# What is a Phase Shift Sine Wave Oscillator?

Anonymous

Question description

An oscillator is used to convert dc power into ac power. In this lab experiment the student will use a phase shift oscillator to convert +/- 15 VDC into a 10๐๐โ๐ sine wave signal in the 8kHz to 10kHz region. The measured actual hardware oscillator frequency is different from that frequency predicted while using an ideal op amp model.

This experiment has multiple goals:

1) The first goal of this experiment is to understand some oscillator concepts and how to use these concepts to predict the output frequency of the oscillator when using an ideal op amp.

2) A second goal is to build the oscillator circuit using a real 741 op amp on a solderless breadboard and to measure the actual performance of the oscillator circuit.

3) A third goal is to show how a slightly more complex model for the 741 op amp can be used to predict the measured oscillator frequency much more accurately.

4) A fourth goal is to see how well Multisim predicts the actual oscillator frequency and other circuit behavior while using Multisimโs very complex model for a 741 op amp. A Multisim virtual op amp is also used to help predict which op amp parameters affect the oscillator frequency.

5) A fifth goal of this experiment is for the student to understand when the op amp circuit is operating as a linear amplifier and when its operation is in a non-linear mode.

6) A sixth goal is to add a single component to the oscillator circuit which drops the frequency of the oscillator to 5kHz. An improved sinusoidal oscillation can be obtained at this lower frequency.

This lab experiment, and the formal lab report, are each divided into four parts:

1) Part A provides the theory of operation for this oscillator. The student is expected to write up the Theory of Operation for the oscillator circuit in his/hers own words as a part of the formal report.

2) Part B1 is a Multisim analysis of the oscillator circuit using a 741 op amp.

Part B2 replaces the 741 with a virtual op amp. Here some op amp parameters can be varied which may, or may not, affect the output frequency of the oscillator.

3) Part C1 is an initial hardware solution without any added circuit refinements. Part C2 includes a diode modification for improved oscillator stability and improved sinusoidal waveform.

4) Part D is the student write up of the formal report. See the last two pages for more details on the formal write up of this experiment. All students must individually explain all sectionsof the experiment as a part of their formal report.

Part D of this report must include a โConclusions:โ section at the end of the report.

Part A. Theory of Operation

Section 10.9.2 of the textbook discusses the Barkhausen Criteria for an oscillation to exist. Oscillation occurs, in a closed loop feedback structure, when the gain around the complete feedback structure is greater than or equal to 1 and the associated phase angle is a multiple of 360 degrees. In addition, if the gain around the complete feedback structure is exactly 1.0 and at some multiple of 360 degrees, the oscillation will be a perfect sine wave. The figure below shows a conceptual example of a linear circuit that might have an oscillatory output.

The above model, shows an inverting op amp circuit whose closed loop gain would be โ๐๐๐/๐ if the op amp was ideal. The model also shows a second feedback network whose complex voltage gain is defined as ๐ฝ. The gain of this feedback network is normally less than one and is often a frequency dependent circuit. Assume that the switch is in position A and the voltage source ๐ฃ๐ ฬ is 1V at an angle of 0ยฐ. Also assume that there exists a frequency, ๐น๐ where the output of the feedback network, ๐ฃ๐ฬ ๐ is also 1V at an angle of 0ยฐ. Then when the switch is moved to position B, the circuit will operate in a self sustaining oscillatory mode at the frequency, ๐น๐. The difference between ๐ฃฬ๐๐ and ๐ฃฬ๐  is often referred to as the โreturn differenceโ. Mathematically this โreturn differenceโ can be expressed by the quantity 1 โ ๐ดฬ ๐ฝฬฬ. The circuit will oscillate in a sinusoidal mode at the point where this complex return difference (i.e. 1 โ ๐ดฬ ๐ฝฬ ) is 0 + j0.

The formal report, submitted by each student, should explain the theory of operation for this oscillator in his/hers own words using information from section 10.9.2 of the text, this material, the characteristics of the โrealโ inverting op amp, and the complex gain of the third order low pass filter (the Beta network). The gain and phase characteristics of this third order low pass filter are studied in the Lab 14 Initial Notes. Schematic diagrams, with all applicable details, must always be included as a part of each section of the formal report.

