Address: Physics Department, Weber State University, Ogden, UT 84408-2508

Phone: 801-626-6202 Fax: 801-626-7445

E-mail: rjeffery@weber.edu

Advanced Laboratory and/or Lecture Demonstration Apparatus

Apparatus Title: **Alternating Current Analyzer for Lecture
or Lab**

Abstract: Voltage drops may be displayed and phase relationships shown between all components in ac circuits with a simple apparatus consisting of differencing op amps. Any combination of series or parallel LCR circuits may be quickly configured to demonstrate a variety of effects, including resonance. The apparatus includes the op amp circuits, a built-in power supply, and optional function generator.

DESCRIPTION:

Alternating current (ac) circuit behavior is difficult for beginning students to visualize. This is partly because of the complex relationship between voltage drops and phase differences for inductors and capacitors relative to the current. It is tempting for instructors in introductory courses to omit or pass lightly over this subject because of the lack of simple demonstration equipment that can illustrate the effects. Since a common ground connection is required for each input to the oscilloscope, it is difficult to view all parts of the circuit at once. This problem is solved with op amps configured to subtract the voltages between two points in the circuit. Thus the differences between the voltages on each side of the various circuit elements give the drops that are desired to be displayed. Using the drop across the series resistor as the input to the oscilloscope trigger establishes the current as the phase reference. Other voltage drops used as inputs to the oscilloscope then show the desired time relationships.

The circuit in this apparatus uses a quad op amp chip for the differencing circuits and an inexpensive built-in power supply. A function generator may be external to the apparatus, or a single-chip generator may be built in. The capacitor, inductor, and resistor plug into banana jacks, making it possible to select series or parallel combinations of the three components, or other combinations. The scope outputs are through BNC connectors.

The apparatus has a variety of applications at different levels in the curriculum. For lecture demonstration, and in ac lab exercises, it can be used to illustrate the fact that the voltage drops across R, L, and C are not in phase, but are different functions (linear, derivative, and anti-derivative, respectively) of the current. The frequency dependence of the magnitudes of the L and C voltage drops, proportional to the inductive and capacitive reactances, is readily demonstrated. The fact that the rms voltage drops do not add to equal the driving voltage may be illustrated with the addition of an ac voltmeter. That the instantaneous voltage drops do add correctly can be seen on the oscilloscope screen, and demonstrate that Kirchhoff's voltage rule still applies. Resonance is demonstrated by the maxima in the R voltage drop, and also by the equality of the L and C drops, at resonance.

In parallel LC (tank) circuits the transition from inductive to capacitive behavior, with corresponding change of the phase shift from +90 to -90 degrees, as the frequency sweeps through resonance, is quite dramatic. Also, the fact that current has a minimum at resonance is seen in the drop across the resistor.

Other effects that can be explored in more advanced laboratories include varying the Q of the circuit by substituting different values of series resistance, and finding the maxima in the L and C voltage drops when R is less than a critical value (or Q is higher than a critical value). Finding that critical value, and solving mathematically for the maxima in the L and C drops make good exercises in complex analysis for the more advanced students. The relationships may then be verified by finding the maxima as functions of R (or Q). Another feature that can be explored is the effect of the finite resistance in the inductor. This results in the phase difference between the drop across the inductor and the current being less than 90 degrees.

In short, this apparatus has a wide range of application in illustrating ac circuit behavior, and can be a valuable tool for improving student understanding of ac circuits.