EEE117L Network Analysis Laboratory 6 Operational Amplifiers

Dr. Milica Markovic Network Analysis Laboratory page 1

EEE117L Network Analysis

Laboratory 6

Operational Amplifiers

Instructor: Dr. Milica MarkovicOffice: Riverside Hall 3028

Email: [email protected]:http://gaia.ecs.csus.edu/˜milica

Figure 1: Courtesy of: http://ohmart.org/

California State University Sacramento EEE117L revised: 11. October, 2012

Contents

1 Purpose and Learning Outcomes 4

2 Step I - Calculations 52.1 Unity Gain Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Calculations for ideal opamp model . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Calculations using non-ideal, frequency-independent model . . . . . . . . . . . 52.1.3 Calculations using non-ideal, frequency-dependent model . . . . . . . . . . . . 6

2.2 Inverting Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Calculations using an ideal opamp model . . . . . . . . . . . . . . . . . . . . . 62.2.2 Calculations using non-ideal, frequency-independent model . . . . . . . . . . . 72.2.3 Calculations using non-ideal, frequency-dependent model . . . . . . . . . . . . 7

3 Step II - Simulations 8

4 Step III - Measurements 114.1 Experimental Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.1 Power Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.2 LM741 Operational Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Discussion Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A Operational Amplifiers 17A.1 Opamp model’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17A.2 Operational Amplifier Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . 19A.3 Ideal Opamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A.3.1 Operational amplifier with a negative feedback . . . . . . . . . . . . . . . . . . 20

B Non-Ideal Op-Amp with Finite Gain 23

C Non-Ideal Op-Amp Frequency Response 27C.1 Non-Ideal, Frequency-Dependent Op-Amp Model of Open Loop Gain . . . . . . . . . 27C.2 Example: Noninverting Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

C.2.1 Open-Loop Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28C.2.2 Closed-loop Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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D Operational Amplifier Data 32D.1 Understanding Operational Amplifier Data Sheets . . . . . . . . . . . . . . . . . . . . 32

D.1.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32D.1.2 Schematic and Connection Diagrams . . . . . . . . . . . . . . . . . . . . . . . 33D.1.3 Absolute Maximum Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.1.4 Electrical Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

D.2 LM741 Data Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


Chapter 1

Purpose and Learning Outcomes

The purpose of this project is to investigate two common active circuits involving operational am-plifiers discussed in class.

At the end of this lab students will be able to:

1. Identify opamp topologies.

2. Read and relate some specificiations to the simulated and measured results.

3. Correctly wire an opamp circuit on the board.

4. Analyze, measure and simulate buffer and inverting opamp.

5. Measure and recognize saturation, bandwidth and offset voltage of an opamp.

6. Calculate gain of all amplifiers.

7. Derive, simulate and measure transfer function (closed-loop gain) of buffer and inverting am-plifier.

8. Compare the transfer functions for the ideal and non-ideal cases.

9. Build the circuits, measure the frequency response, and compare to the simulation results.

10. Become aware of offset voltage, difference between open-loop and closed-loop gain, finite open-loop gain, gain-bandwidth product.

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Chapter 2

Step I - Calculations

In the analysis and design of the circuits below, use standard value resistors and typical values forthe non-ideal op-amp parameters.

2.1 Unity Gain Amplifier

2.1.1 Calculations for ideal opamp model

The schematic diagram for a unity gain op-amp circuit is shown in Figure 2.1. Using the ideal ap-ampmodel, KVL around the outermost loop yields Vin = Vout. This type of amplifier is also referred toas a “voltage follower” since, assuming the ideal op-amp model, the circuit simply reproduces theinput signal at the output, i.e. the output “follows” the input. It is also sometimes called a “buffer”amplifier because it isolates the output from the input, such that Vin = Vout regardless of the valueof RL.

Figure 2.1: Unity gain amplifier circuit

2.1.2 Calculations using non-ideal, frequency-independent model

Using the non-ideal, frequency-independent op-amp model that assumes infinite input resistance, zerooutput resistance and finite open-loop gain, described in Appendix B, derive an expression for the

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closed-loop gain of this circuit. In your model, you may assume that Ri is very large (Ri →∞) andthat R0 is very small (R0 ≈ 0). Assume that the open-loop gain is A0 = 2 × 105, which is typicalfor a 741 op-amp. Derive an expression for the closed-loop gain which is defined output voltageas a function of input voltage (and open-loop gain A). How does the output voltage change whenopen-loop gain increases?