The items in the list below must also be a part of your explanation of the theory of operation for this oscillator:

1) At what frequency does the third order low pass filter circuit, described in the prelab notes, exhibit 180 degrees of phase shift? Explain why this frequency is important and why it is the initially predicted output frequency of the oscillator when using an ideal op amp.

2) What is the ratio of | ๐ฃ๐๐ข๐ก/๐ฃ๐  | for this third order low pass filter at the above predicted frequency of oscillation? Why is the magnitude of this voltage ratio important at the predicted frequency of oscillation?

3) What is the required complex closed loop gain of an ideal inverting op ampto provide a โreturn differenceโ of 0 +j0 at the initially predicted frequency of operation, ๐น๐.

4) What is the calculated value of ๐๐๐ in the feedback loop of an ideal op amp to provide this required closed loop gain?

5) List several reasons why the calculated value of ๐๐๐ may not be the correct value for sine wave oscillation? This question is to be answered after the simulation and hardware portion of this lab are completed and as a part of the conclusions section of your formal report.

6) How can you set the gain around the complete feedback structure to exactly 1+j0 and how might you keep the magnitude of this gain at 1.0? This question, involves the back-to-back diodes, and is to be answered as a part of the conclusions section of your formal report.

7) Explain why the actual frequency of oscillation is not the same as that initially predicted in part A. This question applies to all parts of the lab experiment where the output waveform is sinusoidal, i.e. either when the back-to-back diodes are present or when the oscillation is building up in a sinusoidal fashion. Your answer to this question probably best indicates your understanding of this lab experiment. Your answer should be based on the phase shift of the 741 op amp in the inverting circuit configuration at the actual frequency of operation. An analytical solution is much better than just a measured data solution. This question is to be answered in the conclusions section of your formal report.

Part B1. Circuit Simulation with 741 op amp.

Use Multisim to model the following oscillator circuit. Use the actual values measured in Part C1 of your hardware circuit for resistors ๐1 thru ๐4 and capacitors ๐ถ1 thru ๐ถ3. Each of these three capacitors should be fabricated by connecting two 0.1uF +/- 5% capacitors in series. The student should use the measured series equivalent value for each of the capacitors in their simulation circuit. See part C1 of this lab experiment.

B1.1 Initial Setup

Connect the channel B input of the simulated oscilloscope to the output of your oscillator circuit at pin 6 of the op amp. Set the simulated scope to trigger on the channel B signal. Connect the channel A input of the simulated oscilloscope to theinput of the inverting op amp circuit, i.e. the junction of ๐3 and ๐4. Note that resistor๐4 has a dual purpose in this lab experiment. It is both the load resistance on theโThird Order Low Pass Filterโ and the input resistor of the inverting op amp circuit.

Set ๐๐๐ to 20k ohms and start the simulation. Increase the value of ๐๐๐ in 1k Ohm increments until you obtain a clipped sine wave output at pin 6 of the op amp. If ๐๐๐ is greater than 70k ohms without obtaining an oscillation, some problem exists in your simulation circuit. The easy way to change the value of ๐๐๐ in 1k ohm steps is to replace ๐๐๐ with a 20k ohm fixed resistor in series with a 20k ohm variable resistor. Open the properties window of the variable resistor and change the resolution from 5% to 2%. The computer mouse resolution will probably limit the useful resolution of this variable resistor. Show the Multisim circuit and the dual channel oscilloscope output in section B1.1 of your formal report.

B1.2 Linear and Non Linear Modes of operation

After the simulation has been shown to be able to generate a clipped sine wave output, select the value of ๐๐๐ in 100 ohm increments, or less, until the circuit will just start up and sustain an oscillation. The easy way to change the value of ๐๐๐ in 100 ohm steps is to replace ๐๐๐ with a fixed resistor in series with a 5k ohm variable resistor. The value of this fixed resistor must be small enough so that the circuit will not oscillate when the simulation is started with the variable resistor set to 0 ohms.

