factor 1+AB AB.

11 Current to voltage converter An I to V converter will produce a voltage proportional to I. Eg Resistor V=RI 11 Using Kirchhoff s rule at A: γ Negative feedback: V - =V + =0 Rule #2: I - =0 I p + I = V R = I p R is proportional to the light intensity (I p changes w/ γ) Photodiode operates in reverse bias

12 Summing amplifier Circuit based on an inverting amplifier. Point X is a virtual GND 12 Kirchhoff s X I F = I 1 + I 2 + I 3 = V 1 + V 2 + V 3 R F R 2 R 3 = R F V 1 + R F R 2 V 2 + R F R 3 V 3 Ie the weighted sum of V i, where weight = R F /R i

13 Voltage follower 13 Negative feedback =V in Gain A v =+1 Called buffer : does not amplify or invert the input signal. Instead provides isolation between two circuits. Input impedance very high, while output impedance is low. The follower is a special case of non-inverting amplifier, where: >>R 2 A v =1+R 2 / =1

14 Integrator Recall inverting Op-Amp: 14 Z 2 Generalize this to impedance: Z 1 = R 2 V in = Z 2 Z 1 V in Recall the low-pass filter: f 3dB = 1 2π RC V in = 1 2 when Z c =R Recall, had to keep <<V in to make RC behave like an integrator

15 Integrator (cont.) Attempt to design an active filter: 15 = Z C Z R V in Z c = 1 jωc = 1 jωcr V in Input current V in /R goes through C because V - is virtual GND. I = C dv dt à V in R = C d dt à = 1 RC t V (t) in dt + cst At low frequency, /V in à instead of a constant gain (Z c large at low f).

16 Improved Integrator (cont.) Remedy to the problem of behavior at low f by adding a resistor in parallel to the cap in the feedback loop: 16 Z 2 Z 2 = Z R Z c Z R + Z c = R 2 jωc R jωc Z 2 = R 2 R 2 jωc +1 = Z 2 V in = R 2 1 Z 1 R 2 jωc +1 V in Vin -R 2 / as ωà0 (cst cool!) 1/ω as ωà Example of an active low-pass filter: depends on f

17 Differentiator 17 Recall: in order to make an RC differentiator work, we needed d dt << dv in dt Note: R & C swap with respect to integrator Since V GND, I thru C: I C = C dv in dt Using Kirchhoff s rule at A I C =I R, since I - =0 C dv in dt = V R = RC dv in dt Example of an active high-pass filter

18 Departure from ideal So far, we have assumed ideal behavior of Op-Amps. However, Op-Amp do not quite obey rule #1 & #2. The most important effects are: 18 1) Finite slew rate dv/dt ü Voltage of an Op-Amp can change no faster than the Op-Amp slew-rate. 2) Input bias current I B ü Rule #2 not perfectly satisfied: inputs actually draw non-zero current 3) Input offset voltage V OS ü Rule #1 not completely satisfied: instead of V + =V - at Op-Amp input we have ü Consequence: the circuit will have a constant DC bias of 2V OS V + = V + V OS Slew rate 15 V/µs 0.5 V/µs I B 50 pa pa V OS 0.8 mv (max 2V) 2 mv (max 6mV)

19 Comparator 19 A comparator is a chip that compare two analog voltages to determine which is more positive. Used in: Analog circuits, typically as a trigger Analog to digital conversion Recall the behavior of an Op-Amp if you do not provide any feedback loop: will swing to one of the PS rails ü =+V CC if V + >V - ü =-V EE if V + <V - Comparator is like an Op-Amp but intended to be used without the feedback loop. It will go in saturation (V CC or V EE ) according to the difference of input voltages. Ordinary Op-Amp can be used as comparator, but it is slow. Special IC (LM311) intended to be used as comparator. ~100x faster ü Peculiarity: open collector output - +

20 Examples 20 1) 0V: indicates if input voltage is + or =+V CC if V + >V - =-V EE if V + <V - 2) Using a different voltage as reference (eg +3V) Complications: Noisy input signals crosses threshold multiple times causing glitches

21 Schmitt trigger If we were able to specify a voltage range within which the state of the output can be at either values, then we could build immunity to the noise. Solution: Use a positive feedback V + is related to by the voltage divider: V + = + R F 21 V + depends on and depends on the difference between V + and V - Note: input signal applied to inverted terminal Inverting Schmitt trigger

22 Schmitt trigger 22 Key to analyze this circuit is to assume an initial condition. can be either V H or V L (V H >V L ) Assuming = V H open-loop gain R V + = V 1 R H = A V 1 H V in + R F + R F V + = + R F As long as V in < V H + R F = V TH, remains as V H When V in > V TH, transition occurs: = V L & V + = V L + R F In order for the transition to happen again, V in < V L + R F = V TL V L + R F V in has to change sign, which occurs at V H hysterisis = V TH V TL V TL V TH V in See examples Student Manual p V L

23 RC Oscillator By combining negative and positive feedback and a capacitor for energy storage, we can design a circuit that generates square wave output. 23 varies between V H and V L Assuming =V H V + = βv H β = + R F Thus, the capacitor voltage, V C, shall be charged exponentially towards V H, by a time constant R F C. Once V C > βv H, =V L V + = βv H The capacitor starts discharging exponentially towards V H, until V C = βv H Both charging and discharging occurs with the same time constant R F C. To determine frequency of oscillation, assume the cap ramps as if feed with a constant current V H /R F The resulting square wave has τ H =τ L The cap voltage alternates between βv H and +βv H The output voltage varies between V H and +V H

24 RC Oscillator (cont.) The general equation for V C is: V C (t) = A + Be t RC 24 We can determine A & B t=0, V C (t)= +βv t=τ H, V C (t)= -βv H τ H = τ L = R F C ln 1+ β 1 β β = + R F Thus the period: T = τ H + τ L = 2τ H

25 Backup 25

26 Photodiode 26

27 411: under the hood 27 Simplified schematic Detailed schematic

28 Example: Schmitt Trigger Design a Schmitt trigger with a 311 comparator with an output swing of 0 +5V. Set the hysteresis at ±0.1V and put the threshold close to 1V. 28

29 Example: Schmitt Trigger Design a Schmitt trigger with a 311 comparator with an output swing of 0 +5V. Set the hysteresis at ±0.1V and put the threshold close to 1V. β = R TH R TH + R F Hysteresis at ±0.1V means: V TH -V TL =0.1 Swing of 0 +5V means: V H =+5V V L =0V 29 Output high R V + = V TH = V TH H R TH + R F Output low V = V TL = V L R TH R TH + R F For the voltage divider V TH V TL = R TH R TH + R F ( V H V L ) = β ( V H V L ) β 1 = R TH + R F R TH = 50, R F = 49R TH β = V TH V TL V H V L = = 0.02 R 2 =1k =4k R TH =0.8k R F =39.2k~39k V + = V PS R 2 + R 2 R 2 + R 2 = V + V PS = 1 5, =4R 2 R TH = R 2 + R 2 = 4 5 R 2