Physics Laboratory Report on

Electronic Circuits and Operational

Amplifiers

J. Shapiro and K. ShpundHebrew University of Jerusalem

Racah Institute for Physics

March 19th, 2007

AbstractWhat follows is a survey of basic electronic circuits involving opera-

tional amplifiers and their theory, succeeded by a description of an experi-ment performed as part of the third year undergraduate physics laboratorycourse. We analyze the plethora of circuits which had been employed andelaborate as for their

The experiment itself consisted of connecting various circuits and mea-suring their transfer function. We present the results of the measurementsand our (perhaps trivial) conclusions.

1 Introduction

Operational amplifiers, sometimes referred to as ’op-amps’, are high-gain elec-tronic voltage amplifiers, with differential input and usually a single output.These components can be found in abundance in any consumer electronic de-vice out there nowadays, which is what makes them so important. Since theyare so common, they’ve become very cheap, going for well under $1. In thosedevices, they serve to amplify signals, apply mathematical operations, reduce”‘noise”’, and oscillate. They mainly consist of passive elements such as tran-sistors, capacitors and resistors, which enhance their linearity in a plethora ofphysical situations, such as a wide range of temperatures, frequencies, and soon. However, they are after all only physical and thus they too have cut-offsand saturations.

The basic function of the op-amp is to subtract between the two input volt-ages (denoted by V+ and V−) and direct the result of this operation into theoutput voltage (denoted by VOut), multiplied by a gain. This gain would becalled the open loop gain, and denoted by GOpenLoop. Thus, the the basicfunction of the amplifier can by described with the following equation:

VOut = (V+ − V−)×GOpenLoop (1)

1

1.1 Saturation

The op-amp also has DC power inputs, which enable it to amplify. These aredenoted by Vcc+ and Vcc−. So naturally if VOut > Vcc then the output will beVcc (due to conservation of energy) and this state will be called saturation.

1.2 DC Offset

DC Offset is the voltage that is produced when there is no input to the amplifier.It is also, more generally, the average in time, of the voltage of a system. So, forinstance, if a voltage generator was to produce a sine-signal, then the averageof that sine (which is zero for V = V0 · sin(t)) would be the DC Offset.

1.3 Ideal Op-Amp

The ideal op-amp has the following properties:

• GOpenLoop →∞.

• The impedance for the input →∞.

• The impedance for the output → 0.

• DC Offset → 0.

• The frequency range in which the op-amp operates with the same transferfunction is wide.

The op-amp we used in our experiment, the µA741, is of course not ideal. It is,however, a good approximation for an ideal op-amp.

1.4 Feedback

Generally speaking, feedback systems are systems in which information (or volt-age, for that matter) from the output is directed back into the input. In suchmanners it is possible to either stablize or destablize a system (with either neg-ative or positive feedback).

1.4.1 Negative Feedback

A system that works on the principle of negative feedback is one that directsinformation from its output to its inhibitory input. That is, as the systemgets out of equilibrium, it inhibts itself back into steady-state. In op-amps, thatmeans connecting VOut back into V−. Bearing (1) in mind, we get a relationshipbetween VOut and V+, and by connecting the feedback, we decrease the differencebetween V+ and V−. Specifically, the ratio VOut/V+ is called the closed loopgain sometimes. Connecting a negative feedback enables us to remove unwantedsignals from the system, maintain its linearity, and keep it stable. There is also

2

Figure 1: Diagram[5] of the negative feedback scheme.

Figure 2: High pass (bottom) and low pass (top) filters with their correspondingtransfer functions [4].

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another perspective on this issue. Use Figure 1. In here, from (1) it followsthat:

A(U − βY ) = Y

And thus Y equals:

Y = UA

1 + Aβ

From here the gain of the system is G = A1+Aβ .

1.4.2 Positive Feedback

In the same manner, instead of connecting the output to V−, we connect it toV+, which means we increase the difference between V+ and V−. By increasingthe difference, we attenuate the stability of the system. For more about whenthis feedback becomes oscillatory, see the section about Wein’s oscillator.