2.1.3 Calculations using non-ideal, frequency-dependent model

Using the non-ideal, frequency-dependent op-amp model described in Appendix C, derive an expres-sion for the frequency response of this circuit. In your model, you may assume that Ri is verylarge (Ri → ∞) and that R0 is very small (R0 ≈ 0). Assume that the open-loop gain (at DC)is A0 = 2 × 105, which is typical for a 741 op-amp. Select the values of R and C in the op-ampmodel given in Appendix C so that the resulting single pole model has a pole frequency of 5 Hz(typical value for the 741 op-amp). Find an expression for the 3-dB cutoff frequency of the unitygain amplifier circuit, ωc.

Calculate magnitude and phase of the output voltage for a 500Hz signal with an amplitude of 2Vp-p, then pick several additional frequency points between 10Hz and 10MHz. Repeat the calculationsfor a DC signal of 1V. If both signals are present at the input of the amplifier, what will you expectat the output?

2.2 Inverting Amplifier

2.2.1 Calculations using an ideal opamp model

Using the ideal op-amp model, design an inverting amplifier as shown in Figure 2.2. Design theinverting amplifier so that it has gain of Vout/Vin = −3 and an input resistance of 10 kΩ or greaterwith a 1 kΩ load resistance. Show the schematic diagram of your circuit with all component valueslabeled.

Figure 2.2: Inverting amplifier circuit



2.2.2 Calculations using non-ideal, frequency-independent model

Using the non-ideal, frequency-dependent op-amp model described in Appendix B, derive an expres-sion for the frequency response of this circuit. In your model, you may assume that Ri is very large(Ri → ∞) and that R0 is very small (R0 ≈ 0). Assume that the open-loop gain is A0 = 2 × 105,which is typical for a 741 op-amp.

2.2.3 Calculations using non-ideal, frequency-dependent model

Using the non-ideal, frequency-dependent op-amp model described in Appendix C, derive an expres-sion for the frequency response of this circuit. Again assume that Ri →∞, R0 ≈ 0, and R and C arechosen for a pole at 5 Hz. Find an expression for the gain of the circuit at DC and the 3-dB cutofffrequency.Assume that the open-loop gain (at DC) is A0 = 2×105, which is typical for a 741 op-amp.Select the values of R and C in the op-amp model given in Appendix C so that the resulting singlepole model has a pole frequency of 5 Hz (typical value for the 741 op-amp). Find an expression forthe 3-dB cutoff frequency of the unity gain amplifier circuit, ωc.

Calculate magnitude and phase of the output voltage for a 500Hz signal with an amplitude of 2Vp-p, then pick several additional frequency points between 10Hz and 10MHz. Repeat the calculationsfor a DC signal of 1V. If both signals are present at the input of the amplifier, what will you expectat the output?

In the next step, we will simulate this circuit.


Chapter 3

Step II - Simulations

In this section, you will build in Multisim four circuits: unity-gain amplifier and an inverting amplifer.

1. Unity Gain Amplifier

(a) Simulate the amplifier using AC simulation from 10Hz to about 10MHz.

(b) Explain the behavior of the output voltage as a function frequency. Which spec relates tothis behavior?

(c) What is the 3-dB bandwidth of the circuit? Compare to hand-calculations.

(d) Compare the results to hand-calculated values.

(e) Now perform transient analysis. The input signal should be set at 500Hz. Increase theamplitude of the input signal until you see the change (flattening of the signal) in theoutput voltage. What is the input voltage at which you see distortion? What happens tothe output signal? What does that mean for the vid at the input of the amplifier?

2. Inverting Amplifier

(a) Simulate inverting amplifier in the same frequency range.

(b) What is the 3-dB bandwidth of the circuit? Compare with hand-calculations.

(c) Compare the bandwidth with the bandwith of Unity-Gain Amplifier. Which one hashigher bandwidth and why? Can you relate this to opamps specs?

(d) Increase the input signal to over 9V. What happens to the output signal? What does thatmean for the vid at the input of the amplifier?

(e) Now perform transient analysis. The input signal should be set at 500Hz.Increase theamplitude of the input signal until you see the change in the output voltage. What is theinput voltage at which you see distortion? What happens to the output signal? Recordthe minimum and the maximum voltages of the waveform. What does that mean for thevid at the input of the amplifier?

