Many systems can generate self-sustained oscillations when a source of energy is provided. In this experiment, the self-sustined oscillator is electronic, with a circuit whose principal element is an op amp and whose power comes from plus and minus DC voltage supplies.
Major ideas
1. Oscillation Onset and Frequency: Oscillation occurs because of a combination of gain and positive feedback in an electronic amplifier circuit. An oscillator circuit's behavior be interpreted in terms of an idea called the Barkhausen criterion for oscillation, but it can also be understood in terms of a second-order differential equation describing the time dependence of the output voltage. Linear analysis using Kirhoff's laws and a model of ideal op amp behavior predicts the required gain for oscillation to begin and the expected frequency at the onset of oscillation. The gain and frequency are expressed in terms of values of resistors and capacitors that make up feedback networks in the circuit. This particular circuit is called the Wien Bridge Oscillator and is a modern version of the circuit used in Hewlett Packard's first product!
2. Hopf Bifurcation to Limit Cycle Dynamics in the Modified van der Pol Equation: Analysis of the nonlinear gain-limiting element (the light bulb filament) leads to a model called the Rayleigh - van der Pol equation and this equation predicts that oscillation begins as a Hopf bifurcation. A Hopf bifurcation is a transition at which a steady state (no oscillation) solution to the governing equations becomes unstable and is replaced by a limit cycle solution whose amplitude grows proportional to the square root of a primary dimensionless control parameter. This parameter is defined to be zero at the onset of oscillation and moves the system into oscillation when the parameter is increased to positive values. A second control parameter - one with the dimensions of volts - sets the scale of the oscillation amplitude. Both parameters can be related to values of actual circuit component characteristics.
3. Sources of nonlinearity. Different physical systems have different sources of nonlinearity. In the oscillator circuit, the nonlinearity occurs because a parameter - the resistance of one of the feedback resistors that actually consists of a light-bulb filament - depends on the oscillator amplitude. A physical model of the light-bulb filament and its heat transfer characteristics is needed to understand in detail how the amplitude affects filament resistance. This voltage oscillation amplitude dependence shows up as a nonlinear term in the equations for the oscillator dynamics.
NOTE: there is actually a more complete system of three equations (rather than the van der Pol equation) governing this oscillator that includes the dynamics of the heating and cooling of the tungsten filament of the amplitude-stabilizing light bulb used in the circuit. This is important to properly model initial transient behavior in the circuit: the circuit exhibits a phenomenon called "squegging" in which the oscillation amplitude rises and disappears as pulses. These pulses occur because the time for the filament to respond to energy dissipation is much longer than the time for initial oscillations to grow and the system overshoots its equilibrium amplitude. The pulses eventually even out into steady oscillation and the filament settles into small variations around an equilibrium temperature.
Major equipment
1. Operational amplifier integrated circuit: the time-honored 741.
2. Solderless breadboard method of circuit construction. Consider advantages and limitations, in terms of (a) ease of construction and modification and (b) behavior (or misbehavior) at large voltages, currents, or frequencies. What are alternative methods for prototyping circuits?
3. Digital sampling oscilloscope with capability to do fast Fourier transforms of the data to create a power spectrum plot.
(You should make a clear diagram and list in your lab notebook of all of the equipment, components, and supplies used in the experiment. Any of these is fair game for discussion during your review.)
Data analysis
0. How well do the frequency and variable feedback resistance agree with predicted values? (You can fold these values into your presentation of ideas 1 and 2.)
(Choose 1 of the following per person in your team.)
1. How does the oscillation amplitude vary with the "control parameter" (the ratio of the feedback resistors minus 2)?
2. How does the oscillation frequency vary with the "control parameter" (the ratio of the feedback resistors minus 2)?
3. How do various measures of the pulsating oscillation near onset vary with the "control parameter" (the ratio of the feedback resistors minus 2)?
4. How does "harmonic distortion" of the sine wave vary with the "control parameter" (the ratio of the feedback resistors minus 2) and with amplitude? (Do higher harmonics appear in power spectra and, if so, how do the ratios of peak heights of higher harmonics relative to the fundamental peak vary with control parameter or amplitude?)_
First learn about op amps using the document Signal Conditioning with Op Amps. (1) Do the first exercise on voltage divider analysis.(2) Next reproduce in your own writing the analysis of the op amp circuit under the heading "The Hard Way". Add any intermediate steps or questions that you wish. (3) Finally, read, understand, and if possible execute the Python code under the heading "Calculating values for the example circuit". Do your own hand calculation of what the code is computing using the resistor values shown in the circuit diagram of page 7. The output of this code predicts the output of the op amp circuit that you will build and test in the first week of this lab. This is the circuit that appears on page 7, both as a schematic and as a breadboard layout.
A useful reference for learning about op amp circuits is:
Paul Scherz and Simon Monk, Practical Electronics for Inventors, 4th ed. (McHraw-Hill Education TAB, 2016), isbn 978-1259587542. See chapter 8.
In week 1 you should build the inverting op-amp circuit in the document Signal Conditioning with Op Amps and verify that the output is the level expected from the known input from the voltage divider and the gain set by the resistors in the circuit. Next set up a function generator with an amplitude of a few volts and a frequency of around 1kHz, verified on an oscilloscope. Put this into the voltage divider instead of +12V DC and look at the opamp output using a second input and trace on the oscilloscope. Confirm whether or not the level and phase of the output signal is what is expected and explain why in your lab notebook. Then ramp up the frequency and see what happens to the output. More down what you observe and try to explain it (hint: op amps are characterized by a quantity called "gain-bandwidth product" - find a reference online or otherwise and understand what this implies about the frequency response of the op amp).
In weeks 2 and 3 you will work through the derivation and circuit design in the Jupyter notebook. Download the zip file Wien_Bridge_Oscillator_Theory_v9_2.zip that contains the Jupyter notebook Wien_Bridge_Oscillator_Theory_v9_2.ipynb and a folder of figures that must be placed in the same directory as the Jupyter notebook. For immediate viewing without being able to execute the embedded code, a pdf version of the notebook is Wien_Bridge_Oscillator_Theory_v9_2.pdf. (The zip file also contains this pdf as well as the experiment guide listed below.)
A detailed procedure for the experiment is in the document Wien Bridge Oscillator Experiment.
NOTE: The adjustable feedback resistor is a Leeds & Northrup decade resistance box with exquisite precision (0.1 ohm). Treat this as the precision instrument that it is!
References
Enns, R. H. and G. C. McGuire (2001), Nonlinear Physics With Mathematica For Scientists and Engineers (Birkhauser).
Casaleiro, J., Oliveira, L. B., Pinto, A. C., "Van der Pol Approximation Applied to Wien Oscillators," Procedia Technology 17 (2014) 335-342.
Lerner, L., "The dynamics of a stabilised Wien bridge oscillator," Eur. J. Phys v. 37 (2016) 065807 (22p).