Slowly increase the value of the variable resistor until the oscillation starts to build up. With a sensitive adjustment it might take up to 10 to 20 seconds to build up the oscillation. The student should measure the frequency of sine wave oscillation while the signal is building. The student should also measure the frequency when the output has built up to a clipped sine wave. In the first case the oscillator is operating in a linear mode. After the circuit is generating clipped sine waves, the op amp operation is non-linear. The student will want to repeat this procedure several times to find the value of the total resistance in the feedback path that will just start the oscillation as well as measure the oscillation frequency in both the linear and non-linear mode. Explain why the frequency is different in the two modes of operation in section B1.2 of your formal report. Hint: If in the non-linear mode, some added time delay will exist due to clipping in the 741 output signal.

Print a Multisim oscilloscope picture showing the buildup of the sine wave and record your value of total resistance required to just start the oscillation.

B1.3 Feedback Resistance Value to Start Oscillation

The total feedback resistance to just start the oscillation was measured in Section B1.2. This measured resistance to just start the oscillation is related to the feedback resistance that was originally calculated in question 4 of part A. However, this measured simulation resistance is less than the resistance calculated in question 4 since the โrealโ 741 op amp is not an โidealโ op amp. Explain why this value of resistance is less in Section B1.3 of your formal report. Hint: Use your gain/phase plot for the third order low pass filter and the actual oscillation frequency in the linear mode as a basis of your calculation.

A significant amount of clipping will still exist on the top and/or bottom of the sine wave output signal at the completion of the previous resistance adjustment. Show the circuit diagram and the connections to your Multisim oscilloscope in your formal report. Also Explainwhy the clipping exists on the output waveform in section B1.3 of your formal report.

B1.4 Difference between Part A Predicted Oscillator Frequency and the Multisim Measured Oscillator Frequency

Select the time base to show only about one or two cycles of the output sine waveform while the output is in the process of building up. Measure and record the gain and phase shift of the op amp inverting circuit at the frequency of oscillation. Use this measured closed loop op amp circuit gain and phase, along with the calculated gain and phase characteristic of the third order low pass filter, to explain much of the difference between the original predicted oscillator frequency and the measured oscillator frequency while the oscillation is building up. Include a picture of this dual channel scope waveform in section B1.4 of your formal report. Do not forget to label the input and output scope waveforms along with the vertical sensitivity of each oscilloscope channel.

B1.5 Add the Back-To-Back Diodes to the Oscillator Circuit

Replace ๐๐๐ with the diode network described in the LAB 14 DIODE CHARACTERISTIC NOTES. Select the value of the resistance R in series with the back-to-back diodes (as described in those notes) for a 10 ๐๐โ๐ sine wave output. Measure the peak-to- peak amplitude and the frequency of the output sine wave. Take a scope picture of the op amp inverting circuit input and output voltage for section B1.5 of your formal report. Explain how the back-to-back diodes remove the clipping and improve the sinusoidal wave shape in your formal report in your own words. Note that the back-to-back diodes have also reduced the difference between the frequency initially predicted in part A and the clipped output frequency measured by the simulation in part B1.2. Why do you think the back-to-back diodes have helped to reduce this frequency difference? Hint: The back to back diodes have eliminated the clipped output waveform and have, therefore, forced the op amp to operate in its linear mode. The added delay, mentioned in B1.2, does not exist in the linear mode.

Part B2. Circuit Simulation with virtual op amp.

Replace the 741 op amp with the Multisim OPAMP_5T_VIRTUAL op amp. A virtual op amp is more like the ideal op amp studied in lecture class except some of the characteristics of the virtual op amp are adjustable in the simulation. Open the Properties Window of this virtual op amp to see the adjustable op amp parameters. Start with the default values for the virtual op amp where the โOpen Loop Gainโparameter is 200k and the โUnity Gain Bandwidthโ parameter is 100MHz. Leave the diode network in the feedback loop of the op amp for this part of the lab experiment. Also leave the simulated oscilloscope connected as shown in Part B1.1. Show the circuit diagram and explain what you are doing in your formal report.

The OPAMP_ 5T_VIRTUAL op amp must be connected to a source of +15Vdc and a source of -15Vdc. Its schematic diagram appears as a five terminal network. If you flip this op amp on your simulation vertically, to have the negative input of the op amp at the top of your simulation circuit diagram, make sure you connect the +/- 15V power supplies correctly.