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Figure 3.1: Multisim Circuit Schematic of the Unity Gain Amplifier.



Figure 3.2: Multisim Simulation of a Inverting Amplifier.


Chapter 4

Step III - Measurements

Operational amplifier is an active circuit that requires DC voltage (aka ”bias”) on appropriate termi-nals before it can be used to amplify DC or AC signals at it’s input terminal. You will use Agilent’spower supply as a source of this DC voltage. Your circuits you will be using the Agilent triple outputpower supply. The op-amp you will be using is the National Semiconductor LM741 op-amp. Thedata sheet for this op-amp is given in Appendix D. You will also be using the non-ideal model foran op-amp, which is given in Appendix C.

4.1 Experimental Measurements

In this section, you will construct the two op-amp based circuits and perform experimental measure-ments for comparison to your analytical and simulation results. In order to perform these measure-ments, you must first become familiar with the operation of the power supply and the 741 operationalamplifier, the details of which are presented next.

4.1.1 Power Supply

Adjust the power supply as follows:

1. Turn on the power supply by switching to ON.

2. Press +25V softkey on the Function, Select Menu of the power supply.

3. Press Output On/Off softkey.

4. Adjust the VOLTAGE +9 V, on the +25V power supply output.

5. Press -25V softkey on the Function, Select Menu of the power supply.

6. Adjust the VOLTAGE −9 V, on the -25V power supply output.

7. Using your DMM in DC voltage mode, confirm that you have +9 V between the +9 V andcommon (COM) terminal, and that you have −9 V between the −20 V and common terminals.

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Figure 4.1: http://ohmart.org/



You should make these adjustments each time you begin work at your lab bench. The power supplyshould either be switched off or disconnected from your breadboard when you are wiring up a circuit.The circuit should be checked carefully before power is applied.

4.1.2 LM741 Operational Amplifier

Your circuits will be based on an LM741 operational amplifier in an 8-pin dual in-line package (DIP).The data sheet for this op-amp, with a guide to understanding this information, is given in AppendixD. DIP package is placed in the channel of the breadboard as shown in Figure 4.2.

Figure 4.2: DIP package placed in the channel of the breadboard.

The top view of the LM741 DIP and its internal connections are shown in Figure 4.3. The pin1 end of the DIP is indicated by a semicircular notch or a white band. Pin 1 may also be indicatedby a dot. The ±9 V supply voltages are applied to pin 7 (V +) and pin 4 (V −). To simplify theschematics, these power connections for our op-amp circuits won’t be shown, but you will need toremember to make these connections for each op-amp used in a circuit. The offset null connections(pins 1 and 5) won’t be used in our circuits and there is no internal connection to pin 8.

4.1.3 Measurements

You are now ready to construct the circuits. For all of the circuits, before you apply the supplyvoltages V + and V −, check to make sure that the ±25 V portion of the triple power supply isproducing ±9 V.



Figure 4.3: LM741 operational amplifier dual in-line (DIP) package. The internal connections areshown for reference.

Make your wiring on the breadboard relatively neat, keeping the leads short. Using shorter wireswill reduce the coupling of unwanted signals into your circuits. If wires are available in enoughdifferent colors, use a color code for your wiring, choosing a separate color for the supply voltages,ground, and for signals. This will make your circuit easier to debug.

Figure 4.4: Unity Gain Amplifier.

1. Construct the unity gain amplifier of Section 2.1. Take gain measurements over the frequencyrange 10 Hz to 10 MHz. Use a calculator to convert the gain to dB, then sketch a plot of gainin dB against frequency on a log scale. Compare your plot with your Multisim results.

(a) The input, V1, should be a 500Hz sinusoid with a peak to peak voltage of 2 volts and0 DC offset. The oscilloscope should be used to adjust the function generator to thisvalue before signal is applied to the circuit. (Be sure that the oscilloscope is DC coupled.)Measure the input and output voltages with both the oscilloscope and the DMM. Comparethe two sets of measurements. Calculate the circuit gain for each set of measurements andcompare these calculated gains with the theoretical gain. Describe the phase relationshipbetween the input and output.



(b) Measure the output voltage with the DMM on the DC setting and the DMM on the ACsetting. Record values, and print the scope’s screen.