1.5 Filters

Filters are electronic components consisting of resistors and capacitors whichallow the filtering of certain frequencies. If we were to divide the filters coarsleyinto two different categories, we would have ”high pass filters” and ”low passfilters”. Not very surprisingly, high pass filters filter out the low frequencies,whereas the low pass filters filter out the high frequencies. The frequency atwhich cut-off for the filtering starts is given by: ω0 = 1

RC .

3

1.6 Analysis of Various Circuit Configurations

1.6.1 Inverter Circuit

Take a look at Figure 6. It depicts the inverter circuit. Let us analyze itsfunction with the basic laws of electronics - Kirchoff’s - and (1). This circuitis connected with a negative feedback. Since the op-amp is said to be approxi-mately ideal, its GOpenLoop approaches infinity, and thus, for the output to befinite, the difference between the two inputs should be approaching zero. Thus,if one of the inputs is at ground, then the voltage on the other one shouldapproach zero as well. Thus, as depicted in Figure 6, V− is connected to theground. So V+ should also be approaximately zero. If we try to calculate thecurrent that is flowing near the input of V+ (using the law that I = V/R), thenwe can apply two equations:

VOut − 0R1

= I =0− VIn

R2

and from there arrive to the fact that:

VOut = −R1

R2VIn

The result of this is that VOut is proportional to VIn, but with an oppositephase. This is why this circuit configuration is called ”The Inverter Circuit”- because it inverts the phase of the input. Also notice that now, instead ofhaving infinite gain (that of GOpenLoop) we have a finite gain, which depends onR1 and R2, or more acuurately, on their ratio. Thus we can pick such R1 andR2 to allow any gain we desire, assuming the op-amp still operates linearly insuch conditions1.

1.6.2 Non-Inverter Circuit

The non-inverter circuit is depicted in Figrue 5. Again, we apply the same”‘golden rule”’ as before, that V+ = V− to calculate the current at V−:

VOut − VIn

R1= I =

VIn − 0R2

VOut =(

1 +R1

R2

)VIn

Notice that once again, VOut is linearly proportional to VIn, however, now withthe same phase, which is why it is called ”‘Non-Inverter Circuit”’.

1In fact, in our experiment, which doesn’t put the op-amp under such extreme conditions,we found that we did have to pick such resistors to make the circuits work as we desired.Our instructor explained that as anomalies, but it is evident that the µA741 op-amp does notalways operate as expected

4

1.6.3 Follower Circuit

The voltage follower circuit, depicted in Figure 4, is mostly used as a buffer.Because the op-amp’s input impedance is so high it can be used to take theinformation about what the voltage is, without having to draw any energy fromthe circuit, effectively being isolated from the rest of the circuit. Ideally, sincethe input impedance is very high, there is no current going into V+, and thuswe can assume that V+ is almost VIn, which means V− is almost VIn. And ifR1 is picked to be very small, then VOut ≈ VIn.

1.6.4 Integrator Circuit

As seen on Figure 7, the integrator circuit introduces a new component whichwe haved used before - the capacitor. Apart from that, its analysis is identicalto that of the Inverter circuit, where R1 = ZC and R2 = R. Thus we get:

VOut = −ZC

RVIn

As is known, the impedance of the capacitor is ZC = 1iωC , and so we arrive at:

VOut (ω, t) = − 1iωRC

VIn (ω, t) (2)

But unlike in the case of two resistors, when we have a capacitor, i - the imag-inary number, is suddenly brought into the equations. Because of the natureof the Fourier transforms, from the angular velocity domain into the time do-main, we get effective integration. Moving (2) to the time domain (from thephase domain) is done relatively straight forward. First we define the Fouriertransform:

Fv(t) ≡ 1√2π

∫ +∞

−∞v(t)e−iωtdt ≡ V (ω) (3)

