# OPAMPs I: The Ideal Case

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1 I: The Ideal Case The basic composition of an operational amplifier (OPAMP) includes a high gain differential amplifier, followed by a second high gain amplifier, followed by a unity gain, low impedance, output stage. A simplified model for the LF411 used in class is shown below: (+) = non-inverting input (-) = inverting input V out has the same sign as V (+) - V (-) (the capacitor limits the gain at high frequency, more on this later) To do calculations, we will use a small signal equivalent circuit for the OPAMP 1. Universal assumptions for : 1. Unloaded o =A D where D = (+) (-) A is large, typically r in is large 3. r o is small 0 Note: there are no explicit resistors from inputs to ground or power supplies. This is consistent w/ ignoring input bias currents and having a current source in the Diff. Amp. 1The small signal analysis will avoid imperfections of a D.C. nature, such as input bias current, input voltage offset, and bias current offset. We'll treat these later. 1

2 Negative feedback: Assume that D = (+) (-) 0. The idea is to take a fraction B of o and add it back into the input in order to cancel D. There are two topologies to consider: Voltage Shunt feedback We will derive the Golden Rules from our above assumptions 1. D 0 2. inputs draw no current, D 0 3. Z out 0 Proof of assertion (1) : D 0 Draw the input-output circuit: By superposition: r D = in Z f r o in [r in Z f r o ] A r in D r o Z f r in Use the approximations: r in, Z f and r o Z f Z D f Z in A 1 Z D f Z f D = in1 B where B= BA 1 Z f and D 0 as long as BA >> 1 is called the feedback ratio Proof of assertion (2): no current into inputs follows from (1) since D = D r in 0 2

3 Proof of assertion (3): Z out 0 Use Thevenin's theorem open circuit: o.c. A1 B o =A D = in BA 1 short circuit: s.c. o = 1 s.c. A r D for output shorted to ground D = 1 B in o Z out = o.c. o = r o s.c. o BA 1 therefore Z out 0 for BA 1 As a corollary to assertion (1) it is easy to see that the gain for the entire circuit, including feedback (the closed loop gain ) is: G= o = A d in in = A1 B Z f BA 1 for BA 1 Notice that G is essentially independent of the details of A, the open loop gain. Even if A is non-linear for large signals, e.g. Barn roof distortion, G is still constant. All that matters is that BA>>1. Through use of negative feedback we are trading extremely large gain A for moderate gain ~Zf/Zin with excellent linearity. Voltage Series feedback We will again derive the Golden Rules 1. D 0 2. inputs draw no current, D 0 3. Z out 0 Again proceeding by superposition: (-) = in D = in Z f r o r in [ Z f r o ] A D r in Z f r o r in 3

4 Proof of assertion (1): D 0 Use the approximations: r in, Z f and r o Z f in Z in D A 1 D Z f Z D = 1 1BA so for BA >> 1 D 0 Proof of assertion (2): D 0 in i in = D = 0 for BA 1 r in is effectively boot strapped so that its r in 1BAr in effective impedance r eff in =r in 1BA r in Proof of assertion (3): Z out 0 by Thevenin's theorem open circuit: o o.c. =A D = in A 1BA short circuit: no feedback to input signal, so Closed loop gain in this configuration is: G= o = A d = A in in 1BA 1 B =1 Z f D = in o s.c. = A in r o r o Z out = 0 for BA 1 1BA We have seen that the use of negative feedback improved the desirable characteristics of an amplifier in several ways: 1) increases input impedance 2) reduces output impedance 3) provides a more linear response The preceding analysis was for small signals, but it still holds if we can neglect the small imperfections of the OPAMP (small DC bias currents into the OPAMP, and small voltage offsets at the inputs). 4

5 Next we turn to an analysis of OPAMP circuits using the two most important golden rules (valid when negative feedback is used): I) The opamp's output attempts to do whatever is needed to make the voltage differences between the inputs equal zero (i.e. (-) = (+) ) II)The inputs to the opamp draw no current We will use these simple rules to derive the closed circuit gains for the shunt and series circuits above. We will now drop the small signal notation unless it is explicitly required. Voltage shunt feedback: By rule (I): V (-) =V (+) =0 By rule (II): I 1 = V in 0 =I Z f = 0 V out 1 Z f Therefore V out = V in Z f Voltage series feedback: By rule (I): V (-) =V (+) =V in By rule (II): I 1 = V in =I f = V out V in Z f V in 1 1 Z f =V out Z f therefore V out =V in 1 Z f Notice: these solutions are general. As long as there is a closed circuit providing negative feedback, the Golden Rules will apply. Subject to this condition arbitrary circuits can be 5