(c) As the final step, add a 1V DC offset to the input voltage (function generator has a DCoffset setting). Measure output voltage for the 1kHz signal.

(d) Use the oscilloscope to measure the DC and the amplitude of the output voltage whenthe oscilloscope channel is AC coupled and when it is DC coupled. (This is four measure-ments.) What is the difference between these two measurements?

2. Construct the inverting amplifier of Section 2.2. Take gain measurements (record both ampli-tude and phase) over the frequency range 10 Hz to 10 MHz. Use excel to convert the gain todB (20 log vout

vin), then sketch a plot of gain in dB against frequency on a log scale. Compare

your plot with your Multisim results. In addition perform the following two measurements:

Figure 4.5: Unity Gain Amplifier.

(a) Saturation of the Inverting Amplifiers. Increase the applied peak to peak voltage untildistortion is observed. Describe the distortion. Record the maximum and minimumvoltages observed. Now reduce the applied input voltage back to 2 volts p-p. Apply a DCoffset voltage and increase it until distortion is observed. Describe the distortion. Recordthe maximum and Minimum voltages. Compare to simulations.

(b) Offset voltage. To measure offset voltage, ground the inverting opamp input (you willhave no sources in your circuit) and record output voltage. Be sure to use a sensitivescale of a DMM to measure Vout to at least two significant figures. Record the measuredoutput voltage, and divide it by the gain of the amplifier. The result usually labeled asVOS , is the input offset voltage of the amplifier. Check the datasheet for the op-amp andmake sure that your offset is within the value given in the datasheet. Consult with yourinstructor and other groups and record the values other groups recorded.



4.2 Discussion Topics

• Compare the calculated, theoretical and measured gains.

• Compare the measurements by the oscilloscope and by the DMM.

• Describe the distortion. How close to the t supply was the distorted voltage?

• Discuss the phase shifts observed, calculated and simulated.

• Discuss the various voltage measurements.


Appendix A

Operational Amplifiers

Opamps can be used to:

1. Amplify Signals

2. Filter Signals

3. Perform Arithmetic Operations

4. Compare Signals

5. Construct a Current Source

6. Construct a Voltage Regulator

7. Construct oscillators

The first opamp was developed in 1964 by R. Widlar, who was at the time with Fairchild Semi-conductor (µ709)1. The schematic diagram of an opamp is shown in Figure A.1. The voltages V−,V+ and Vout are single-ended voltages. This means that the voltages are measured with respect tothe same point, the ground. Vd is called the differential voltage, because it is the difference betweenthe voltages V+ and V−.

Vd = V+ − V− (A.1)

A.1 Opamp model’

Operational amplifiers have many important charachteristics that are modeled 2. The three mostimportant characteristics for us now as shown in Figure A.2 are

1See the paper Some Circuit Design Techniques for Linear Integrated Circuits2See the 741 data sheet at the end of this lab

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Figure A.1: Operational Amplifier Schematic Diagram



1. open loop gain,

2. input and

3. output resistance.

Open loop gain represents the gain of the operational amplifier when it is not connected in acircuit. For example, if we connect 5 volts at the input of the operational amplifier and the openloop gain of the amplifier is 100000, then the output of the amplifier would be 600kV. It would benice to have a device like that but we already know that with electronic circuits it is very likely thatsomething will burn before we reach the 0.6MV.The output of the operational amplifier is limited bythe DC battery voltage Vcc, which is usually of the order of 15V. Typical value for the open loopgain is 106.

Input resistance is resistance that a generator connected between the + an - terminals would seewith all other independent sources switched off. Typical input resistance is 2MΩ.

Output resistance is the resistance that a generator connected between the output terminal andthe ground would see with all other independent sources switched off. Typical output resistance is75Ω.

Figure A.2: Important properties of an operational amplifier

A.2 Operational Amplifier Equivalent Circuit

The most basic equivalent circuit for an operational amplifier consist of three characteristics describedin the previous section. Equivalent circuits are simplified models of real circuits that allow us toapproximately predict the behavior of a complex circuit. A common way to represent any electroniccircuit is with two port networks. An equivalent circuit of an opamp is shown in Figureopampjpgd2.

A.3 Ideal Opamp

An ideal opamp is a simplification (zero-order model) of the opamp. It has the following characteris-tics: infinite open-loop gain, infinite input resistance, zero output resistance. Infinite input resistance



Figure A.3: Important properties of an operational amplifier

yields zero current at the input of the amplifier, currents flowing into the opamp positive and negativeterminals I+ = I−.