F−1V (ω) ≡ 1√2π

∫ +∞

−∞V (ω)eiωtdω ≡ v(t) (4)

Then we recall how to integrate by parts:∫fdg = fg −

∫dfg (5)

Now suppose that our function for Fourier transform is v(t) =∫

f(t)dt. Puttyingit into (3) results in:

Fv(t) =1√2π

∫ +∞

−∞v(t)e−iωtdt

Re-arranging to be more adequate:

1√2π

∫ +∞

−∞v(t)d

(e−iωt

−iω

)

5

To that we apply (5) and get:

1√2π

[v(t)

(e−iωt

−iω

)]+∞

−∞−

∫ +∞

−∞d(v(t))

e−iωt

−iω

The first part is zero since we assume v(t) is zero at infinity and we are leftwith:

− 1√2π

∫ +∞

−∞d(v(t))

e−iωt

−iω

Which is, by definition:

− 1√2π

∫ +∞

−∞f(t)

e−iωt

−iω

Or, more elegantly:F (ω)iω

The conclusion for the reverse Fourier transform holds:

F−1

F (ω)iω

=

∫F−1 F (ω) dt (6)

Applying (6) to our circuit, given by (2), we arrive at:

vOut (ω, t) = − 1RC

∫vIn (ω, t) dt

And finally we arrived at the fact that the output is indeed proportional to theintegral of the input voltage (with a phase difference as well)2.

1.7 Nyquist Criterion

A criterion that is useful in assessing the stability of a system with negativefeedback. It identifies points of singularity which indicate instability in thefeedback system.

1.7.1 Nyquist Plot

This plot depicts the gain of the circuit and its phase difference, presented in thecomplex plane. The gain is the absolute length of the vector (the norma) andthe phase difference is the angle of the vector with respect to the real positiveaxis - Gain = <c, Phase = =c. Therefore we could get the information ofthe Bode plots in one plot, that of Nyquist. The information on this plot tellsus if the aforementioned circuit is stable, or not. The circuit will be unstable ifthe curve encircles the coordinate of (1, 0).

2One might think it is redundant to include this full explanation in a physics lab report.However, we felt that too many people throw off pompous terms such as ”‘Integrator”’ withoutactually realizing why it is true that these circuits integrate. It is for those readers that weincluded this, perhaps extranouse, elaboration.

6

1.8 Wein’s Oscillator

An oscillator is a circuit that is designed to be unstable in the first place, thatis, intentionally unstable. The characteristic equation of the gain is:

AGain =VOut

VIn=

A

1 + A · β(7)

Thus one can see a singularity when the denominator - A · β approaches −1.This is the basic instability of which the outcome is that the gain goes to infinity.Nevertheless, the circuitry is bounded by the energy supplier and it amplifiesthe ”null signal” of the feedback unit of the circuit. When the circuit reachesthe boundary of the voltage supplier there are a couple of scenarios that cantake place. In our case the circuitry remains linear until it reaches the voltageboundary, in which stage it changes its phase direction and a sinusoidal signalis created.

1.8.1 Analysis of the Circuit

The circuit consists of a resistor and capacitor in serially connected, parallel tothose is another set of a resistor and a capacitor connected in parallel themselves.This connection functions as a high pass filter and a low pass filter. We willdivide the circuit into two sections, one of which is the filter’s circuitry (β). Thesecond is the resistors’ feedback - A. The null signal of the circuit is reached byconnecting the output to the inverting and non-invrting modes, which leads toa sinusoidal signal. To elaborate on (7), we’ll present here the values of A andβ:

A = 1 +R2

R1

β =ZParallel

ZSerial + ZParallel

ZSerial = R4 +1

iωC4

ZParallel =1

1R3

+ iωC1

Where Ri, Ci, are the ones shown in Figure 8. It can easily be shown that ifwe require that β ∈ < then we arrive at β = 1/3. This requirement is employedbecause the band-pass filters have a cut-off angular frequency of ω = 1/(RC).If we require this frequency to be the same for the two filters in our circuit, thenthe notion that β is real follows directly. From here, it is trivial that for Aβ tobe equal to −1 (so that there will be a singular point), A is required to be equalto 3. Thus, the ratio between the resistors R2/R1 must equal 2.