6 placed into the feedback loop. 6

7 Some applications of Analog adder Rule 1) V - =0 I 1 =V 1 / R 1 and I 2 =V 2 / R 2 Rule 2) I f =I 1 I 2 V out =V - I f R f = [ V R f R 1 V f R 2 1 R 2 ] Adds, but also inverts Differential Amplifier Rule 1) Rule 2) R V - =V + =V 2 2 R 1 R 2 I 1 = 1 R 3 V R 2 1 V 2 R 1 R I 1 =I 4 I 2 R V out =V - IR 4 =V 2 2 R 1 R 2 1 R 4 R 3 R 4 V R 1 3 = 1 R 2[ 2 R R 1 R4 R 4 R R ] V DIF G DIF R [ 2 R R 1 R4 R 4 R R ] G CM V CM Setting [ i.e. R 2 / R 1 =R 4 / R 3 R 2 R 1 R 2 1 R 4 R 3 = R 4 R 3] gives G CM =0 and G DIF = R 2 R 1 this requires precise resistors, difficult to achieve in practice 7

8 Logarithmic amplifier Used to compress a signal that varies over many orders of magnitude into a few volt range. To analyze this circuit: 1) Don't panic 2) Find Vout, first find VA, VB V A = V 1 BE V T ln I in 1 I S from Ebers Moll Eqn. = V ln T [ I in I 0 ] V B = V 1 BE V 2 BE = V T ln[ I 2 in I S ] 1 I S I 0 Assuming Q1 and Q2 are identical, IS cancels... 3) The gain for the second OPAMP is 1 R 3 R 2 then V out = V T ln I in I 0 [ 1 R 3 R 2] ln V in Note: since V T =kt /q the gain is temperature dependent. A solution is to put a temperature variable resistor in series w/ R2 dr S dt = R S. 8

9 9

10 V o = 1 B [V OS I 1 R 1 I 2 R B R 2 ] for OPAMPS II: Imperfections of Three D.C. imperfections are common to Offset voltage VOS: Vout = A(VD-VOS) Input bias current Ib: Ib = 1/2(I1+I2) Bias current offset IOS: IOS = (I1-I2) For the configuration shown: define R B =R f R' f V D =V (+) V (-) = I 1 R 1 BV out I 2 R B R 2 AB 1 = 1 B [V OSI b R B R 2 R I OS R 1 R 2 R B ] To measure VOS: Choose R B R 2 R 1 and R 1 R 2 R B as small as possible Also choose 1/B ~ 1000 so VOS = BVout To measure Ib, IOS: Choose R 1 R 2 so large that I n R n V OS. (usually 10Meg is good enough for a bipolar OPAMP). Also make R B 10 M for simplicity. Then i) I OS = 2 BV out /R 1 R 2 ii) Short across R1, then I b =BV out 1 2 I OS R 2 / R 2 10

11 dv o G I D I D =V dt C M C (+) V (-) g m C C C Slew rate max dv o dt = I s C C Max rate at which output voltage changes dv In order to achieve o Slew Rate there must be a large difference voltage, VDIF. dt This is not the region in which the amplifier operates when negative feedback is used. 11

12 OPAMPS III: Comparators, Oscillators, etc. The comparator is a non-linear circuit similar to an OPAMP: Large gain, ~high input Z, differential amp front end. Differences: No Miller Compensation capacitor, this means large gain even at high frequency. This device is not meant to be used with negative feedback since it would be unstable. Output impedance not necessarily small, sometime open collector out (a current source). V+ > V_ Vo = VH high level (usually VH, VL V+ < V_ Vo = VL low level are power supply rails ) Comparators produce a binary output (VH or VL) and are used to interface analog signals (eg from a sensor) to a digital system (like a computer). A simple circuit using the 311. Multiple transitions can be cured by hysteresis --- making the threshold dependent on Vout Suppose Vin<VThr and increasing True threshold = V + * =V Thr 1 BVB o where B= R 2 R 1 R 2 Suppose initially Vin<VThr and is increasing: V o =V H and V * + =V Thr BV H When Vin * * exceeds V +, V o V L and V in V + is now a larger difference than previously (ie positive feedback). Output will not switch back to V o =V H (which is less than V Thr BV H ) until V in V Thr BV L Make B(VH-VL) larger than noise to avoid multiple transitions 12

13 Relaxation oscillator square wave output Assume: VH = +VCC = +15V VL = -VCC = -15V Suppose at t=0, V-=Vcap=0, Vo=+15V Capacitor will charge towards +VCC w/ time constant RC. When V->BVCC, Vo goes to -VCC and the capacitor charges towards -VCC w/ time constant RC. When V-<BVCC, Vo goes to +VCC and the cycle repeats. B= R 2 R 1 R 2 The half period T1/2 is given by : V CC 1B1 e T 1/2 / RC =2 BV CC 1 B 1B =et 1/2/ RC Full period = 2 T1/2, T =2 RC ln 1B 1 B For a 311 the square wave is slightly asymmetric. Zout=pullup R when VO=VH and Zout~0 when Vo=VL. This it spends more time at Vo=VH 13

14 One-shot or Monostable The relaxation oscillator was astable (no stable state). The circuit below has one stable state to which it returns after spending a brief time in a quasi-stable state. VCC=-VEE=15V V _=V D 0.6V D 1,D 2 on Initially take VO=VCC then V Thr =V + V D = V CC V D 1R 3 / R 2 1R 3 / R 1 R 3 / R 2 When in V Thr then V O V EE D1,D2 shutoff, V_ changes towards VEE and V + = R 2 V R 2 R EE =BV EE 3. When V_<V+ then V O V CC and the device returns to its initial state. 14

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