A.3.1 Operational amplifier with a negative feedback

Let’s look at what happens if we add negative feedback to an opamp as shown in Figure A.4 If theoutput voltage increases, the voltage at the negative terminal will increase too. Since the outputvoltage is equal to Vout = A(V+−V−) the output voltage will decrease. To look at how is this appliedto solve opamjpg circuits we can re-write the above formula as:

V+ − V− =VoutA

(A.2)

All opams have very large open-loop gain A (order of 106). The value of the output voltage voutis limited between 0 and 15V. Therefore the ratio of Vout

Ais a very small number (0.000001). In all

practical cases we can say that this ratio is equal to zero. Since the ratio is equal to zero. This inturn means that V+ − V− is also equal to zero from Equation A.3.1, and that V+ = V−.

Some typical opamp circuit configurations are shown in Figures A.5 and A.6.



Figure A.4: Negative feedback

Figure A.5: Inverting Amplifier



Figure A.6: Noninverting Amplifier


Appendix B

Non-Ideal Op-Amp with Finite Gain

In this section we will assume that the input resistance of an opamp is infinite, output resistance iszero, and the only non-ideal parameter is the open-loop gain. The amplifier model shown in FigureB.2 is placed in a non-inverting circuit configuration in Figure B.3. We can write the followingequations for this circuit

v1R2

+v1 − voR1

= 0 (B.1)

vo = Avid (B.2)

vs = vid − v1 (B.3)

By solving above equations we get the expression for closed-loop gain

vovs

= G =1 + R1

R2

1 + 1A

(1 + R1

R2)

(B.4)

When A → ∞ we get the ideal closed loop gain G = 1 + R1

R2. Agilent Advanced Design System

(ADS) was used to simulate the effect of finite open-loop gain to closed-loop gain of the amplifierin Figures B.3 and B.4. You can see that when the open-loop gain increases, the closed loop gainrapidly approaches the value for the ideal operational amplifier (with infinte open-loop gain model).

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Figure B.1: Non-ideal op-amp circuit model with finite gain, AC simulation



Figure B.2: Non-ideal op-amp circuit model with finite gain, DC simulation.

Figure B.3: Non-ideal op-amp circuit model with finite gain in an non-inverting amplifier.



Figure B.4: Non-ideal op-amp circuit model


Appendix C

Non-Ideal Op-Amp Frequency Response

The next section describes the non-ideal frequency response properties of an op-amp. This is animportant consideration when designing op-amp circuits, since it limits the maximum frequency ofoperation of the circuit.

C.1 Non-Ideal, Frequency-Dependent Op-Amp Model of Open

Loop Gain

For the ideal model, we assume that the open-loop gain of the op-amp, A0, is a constant. In actuality,the voltage gain of an op-amp is a function of the frequency of the excitation signal. We will denotethe frequency-dependent open-loop gain as A(jω) or A(f). To a reasonable approximation, theop-amp can be modeled as a first-order low pass filter having a gain of

A(jω) =A0

1 + jω/ω1

(C.1)

where A0 is the open-loop gain at DC and ω1 is the open-loop cutoff frequency. Typically, A0 is greaterthan 104 and ω1 is less than 100 rad/s. Notice that the gain decreases as the frequency increases.A circuit model, incorporating a finite input resistance Ri and a non-zero output resistance Ro inaddition to the frequency-dependent gain is shown in Figure C.1. In the model, the voltage Vp−n isthe differential voltage between the noninverting and inverting op-amp inputs.

Other important parameters pertaining to the non-ideal op-amp model are the unity gain fre-quency, fu, and the gain-bandwidth product, GB. The unity gain frequency is the frequency at whichA(fu) = 1, or where the gain is equal to one. For the 741 op-amp, fu ≈ 1 MHz. The gain-bandwidthproduct is the product of the frequency and open-loop gain at that frequency:

GB = A(f) · f (C.2)

For the 741 op-amp, GB ≈ 106 Hz. Because the open-loop DC gain of the 741 op-amp is A0 ≈ 2×105,the open-loop cutoff frequency can then be approximated as

f1 ≈GB

A0

=106

2× 105= 5 Hz (C.3)

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Figure C.1: Non-ideal op-amp circuit model

C.2 Example: Noninverting Amplifier

To gain an understanding of how the frequency-dependent model is used, let us consider a noninvert-ing amplifier operating both open-loop and closed-loop (without and with negative feedback). Ourgoal is to compare the frequency response of the two circuits. For simplicity, let us assume that theop-amp we choose has an open-loop DC gain of A0 = 105, an open-loop cutoff frequency of ω1 = 10rad/s, an infinite input resistance (Ri →∞), and negligible output resistance (Ro → 0).