In our circuit we added two diodes that helped us sustain a reasonable am-plitude without getting to the saturation limit.

7

Figure 3: An image taken off the web [3] which depicts a 741 operational am-plifier inside a circuit-matrix - just like in our experiment.

1.8.2 The Diodes

The diodes are added in order to prevent saturation and bring the oscillatorback into equilibrium whenever it diverges away from its characteristic voltage.

2 Apparatus

Our instruments included an array of devices, both for measurements and forassembly of the circuits:

1. TTi TG 230 2Mhz Sweep/Function Generator

2. Tektronix TDS 210 Oscilloscope

3. Digital Multimeter DT9208A Potential-Meter.

4. A circuit-matrix of an unidentified manufacturer (See Figure 3).

5. A 741 operational Amplifier.

6. Generic inductors, capacitors, resistors, and conducting wires.

3 Method

The experiment was divided into sections, each dealing with a different type ofcircuit involving the operational amplifier. In each section, that is, for each typeof circuit, we would first connect all the components of the circuit according to a

8

Figure 4: A diagram[6] of the voltage-follower circuit.

Figure 5: A diagram of the non-inverter circuit drawn by hand.

diagram, check with the oscilloscope that we are indeed observing the behaviorwe have expected to observe, and then systematically measure the voltages andfrequencies of different places on the circuit. Below is a list of these circuits:

1. Voltage follower (Figure 4).

2. Inverter (Figure 6).

3. Non-inverter (Figure 5).

4. Itegrator (Figure 7).

5. Wein’s Oscillator (Figure 8).

• Without diodes.

• With diodes (IN4004 MiC).

• With a power source.

• Driven-circuit for a Nyquist plot.

• With a lower-than-required power source.

9

Figure 6: A diagram of the inverter circuit drawn by hand.

Figure 7: A diagram of the integrator circuit drawn by hand.

Figure 8: A diagram of the Wein oscillator circuit drawn by hand.

10

Table 1: Raw Data for Voltage Follower.

f (kHz) VIn (mV ) VOut (mV ) ∆T (µsec) T (µsec)0.2 616 616 1 5.00E+030.5 616 616 0 2.00E+03

1 616 616 0 1.01E+035.01 640 640 2 2.00E+02

206.6 632 632 0 2.20E+00460 632 616 2.80E-01 2.20E+00

509.1 632 592 3.20E-01 2.00E+00604 632 548 3.60E-01 1.68E+00702 628 508 3.80E-01 1.42E+00803 628 452 4.00E-01 1.24E+00929 616 396 3.80E-01 1.08E+00

1.49E+03 600 230 3.00E-01 6.70E-012000 616 170 2.60E-01 5.00E-01

4 Results

4.1 Voltage Follower

For this circuit, we took wires without any resistors, hoping that the impedanceof them would approach zero. We tried to measure the impedance with apotentio-meter and it wasn’t nice enough to show anything beside zero. ABode plot of the results from this circuit is presented in Figures 9 and 10. Asshown in those figures, it can be seen that the voltage remains constant (mean-ing a gain of 1), until the cutoff frequency. Beyond this frequency, the gainattenuates. The slope of the cut-off line is approximately −7.59 dB / decade- way off than what the theory predicts. The phase remains infinitesimal untilwe arrive at the same cut-off frequency, upon which the phase difference beginsto increase. This cut-off frequency is calculated to be around 460kHz.

