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TRANSIENT BEHAVIOR OF R-C SERIES CIRCUITS

In this experiment we shall use the cathode ray oscilloscope to study the transient behavior of an electrical circuit containing capacitance and resistance. We shall use a square wave generator to supply an instantaneous switching between two voltages.

The capacitor is a device for storing electric charge. As a switch is closed and the voltage builds up (nearly instantaneous for the source) we can calculate the time required for the charge to build up in the capacitor. In particular, the source voltage looks something like:

Figure 1: Source voltage from audio-generator

  1. Set the squarewave generator at about 50 hertz. Apply this signal to the input of Channel 1 (or X) of the oscilloscope.

  2. Draw this signal below so you have a reference source for comparison later. Be sure to label it clearly:

  3. Be sure you know the value of the sweep frequency for the squarewave. You should expand the scale so you will have just one or two steps. The bigger the picture, the easier to make measurements.

  4. Disconnect the signal generator from the oscilloscope and make the following connections: Let R be 100 omega and C 1 µf.

    Figure 2: Schematic of Experimental Circuit. A simple R and C are added in series to the voltage source. The Oscilloscope is used as a voltmeter to measure voltage V across the capacitor C.

  5. The circuit is now a series circuit with a resistor, a capacitor and the oscilloscope acting as a voltmeter. The signal generator acts as an emf. There are two values for V, depending on time; i.e. V = V(t).

    The time constant tau is the time after the voltage has switched from value to another, V(t) will have gone all but 1/e = 0.368 of the way from one value, Vo to the final value, Vf ; i.e.
    V(t) = .368 Vo (1)
    Be sure the oscilloscope is in the DC mode.

  6. Draw carefully the graph of the capacitor voltage. This can be done using Excel and plotting voltage values on the Y-Axis and time values on the X-Axis, just like you see on the oscilloscope. From this, you can directly "read" the value of the time constant right off the oscilloscope. It is the time (in cm) when the value drops from a starting point to .368 of that value. The most accurate way, however, to determine tau is from the slope of a semilogarithmic graph. To do this, use semi-log graph paper. This is available from your "friendly" bookstore.

  7. Looking closely at the oscilloscope, the voltage waveform will look something like:

    Figure 3: Voltage waveforms for both charging and discharging of the capacitor. You will select to focus on the discharge case only. The charging case is a bit more complex mathematically and gives us no more information.

  8. Using the upper potential as a reference, V = Vo we get:
    V(t) = Voe- t/tau (2)
    for the discharging case. Set your oscilloscope to include only the discharge.

  9. Take the logarithm of both sides:
    ln V(t) = ln Vo + ln e- t/tau (3)
    This reduces to:
    ln V(t) = ln Vo - t/tau ln e (4)


  10. Since ln e = 1, we can write this as a simple linear equation:
    ln V(t) = ln Vo + (-1/tau) t (5)
    This can be plotted on semi-log graph paper to yield the time constant as a function of the slope of the straight line.

    Plot t on the linear axis (x-axis) and ln V(t) on the log-scale, the y-axis. Ln Vo is a constant, or the y-intercept. On the semi-log graph paper, the Y-scale automatically takes the natural log so you really just plot the voltages. You may also do this through Linear Regression on the computer to determine the slope. The slope is -1/tau.

  11. Find the time constant first from the oscilloscope, secondly by using the semi-log graph paper, and finally, calculate what it should be using tau = RC. Compare your results, (i.e., find % differences). Remember, there are 3 ways to obtain tau:
    1. Value of tau when V = (1/e) Vo
    2. tau = RC
    3. Slope of curve on semi-log graph paper.
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