C.2.1 Open-Loop Operation

The schematic diagram of the noninverting amplifier operating open-loop is shown in Figure C.2(a)and the corresponding circuit model is shown in Figure C.2(b). Notice the lack of a feedback pathfrom the op-amp output to the inverting input. Analyzing the equivalent circuit model yields:

Vout = A(jω)Vp−n = A(jω)Vin =A0

1 + jω/ω1

Vin (C.4)

The open-loop frequency response is then

H(jω) =Vout

Vin

=A0

1 + jω/ω1

=105

1 + jω/10(C.5)

The Bode straight-line magnitude plot of H(jω) is shown in Figure C.3, indicated as “open-loop.”Note that the DC gain in dB is 20 log10 105 = 100 dB and the cutoff frequency is ω1 = 10 rad/s. Theopen-loop response has a large DC gain but a low cutoff frequency. The magnitude is equal to 1 (0dB) at ω = 106 rad/s, confirming that the gain-bandwidth product of the op-amp is 106 rad/s.

C.2.2 Closed-loop Operation

Now consider the case of a noninverting op-amp circuit operating closed-loop. The schematic of thiscircuit is shown in Figure C.4(a) and the equivalent circuit model is shown in Figure C.4(b).



Figure C.2: (a) Open-loop-amplifier schematic; (b) Equivalent circuit model

Figure C.3: Bode straight-line magnitude plots for the noninverting op-amp circuit, operating open-loop and closed-loop

Figure C.4: (a) Closed-loop noninverting amplifier schematic; (b) Equivalent circuit model



The analysis of the circuit model proceeds as follows. Denote the node voltage at the invertingterminal of the op-amp as Vn. Writing a node voltage equation at the inverting terminal yields

Vn

R1

+Vn −Vout

R2

= 0 (C.6)

Solving for the node voltage

Vn =R1

R1 +R2

Vout = kVout (C.7)

where we have defined

k =R1

R1 +R2

(C.8)

The op-amp differential input voltage is

Vp−n = Vin −Vn (C.9)

and the output voltage can then be expressed as

Vout = A(jω)Vp−n

= A(jω)[Vin −Vn]

= A(jω)[Vin − kVout] (C.10)

Rearranging the equation yields

[1 + kA(jω)]Vout = A(jω)Vin (C.11)

Solving for the closed-loop frequency response

H(jω) =Vout

Vin

=A(jω)

1 + kA(jω)

=A0/(1 + jω/ω1)

1 + kA0/(1 + jω/ω1)

=A0

1 + jω/ω1 + kA0

=A0

(1 + kA0) + jω/ω1

=A0/(1 + kA0)

1 + jω/[ω1(1 + kA0)]

=Ac

1 + jω/ωc

(C.12)

where the closed-loop DC gain, Ac, is related to the open-loop DC gain, A0, by

Ac =A0

1 + kA0

(C.13)



and the closed-loop cutoff frequency, ωc, is related to the open-loop cutoff frequency, ω1, by

ωc = (1 + kA0)ω1 (C.14)

As an example, assume that we choose the resistor values R1 = 10 kΩ and R2 = 100 kΩ. In thiscase, the parameter k is

k =10

10 + 100= 0.0909 (C.15)

Assuming an open-loop DC gain of A0 = 105 and an open-loop cutoff frequency of ω1 = 10 rad/s,the closed-loop DC gain is

Ac =105

1 + 0.0909 · 105= 11.0 (C.16)

or20 log10Ac = 20 log10 11.0 = 20.83 dB (C.17)

The closed-loop cutoff frequency is

ωc = (1 + 0.0909 · 105) · 10 = 9.90× 104 rad/s (C.18)

The Bode magnitude plot of H(jω) using the resistor values R1 = 10 kΩ and R2 = 100 kΩ is shownin Figure C.3, indicated as “closed-loop.” The Bode magnitude plot shows a decreased DC gainand increased cutoff frequency, in comparison with the open-loop response. An important designconsideration for op-amp circuits is this trade-off between DC gain and low-pass cutoff frequency.Notice that the open-loop response approaches the closed loop response above the closed-loop cutofffrequency, ωc.