4.2 Voltage Inverter

A Bode plot of the results from this circuit is presented in Figures 11 and 12. Thecut off slope this time, is just a little better, still way off, at −8.519 dB / decade.In this circuit, we wanted the gain to be 10, thus we tried to pick resistors withratio approaching to that number. We measured R1 = 0.985kΩ± 0.002kΩ andR2 = 9.94kΩ± 0.1kΩ. Thus, the expected gain was approximately 10 - which iswhat indeed received, as the results show. The gain is 20dB before the cut off- which is approximately 28.5kHz. The phase blot shows a lot of variance.

11

Figure 9: Magnitude Bode plot of the voltage follower circuit.

1 0 0 0 0 0 1 0 0 0 0 0 0 1 E 7

- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

D a t a : D a t a 1 _ CM o d e l : l i n e a r _ f o r _ b o d eW e i g h t i n g : y N o w e i g h t i n g C h i ^ 2 / D o F = 0 . 5 3 1 4 7R ^ 2 = 0 . 9 7 0 8 9y = a * l o g ( x ) + b a - 7 . 5 8 9 9 7 0 . 5 3 6 5 8b 1 1 3 . 6 1 5 8 9 8 . 3 0 1 9 7

Gain

(dB)

A n g u l a r F r e q u e n c y ( r a d / s e c )

Figure 10: Phase Bode plot of the voltage follower circuit.

1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0

0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

Phas

e (De

grees

)

F r e q u e n c y ( r a d / s e c )

12

Table 2: Raw Data for Inverter Circuit.

f1 (kHz) f2 (kHz) VIn (V ) VOut (V ) ∆T (µsec) T (µsec)0.204 0.204 1 9.8 4.90E+03 2.60E+03

2 2 1 9.8 500 25510.1 10.1 1.08 10.3 99 52

14.79 14.79 1.04 10.2 70 3620.2 20.2 1.04 10.2 50.5 26.520.9 20.8 1.06 10 48.5 26

21.88 21.86 1.06 10 46 2522.85 22.83 1.04 10 43.5 2623.8 23.9 1.06 10 40 22.5

25 25 1.04 9.9 40 22.528.5 28.5 1.04 9.7 35 20

30 30 1.04 9.6 34 2033 30 1.04 9.4 30 18

36.4 36.4 1.06 9.2 28 1737.5 37.5 1.06 8.9 26 16

40 40 1.06 8.7 24.5 1542.3 42.3 1.04 8.4 23.5 1450.4 50.4 1.04 7.6 20 1355.3 55.3 1.06 7.1 18 11.560.1 60.1 1.06 6.6 16.5 11.565.3 65.3 1.04 6 15 1075.1 75.1 1.04 5.4 13 987.9 87.9 1.04 4.8 11 8100 100 1.04 4.3 10.2 7504 504 1.03 0.79 1.95 1.7

13

Figure 11: Magnitude Bode plot of the voltage inverter circuit.

1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0- 5

0

5

1 0

1 5

2 0

2 5

D a t a : D a t a 1 _ CM o d e l : l i n e a r _ f o r _ b o d eW e i g h t i n g : y N o w e i g h t i n g C h i ^ 2 / D o F = 0 . 1 4 2 2 4R ^ 2 = 0 . 9 9 6 9 5 a - 8 . 5 1 9 3 1 0 . 1 9 2 2 6b 1 2 5 . 4 7 8 6 9 2 . 5 4 5 9 7

Gain

(dB)

A n g u l a r F r e q u e n c y ( r a d / s e c )

Figure 12: Phase Bode plot of the voltage inverter circuit.