B2.1 First Test with Virtual Op Amp

Find the value of the resistance R in series with the back-to-back diodes for a 10 ๐๐โ๐ output from the simulator. Record the value of this series resistance in your formal report. Note that the oscillator frequency now quite close to the frequency initially predicted in part A of this experiment.

B2.2 Gain and Phase Shift with Virtual Op Amp

Select the time base to show only about one or two cycles of the oscillator waveform. Measure and record the gain and phase shift of the op amp circuit at the frequency of oscillation. Use this closed loop op amp gain and phase, along with the calculated gain and phase characteristic of the third order low pass filter to explain why the simulated oscillator is now very close to the original predicted oscillator frequency. Include a picture of this dual channel waveform in section B2.2 of your formal report.

B2.3 Change the DC Open Loop Gain of the Virtual Op Amp

Change the value of the โOpen Loop Gainโ parameter of the virtual op amp from 200k to 20k. If necessary, readjust the series feedback resistor R, to obtain a 10 ๐๐โ๐ output. This change in the โOpen Loop Gainโ parameter does not significantly affect the frequency of the sine wave output and should not significantly affect the value of the series feedback resistor.

Set the time base of the oscilloscope to show only about one or two cycles of the oscillator waveform. Measure and record the gain and phase shift of the op amp circuit at the frequency of oscillation. Take a picture of this dual channel waveform for section B2.3 of your formal report. Use the measured phase shift of the op amp circuit to explain why this 10 to 1 change in the โOpen Loop Gainโ parameter does not affect the oscillator frequency.

B2.4 Change the Unity Gain Bandwidth Parameter to 800kHz

Return the value of the โOpen Loop Gainโ parameter of the virtual op amp to 200k. Decrease the โUnity Gain Bandwidthโ parameter of the virtual op amp to that of a 741 op amp, i.e. about 800kHz. Readjust the series feedback R to obtain a 10 ๐๐โ๐ output. Note that this op amp parameter affects the frequency of the sine wave output. Measure and record the gain and phase shift of the op amp circuit at the frequency of oscillation. Use this closed loop op amp gain and phase, along with the calculated gain and phase characteristic of the third order low pass filter, to explain much of the difference between the original predicted oscillator frequency and the measured oscillator frequency. Include this analysis and your dual channel waveform in section B2.4 of your formal report.

Part C1. Experimental Circuit without back to back diodes.

Fabricate the oscillator circuit shown below on your solderless breadboard. Measure the values of resistors ๐1 thru ๐4 and capacitors ๐ถ1 thru ๐ถ3 before connecting them into the circuit. These four resistors should each have a nominal value of 1000 ohm. The three capacitors should each have a nominal value of 0.05๐F. Remember that you will fabricate each 0.05๐๐น capacitor by connecting two 0.1๐๐น capacitors in series. Give these measured values and their location in the circuit to your Multisim partner.

The two 10uF capacitors may not be required and are only significant if the Thevenin output impedance of the dual power supply, including its power supply lead wires, is not close to zero ohms in the frequency region above a few kHz. It is better to add these two 10uF capacitors to avoid possible problems.

Note that resistor ๐4 has a dual purpose in this lab experiment. It is both the load resistance on the โThird Order Low Pass Filterโ and the input resistor of the inverting op amp circuit.

Connect channel 2 of the oscilloscope to the sine wave output (pin 6) of your oscillator. Connect channel 1 of the oscilloscope to the junction of ๐3 and ๐4. This junction point is both the input to the inverting op amp circuit and the output of the third order low pass filter circuit. Have the instructor inspect your circuit before applying power to the circuit. Set ๐๐๐ to the value calculated in part A of this experiment. If the circuit does not oscillate, increase the value of ๐๐๐ to the point where the circuit just oscillates. If the circuit does oscillate, decrease the value of ๐๐๐to the point where the circuit no longer oscillates. Then increase the value of ๐๐๐ to the point where the circuit just does oscillate. Replacing ๐๐๐ with a 10k ohm potentiometer and a series connected fixed resistor will speed up this adjustment process.

Measure the total value of ๐๐๐ for use in your formal report when the oscillation just starts to build up. Repeat this procedure several times to get an average for this resistance value. Make sure you remove power from the circuit and open the feedback circuit before you measure the total resistance of the feedback circuit. Otherwise, your measured resistance value will probably not be correct.