Appendix D

Operational Amplifier Data

This appendix provides design data for the operational amplifiers you will use in the laboratory.Information is first provided on how to understand this data.

D.1 Understanding Operational Amplifier Data Sheets

To be able to design circuits and systems from integrated circuits, you will need to learn how to readand understand the data sheets provided by the IC manufacturers. In many cases, this will be theonly information on which to base your design. In this section, a short guide to understanding ICdata sheets is provided. The guide primarily provides definitions of the terms and symbols used foranalog ICs (based on the 741 operational amplifier). In reading this description, refer to the datasheet from National Semiconductor for the LM741 given in Section D.2

You must be warned that data sheets are often incomplete or incorrect. They may have been basedon an early prototype of the device and not have been changed when the final device specificationswere established. They are often written hurriedly and with little attention to detail by someone inmarketing who was given incomplete or incorrect information by the designer. Unfortunately, thesedata sheets are not often revised when a new edition of a data book is published.

Thus, you must read the data sheets critically. When the information is incomplete, you mayneed to make some measurements on the device to determine its true characteristics. Be alert forerrors. Some data sheets have even given the wrong pin assignments for a device, which can lead todamage to the device if connected in this way. If there is a second source for the device, check thedata sheet from this source for inconsistencies.

D.1.1 General Description

The general description provides a short description of the basic characteristics of the device. Thisis read to determine if the device has characteristics that suggest it should be considered for use ina design that you are developing.

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D.1.2 Schematic and Connection Diagrams

Complete schematics are given for many analog ICs, which also provide notation and pin assignmentsfor inputs and outputs. Analog ICs are often provided in several packages, so the connection diagramsare provided for each of these.

D.1.3 Absolute Maximum Ratings

The absolute maximum ratings are values of operating parameters beyond which the safety of thedevice cannot be guaranteed. The device should not be operated at these limits, but below them bya reasonable safety margin. The device can usually be expected to rapidly fail if this warning is notheeded.

D.1.4 Electrical Characteristics

The table of electrical characteristics provides information on input and output voltages and currents.The columns labeled “min” and “max” specify the minimum and maximum voltages or currents thatthe device will provide. The column labeled “typ” lists typical or average values. The following aredefinitions of the most important parameters listed in the table.Input offset voltage: That voltage which must be applied between the input terminals throughtwo equal resistances to obtain zero output voltage. See Figure D.1(a).Input offset current: The difference in the currents into the two input terminals when the outputis at zero. See Figure D.1(b).Input bias current: The average of the input currents. See Figure D.1(b).Input resistance: The ratio of the change in input voltage to the change in input current on eitherinput with the other grounded. See Figure D.1(a).Input voltage range: The range of voltages on the input terminals for which the amplifier operateswithin specifications.Large signal voltage gain: The ratio of the output voltage swing to the change in input voltagerequired to drive the output from zero to this voltage.Output voltage swing: The peak output voltage swing, referred to zero, that can be obtainedwithout clipping.Output short circuit current: The maximum output current available when the output is shortedto ground or to either supply.Common-mode rejection ratio: The ratio of the input common-mode voltage range to the peak-to-peak change in input offset voltage over this range.Supply voltage rejection ratio: The ratio of the change in input offset voltage to the change inpower supply voltages producing it.Transient response: The closed-loop step-function response of the amplifier under small-signalconditions.Bandwidth: That frequency at which the voltage gain is reduced to 0.707 times the low frequencyvalue.Slew rate: The internally-limited rate of change in output voltage with a large amplitude stepfunction applied to the input.



Figure D.1: (a) An equivalent circuit for an operational amplifier showing the effect of input resis-tance, output resistance, and input voltage offset. (b) Equivalent circuit for the input currents to anoperational amplifier.



Supply current: The current drawn from the power supply by the operational amplifier when theoutput current and voltage are essentially zero.



D.2 LM741 Data Sheet1

1Courtesy of Texas Instruments.


EEE117L Network Analysis Laboratory 6 Operational Amplifiers

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