1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0

1 8 0

2 0 0

2 2 0

2 4 0

2 6 0

2 8 0

3 0 0

3 2 0

Phas

e (De

grees

)

F r e q u e n c y ( r a d / s e c )

14

Table 3: Raw Data for Non-Inverter Circuit.

f (kHz) VIn (V ) VOut (V ) ∆T (µsec) T (µsec)0.099 1.02 10.2 1.00E+04 400

0.5 1.02 10.2 2.00E+03 200.9 1.02 10.2 1.00E+03 0

10.08 1.06 10.6 1.00E+02 220 1.04 10.4 4.95E+01 1.5

25.1 1.05 10.5 3.94E+01 226.1 1.04 10.4 3.84E+01 3.2

28 1.04 10.2 3.64E+01 3.230 1.04 10 3.32E+01 2.4

31.9 1.04 9.8 3.16E+01 2.835.8 1.04 9.4 2.80E+01 3.240.2 1.16 9.6 2.64E+01 3.545.1 1.18 9 2.20E+01 450.4 1.18 8.2 2.02E+01 3.654.9 1.19 7.6 1.80E+01 3.959.4 1.18 7 1.66E+01 3.465.1 1.18 6.6 1.52E+01 3

71 1.16 6.2 1.44E+01 3.275.3 1.18 5.8 1.40E+01 3100 1.18 4.16 1.00E+01 2.4130 1.16 3.2219 1.16 2427 1.16 0.99860 1.15 0.5

1.75E+03 1.15 0.195

15

Figure 13: Magnitude Bode plot of the voltage non-inverter circuit.

1 0 0 0 0 0 1 0 0 0 0 0 0 1 E 7- 2 0

- 1 5

- 1 0

- 5

0

5

1 0

1 5

2 0

2 5

D a t a : D a t a 1 _ CM o d e l : l i n e a r _ f o r _ b o d eW e i g h t i n g : y N o w e i g h t i n g C h i ^ 2 / D o F = 0 . 3 0 3 9R ^ 2 = 0 . 9 9 7 3 9y = a * l o g ( x ) + b a - 8 . 8 0 5 4 6 0 . 1 3 0 1 5b 1 2 8 . 5 7 0 6 8 1 . 7 7 1 1 7

Gain

(dB)

F r e q u e n c y ( r a d / s e c )

4.3 Voltage Non-Inverter

A Bode plot of the results from this circuit is presented in Figures 13 and14. Again, the cut-off slope is approximately −8.8 dB / decade. And in thiscircuit we wanted the gain to be 10 as well, and as the graph shows, the gainis 20dB - which is indeed what we wanted. We measured R1 = 97kΩ ± 2kΩ,R2 = 10.8kΩ± 0.2kΩ. R1/R2 + 1 = 8.90 + 1 = 9.9 ≈ 10. The cut-off frequencyfor this circuit is around 33kHz.

4.4 Integrator

In the integrator circuit, beyond the Bode plots, we also had preliminary testing,which involved inputting a sinus voltage, and getting a cosinus in the output.Another input was a straight line, which became a linear slope in the output,or a saw, which became a parabola - indeed we got integration. A Bode plotof the results from this circuit is presented in Figures 15 and 16. We measuredR = 1.97kΩ± 0.01kΩ and C = 109nF ± 2nF .

4.5 Wein’s Oscillator

For Wein’s oscillator, we measured the following: R1 = (1.97 ± 0.1)kΩ, R2 =(5.02 ± 0.1)kΩ, R4 = (502 ± 1)Ω, R3 = (504 ± 2)Ω, C1 = (85.8 ± 1)nF , C2 =(86± 5)nF .

16

Figure 14: Phase Bode plot of the voltage non-inverter circuit.

1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

2 0

4 0

6 0

8 0

1 0 0

Phas

e (De

grees

)

F r e q u e n c y ( r a d / s e c )

Table 4: Raw Data for Integrator Circuit.

f (kHz) VIn (mV ) VOut (mV ) ∆T (msec) T (msec)0.5 6.00E+02 2.60E+03 2 1.48

1 6.00E+02 1.46E+03 1 0.751.99 6.00E+02 8.40E+02 0.5 0.672.49 6.00E+02 7.60E+02 0.4 0.33.03 6.00E+02 6.40E+02 0.38 0.2443.48 6.24E+02 6.00E+02 0.288 0.212

17

Figure 15: Magnitude Bode plot of the voltage integrator circuit.