If you get to 70k ohms without obtaining an oscillation, some problem exists in your circuit. The problem must be fixed before you can proceed. See your instructor if you need help.

After you get the oscillator circuit working take a cell phone picture of the circuit for section C of your formal report. Also do not forget to add a schematic diagram to section C of your formal report including all the test equipment.

Select the time base of the scope to show only about one or two cycles of the oscillator waveform. Measure the gain and phase shift of the op amp circuit at the frequency of oscillation as the oscillation is building up with the oscilloscope. Print a picture of this dual channel oscilloscope output for your formal report. Use this closed loop op amp gain and phase, along with the calculated gain and phase characteristic of the third order low pass filter to explain some of the difference between the original predicted oscillator frequency and this measured oscillator frequency in your formal report.

Also use the oscilloscope to measure the gain and phase shift of the op amp circuit at the frequency of oscillation after the oscillation has built up to a clipped sine wave. Explain why this frequency is lower after the oscillation has built up to a clipped sine wave in your formal report. See the Lab 14 notes titled โUsing the Magnitude andPhase Angle of a 741 Inverting Op Amp Circuit to Predict the Oscillator Frequencyโ for more details.

Also Explain why clipping exists on your output waveform at pin 6 of the op amp in section C1.1 of your formal report.

Part C2. Experimental Circuit with added back-to-back diodes.

Replace ๐๐๐ with the diode network described in the LAB 14 DIODE CHARACTERISTIC NOTES. Select the value of the series resistance R (described in those notes) for a 10๐๐โ๐ output at pin 6 of the op amp. Measure the peak-to-peak amplitude and the frequency of the output sine wave. Record a picture of the oscilloscope output for section C2 of your formal report. Also do not forget to add a schematic diagram to section C2 of your formal report including the back-to-back diodes and all the test equipment.

C2.1 Explain the Operation of the Back-to Back Diodes

Explain the operation of the back-to-back diodes in your formal report in your own words. Note that the back-to-back diodes have also reduced the difference between the initially predicted frequency in part A and the measured oscillator frequency.

Explain, in your formal report, what component should be modified in order to get better agreement between the initially predicted frequency and the measured frequency of your oscillator. Your Multisim analysis of the virtual op amp parameters should help in your selection of the characteristics of a different op amp.

C2.2 Switching Between Linear and Non-Linear Operation

Note that the output of your hardware oscillator circuit switches from a sine wave output to a clipped output somewhere in the region from about 12Vpp output to about 16Vpp output. This change in waveshape occurs when the slew rate limit of the op amp is reached. If your Multisim circuit simulator is accurate, Multisim should approximate this switching characteristic when using a 741 op amp with the back-to- back diodes. Try it with Multisim!Based on this test, do you think Multisim provides an accurate simulation for this Phase Shift Oscillator? Answer in section C2.2 of your formal report.

C2.3 Add a Capacitor to Change the Oscillator Frequency

Add a capacitor from pin6 of the op amp to pin2 of the op amp to drop the oscillator frequency to about 5.0kHz. The value of this capacitor should be less than 1500pF. The value of ๐๐๐ will have to be changed to maintain a 10Vp-p output. Explain why this capacitor reduces the output frequency of the oscillator in section C2.3 of your formal report. Hint: This capacitor increases the phase lag in the op amp circuit which, in turn, reduces the phase lag in the Third Order Low Pass Filter circuit in order to maintain a total of -360ยฐ of phase shift. A mathematical derivation of the value of capacitance is worth more credit than just a verbal description. See the Lab14 notes titled โUsing the Magnitude and Phase Angle of a 741 Inverting Op AmpCircuit to Predict the Oscillator Frequencyโ for more details.

Note that you can now turn the output up to 20Vpp at 5kHz without jumping to the clipped mode. Explain why in your formal report. Hint: Consider the effect of op amp slew rate limiting at a frequency of 5kHz and at 10kHz.

Note also that the sine wave at 5kHz appears to look somewhat more like a real sine curve than the previous 8kHz to 10kHz waveform. The 5kHz waveform is said to have less distortion.