1 0 0 1 0 0 0 1 0 0 0 0- 2

0

2

4

6

8

1 0

1 2

1 4Ga

in (dB

)

A n g u l a r F r e q u e n c y ( r a d / s e c )

Figure 16: Phase Bode plot of the voltage integrator circuit.

1 0 0 1 0 0 0 1 0 0 0 02 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

Phas

e (De

grees

)

A n g u l a r F r e q u e n c y ( r a d / s e c )

18

4.5.1 No Diodes

When there were no diodes, we measured VA = 11.2V ± 0.5V (VA as shown inFigure 8), fA = 3.05kHz±0.1kHz, VB = 29.4V ±0.5V , fB = 3.06kHz±0.1kHz.Both measurements were sinusoidal, although a little bit distorted, we assumedthat was due to the lack of diodes.

4.5.2 With Diodes

Then we added the diodes and measured: VA = 496mV ±8mV , fA = 3.73kHz±0.2kHz, VB = 1.44V±0.2V , fB = 3.72kHz±0.2kHz. Now the sinusoidal signalswere beautifully shaped. We denote this particular VA as VA∗ ≈ 496mV

4.5.3 External VA and fA

Now we input the values of VA and fA into the circuit, instead of the positivefeedback. We noticed indeed that the circuit still acts as an oscillator (bymeasuring VB and fB - although we have no empirical way to prove this),depicting sinusoidal signals.

4.5.4 Forcing different VA to build a Nyquist plot

Now we forced the circuit with voltages below different than VA∗ ≈ 496mVand measured the resulting VB and fB . The resulting Nyquist plot is presentedin Figure 17. We can see that the curve does not encircle the point (1, 0) aswe would expect from an oscillatory circuit, which would consequent a singularpoint.

4.5.5 Inputting VA below VA∗

Now we inputted into the circuit voltage that is below that of its null signalvoltage. Bode plots of this stage are shown in Figures 18 and 19.

5 Discussion

The biggest problem we had with the experiment is that we have received aconsistent (among the various circuits) cut-off slopes of below −20 db/Decade.We calculated around −8.5 db/Decade. What is interesting about this result,is that it is consistent among different circuits and experiment sessions we had.This could be the consequence of either two things: a) Our fitting program ormethod is mistaken. b) We had a systematic error in executing the experiment.We prefer the former option, because in order to try and test our hypothesisthat the fitting program has gone bananas, we took two points from the inverterBode plot and calculated just these two points’ slope (the two points were theextreme ones). The slope we received was about −21, which substantiates ourhypothesis.

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Table 5: Raw Data for Wein’s Bridge Circuit with External Output for NyquistPlot.

f (kHz) VA (mV ) VB (mV ) ∆T (µsec) T (µsec)0.2 592 108 5.10E+03 1.10E+030.4 592 204 2.48E+03 2.00E+030.6 592 280 1.60E+03 1.30E+03

1 592 396 1.00E+03 8.40E+021.508 592 476 6.60E+02 5.70E+022.016 592 504 5.00E+02 4.50E+022.304 592 516 4.30E+02 3.90E+022.59 592 520 3.80E+02 3.60E+02

3 616 548 3.36E+02 3.12E+023.2 616 552 3.12E+02 2.96E+02

3.397 616 552 2.96E+02 2.80E+023.684 616 544 2.72E+02 2.64E+02

4 616 544 2.52E+02 2.44E+024.2 608 538 2.40E+02 2.36E+02

4.45 608 538 2.28E+02 2.20E+024.82 616 538 2.38E+02 2.20E+024.99 616 528 2.00E+02 2.04E+02

5.507 616 524 1.84E+02 1.92E+026.053 616 520 1.68E+02 1.76E+026.53 616 512 1.56E+02 1.64E+02

7 616 508 1.44E+02 1.54E+027.5 616 500 1.32E+02 1.44E+02

8.078 616 490 1.24E+02 1.36E+028.51 616 484 1.20E+02 1.32E+02

9.025 616 472 1.12E+02 1.24E+029.51 616 468 1.06E+02 1.18E+02

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Table 6: Raw Data for Wein’s Bridge with Forced Below VA∗ Voltage Input.

fA (kHz) VA (mV ) VB (mV ) ∆T (µsec) T (µsec)1.1 260 188 1.00E+03 8401.5 260 240 670 600

2 260 268 510 4702.5 260 284 400 380

3.01 274 304 332 3203.51 274 310 284 2803.71 274 308 278 268

4.045 272 310 248 2484.51 272 308 220 2245.01 272 308 200 2046.12 272 300 172 1807.12 272 294 140 152

8.091 272 284 122 132

Figure 17: Nyquist plot of the Wein oscillator circuit we assembled.

0 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0

0

3 0

6 09 0

1 2 0

1 5 0

1 8 0

2 1 0

2 4 02 7 0

3 0 0

3 3 0

0 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0

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Figure 18: Gain Bode plot of the inputting lower voltage than VA∗ into thebridge.

1 0 0 1 0 0 0 1 0 0 0 0

- 3

- 2

- 1

0

1Ga

in (dB

)

A n g u l a r F r e q u e n c y ( r a d / s e c )

Figure 19: Phase Bode plot of the inputting lower voltage than VA∗ into thebridge.

1 0 0 1 0 0 0 1 0 0 0 0

3 0 0

3 2 0

3 4 0

3 6 0

3 8 0

4 0 0

Phas

e (De

grees

)

A n g u l a r F r e q u e n c y ( r a d / s e c )

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Figure 20: Gain combined Bode plot of the follower, inverter and non-invertercircuits.

1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 E 7- 2 0

- 1 5

- 1 0

- 5

0

5

1 0

1 5

2 0

2 5Ga

in (dB

)

A n g u l a r F r e q u e n c y ( r a d / s e c )

In order to further investigate this, we had decided (by the initial idea of ourinstructor) to put all the magnitude Bode plots on each other. Figure 20 is thiseffort. In this Bode plot, which includes the follower circuit, the non-inverterone, and the inverter one, we can clearly see how the different experiments areconsistent with each other.

Another problem with our results has been the fact that the Nyquist plotfor the Wein bridge did not encircle the point (1, 0) in the complex plane. Ofcourse, we expected it to encircle it, for this is what the Nyquist criterion saysis the condition for a singularity (oscillations). Another problem with this plotis that it is not continuous - it is cut in the middle without reaching a fullcircle. The reason for the latter problem is simple - we simply did not sampleenough points from a large enough range. Had we done that, we would’ve seena complete circle. The former problem is a one harder to tackle. We could ofcourse relate it to some fundamental error in the experiment conduct, but thatis the trivial and uninteresting answer.

A last minor problem is that the Bode plot for the integrator circuit looks alittle bit odd. The phase looks like a sporadic collection of dots. The magnitudeis a little bit better, and could be explained as a cut-off slope with no stablepart at all. Since there is only cut-off, we would expect the phase to increasein that range. However, excluding two divergent points, it is actually constant.We fail to explain these data.

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References

[1] http://www.phys.ualberta.ca/∼gingrich/phys395/notes/node1.html

[2] ”Student Manual for The Art of Electronics” - Hayes and Horowitz

[3] http://www.feedshow.com/show items-feed=979fe0611d90340db7af0646c740030b

[4] http://www.intertent.net/pphlogger/dlcount.php?id=spin&url=amitay/

Physics/B.ScC/LabC/Electronics/Elec2.pdf

[5] http://www.macs.ece.mcgill.ca/∼roberts/ROBERTS/COURSES/

AnalogICCourse/IC Components Ccts HTML/sld055.htm

[6] http://hyperphysics.phy-astr.gsu.edu/HBASE/electronic/opampvar2.html

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