Part D Write up of formal report:

Use your lab notebook, in this experiment, to clearly record all the information you may need to write up your formal report. This information will be very important as you prepare your formal report. I will not grade your lab notebook, but you should use it as a source of information as you write up your formal report.

The ability to communicate the results of observations is essential to good experimental technique. It is an indispensable tool to the practicing engineer. The lab notebook is intended primarily as a record of results and not as a method of formal communication. This formal report must be typed using good grammar and without spelling errors, double spaced, and should include the following, sequentially:

1) Number of the experiment, title of the experiment, and number of your lab station.

2) Date of the report and the date the test data was measured.

3) Name of the author of formal report and name of all students in the team including the primary responsibility of each student, i.e. responsibility must be either theory and circuit simulation or theory and hardware fabrication/test.

4) Objectives of the experiment. This should be limited to two or three sentences and must include a statement regarding the purpose of this experiment and what you expect to learn regarding correlation between the simulation oscillator frequency and the experimentally measured oscillator frequency.

5) Detailed discussion of the theory of operation including all applicable schematic diagrams and expected theoretical results. Use your own words to describe the theory of operation. See Part A for more detail.

6) Detailed discussion of the Multisim analysis including applicable circuit diagrams, Multisim control settings, units of measurement, version of Multisim used by the student, Multisim oscilloscope waveforms, and applicable results. Also answer all questions posed in each section of the Multisim portion of the experiment. Label and define the purpose of each section of part B as you answer related questions in your formal report.

7) Detailed discussion of the hardware portion of the experiment including applicable circuit diagrams with power supplies, test equipment settings including the power supply settings, units of measurement, list of test equipment used for experiment, oscilloscope waveforms, and measured results. Also answer all questions posed in each section of the hardware portion of the experiment. Label and define the purpose of each section of part C as you answer related questions in your formal report.

8) Conclusions should include a comparison of the initial predicted oscillator frequency versus various Multisim results versus various experimental results. The conclusions should also discuss items 5, 6, and 7 listed in the theory of operation. Also discuss what was learned, difficulties, problems, etc.

9) An outstanding lab report (meaning outstanding technical content with good presentation) will be rewarded with up to 40 lab points, rather than the normal maximum of 30 points for this formal report.

My idea for an outstanding report would be to add a section to the theory of operation explaining mathematically how the 741 op amp, with its limited โUnity Gain Bandwidthโ parameter of, say 800kHz, affects the output frequency of this oscillator. This would include the following steps:

1) Use of the op amp gain/phase equation in the inverting mode to predict the value of ๐๐๐ at the inverting input of the 741 op amp. For this ๐๐๐ calculation, use the actual measured oscillator frequency. See the Lab 14 notes titled โUsing the Magnitude and Phase Angle of a 741 Inverting Op Amp Circuit to Predict the Oscillator Frequencyโfor more details.

2) Revise the RREF gain and phase analysis of the Third Order Low Pass Filter at the measured output frequency. Use the measured component values and with the op amp ๐๐๐ signal applied to the bottom of ๐4. (Rather than zero volts as was done in the Initial analysis of the Third Order Low Pass Filter.) This is a more accurate calculation for the value of ๐ฝ at the frequency of oscillation.

3) Calculate the inverting op amp circuit gain and phase while including the effect of๐๐๐. This calculation would be from the input of R4 to the output of the op amp. This is a more accurate calculation for the value of ๐ด at the frequency of oscillation.

4) The phase angle for the ๐ด๐ฝ product should now be very close to 0ยฐ (or -360ยฐ) at the measured oscillator frequency.

The student should be able to calculate the C2.1 oscillator frequency within about 1% by taking the above factors into account and using an iterative solution.

The process of correlating theory with practice will enhance your understanding of the circuit, and is essential to becoming a good RF or analog circuit designer.

Check that all students, of a given team, have all the test data, oscilloscope pictures, and the cell phone pictures required to write up all parts of their individual formal report.

Carefully label (i.e. the section of the experiment, test point location for each scope plot, v/div for each scope plot, usec/div, cursor data, etc) all your scope pictures as you take them, otherwise, it is easy to mix them up when you write up your formal report.

DO NOT PLAGIARIZE

INCLUDE EVERYTHING LISTED ABOVE

Studypool has helped 1,244,100 students
flag Report DMCA

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors