Saturday, June 1, 2013

RL, RC, and RLC Circuits

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The primary goal of this assignment is to quickly review what you already know about capacitors, inductors, and AC circuits and to extend your new circuit analysis skills to cover sinusoidal signals. The assignment draws from Chapters 6-10 of your text. You will also learn how to use Spice to simulate time dependent effects and to analyze frequency dependent response in AC circuits.

We increasingly conceive of circuits in terms of terminal behavior. That is, we label the source signal $v_{in}$ and the potential difference between some other pair of nodes $v_{out}$, and we think of the network of devices between the input and output terminals to be a filter that transforms $v_{in}$ into $v_{out}$.

Measuring C and L : Natural/Step Response

  1. Experiment

    Figure 5: Series RC and RL circuits.
    (a) \includegraphics{lab2-rc.eps} (b) \includegraphics{lab2-rl.eps}
    You have been provided with a breadboard, a resistor, a capacitor, an inductor, a function generator, and an oscilloscope. First, measure the resistance of the resistor by any means you like. Then construct the series LC and RC circuits shown in Figure 5. Drive the circuits with a square wave signal from the function generator.
    For each circuit, pick a good frequency, and find the capacitance/inductance by measuring the time constant of the response with the oscilloscope. Explain what made your choice of frequency ``good.'' Use one channel of the oscilloscope to measure $v_{in}$, and trigger on that channel. Measure $v_{out}$ with the other channel to look at the natural and step response of the circuit. Measuring the half time and converting to the time constant may be easier than trying to measure the time constant directly. Use the $\times10$ time base magnification feature of the oscilloscope to improve the precision of your time measurements. Give the uncertainties in your results.
  2. Simulation
    Use a transient analysis (.TRAN analysis with a PULSE signal) to simulate one of your experiments of Section 1. You can measure the voltage levels and rise and fall times of the function generator with the oscilloscope to (possibly) produce a more realistic simulation. Compare and contrast the voltage plot from your simulation with your measured oscilloscope traces.


Measuring L : Sinusoidal Response

  1. Exercise
    Consider the series RL circuit of Figure 5 driven by a sinusoidal voltage. Using your favorite method, derive an expression for the ratio $S = \left\vert \frac{V_{out}}{V_{in}} \right\vert$ of the magnitude of the voltage across the capacitor/inductor to the magnitude of the voltage across the voltage source as a function of frequency. To model a real inductor, include an internal resistance $R_L$ in series with the inductor, and include this resistor in the voltage drop across the inductor. Use your expression for $S$ to find an expression for the inductance $L$ in terms of $S$, $R$, $R_L$, and $\omega$. It should be noted that there is nothing about this general approach that is specific to inductors. It works just as well for measuring capacitance.
  2. Simulation
    Use a small signal AC analysis to generate a plot of $S$ vs. frequency for the RL circuit you analyzed in the Exercise above. Include the internal resistance of the inductor. If you use a source amplitude of 1 V, then values of the voltage across the inductor and its internal resistance (with units removed) are also values of $S$. Set up the simulation, guessing values for $L$ and $R_L$. You will refine it as you work through the following experiment.
  3. Experiment
    First, measure the internal resistance $R_L$ of the inductor. Give an uncertainty with your result. (Presumably, you know the resistance $R$ of the resistor you are using. If not, measure that, too.) Then, use the function generator to drive your series RL circuit with a sinusoidal wave form. Use one channel of the oscilloscope to measure $V_{in}$ and the other to measure $V_{out}$. Compare $V_{in}$ and $V_{out}$ at various frequencies. How well do your measurements compare with your theoretical calculation and spice simulation? Remember that $\omega = 2 \pi f$. Use a measurement of $S$ in your theoretical calculation to find $L$. Report an uncertainty with your result. Be sure to measure your $S$ value at good frequency. What does ``good'' mean here?2 How far off were you in your measurement of $L$ of Section 1? Can this discrepancy be attributed to ignoring the internal resistance of the inductor? Explain.

RLC Resonance


  1. Exercise
    At resonance, the capacitive and inductive reactances cancel, the phase angle $\phi$ is 0, and the voltage across the resistor is maximal (as are the current and $S$). Derive an expression for the resonant frequency $f_o$ of a series RLC circuit in terms of $L$ and $C$.
  2. Simulation
    Figure 6: Series RLC circuit.
    \includegraphics{lab2-rlc.eps}
    Set up an AC analysis of the circuit shown in Figure 6. Produce plots of $S$ and $\phi$ as a function of frequency, and note the predicted resonant frequency. Investigate and report on the dependence of the plots and the resonant frequency on the resistance $R$, and explain.
  3. Experiment
  4. Set up the circuit shown in Figure 6. Measure the voltage across the source $V_{in}$ with one channel the oscilloscope and $V_{out}$ with the other. Dial up X-Y mode on the time base knob to plot $V_{in}$ vs. $V_{out}$. At resonance, these two signals are in phase. In X-Y mode, this looks like a straight line. At any other frequency, the oscilloscope trace resembles an ellipse.
  5. In time base mode, you can look for the frequency at which the voltage across the resistor $V_{out}$ is maximal. Report an uncertainty in your result. Compare your result with your theoretical prediction using your best measured values of $C$ and $L$.

RC Low/Hi/Band Pass Filters

Figure 7: Signal filters.
(a) \includegraphics{lab2-lpf.eps} (b) \includegraphics{lab2-hpf.eps}
(c) \includegraphics{lab2-bpf.eps}

  1. Exercises
    The networks of resistors and capacitors shown in Figure 7 are called sometimes called filters. One is a low pass filter, meaning that it transmits low frequency signals to its output terminals favorably. Another is a high pass filter, transmitting high frequency signals favorably. There is also a band pass filter that transmits signals in a frequency band. Draw the low frequency and high frequency equivalents of these networks, and also for the series RL and RLC circuits you considered in Sections 2.2 and 2.3. On this basis, classify each of these networks as a low pass, high pass, or band pass filter. Derive general expressions for $S$ and the phase angle for the band pass filter in Figure 7 as a function of the resistances, capacitances, and signal frequency. Use this expression to produce a plot of $S$ vs. frequency. Predict the cutoff frequencies (3 dB points) of the filter.
  2. Simulation
    Produce $S$ vs. frequency and phase angle vs. frequency plots for the filter circuits of Figure 7 using small signal AC analyses. Does your simulation agree with your calculation of Section 1?
  3. Experiment
    Set up the band pass filter, driving it with a sinusoidal wave form from the the function generator. Measure $S$ and the phase angle for the circuit over a frequency range 100 Hz - 1 MHz. Also measure the cutoff frequencies of the filter. Produce plots comparing your measured $S$ and phase values with your theoretical calculation / spice analysis.

AC to DC Conversion


Half Wave Rectifier


Figure 8: A half wave rectifier.
\includegraphics{lab2-halfa.eps}
An important step in converting the 120 V AC power supplied by a standard wall outlet into 5 V DC power for your computer is voltage reduction. You have been provided with a variable transformer that can step the 120 V signal from the wall socket down to 7 V across the two red terminals, or 3.5 V across either black/red pair. In the following exercises, use the 7 V option.
In order to convert an AC current that spends equal amounts of time flowing in opposite directions into a DC current flowing in only one direction, we need a device that responds differently to applied voltages of different polarities. The standard device used for this purpose in modern electronics is the semiconductor diode. The circuit of Figure 8 is called a half wave rectifier, because the diode simply blocks the segments of the signal with negative polarity.
Set up the circuit of Figure 8 using a 10 k$\Omega $ load resistor, and compare the voltage across the load resistor to the input signal. Are you surprised by the peak voltage across the load? Make note of this. Also compare your observations with a Spice simulation (a transient analysis). If they agree, just hand in a graph of the simulation. Otherwise, also sketch your observed oscilloscope traces.
In your Spice simulation, do not explicitly include the transformer. Simply treat it as a sinusoidal voltage source. In order to use diodes in a circuit, include the following statement somewhere in your circuit file.
.MODEL D1 D
This tells Spice that device type D1 is a generic silicon diode. Then, diodes are specified with the syntax
[Name] [+] [-] D1
where the [Name] field must begin with a D, and current is allowed to flow from the [+] node to the [-] node.

Figure 9: A half wave rectifier with a filter.
\includegraphics{lab2-halfb.eps}
You should have found that the load is not being driven by a particularly steady DC signal, although the current always moves in one direction. (If you did not, perhaps we should look at what you discovered together.) The next step is to use a large capacitor to filter out most of the unwanted variation or ripple in the signal across the load resistor. The circuit of Figure 9 includes a filter capacitor in parallel with the load. Since the diode does not allow current to flow back through to the transformer, the capacitor discharges through the load resistor with a time constant $R_{Load}C$. Assuming $R_{Load}C >> \frac{1}{f}$ derive an expression for the peak to peak magnitude of the ripple in the output signal in terms of $C$ and $R_{Load}$. Assuming a 10 k$\Omega $ load, find the capacitances you would need to use to reduce the ripple to 10% and 1% of the DC output voltage.
Using the best filter capacitor you can find 3, measure the DC level and the magnitude of the ripple across the load. Use the DC (direct coupling) mode of the oscilloscope or a DVM to measure the the DC level, and use the AC (active coupling) mode of the oscilloscope to remove the DC component of the signal and zoom in on the ripple. Then change the load resistance, and determine whether the ripple changes as expected.

Full Wave Rectifier


Figure 10: A full wave rectifier with a filter.
\includegraphics{lab2-bridgeb.eps}
The configuration of four diodes in the circuit shown in Figure 10 is called a bridge rectifier. It is also a full wave rectifier, because it does not throw away any part of the input signal. Consequently, the filter for this circuit requires roughly half the time constant of the half wave rectifier.
Build the circuit of Figure 10 using a 10 k$\Omega $ load resistor and a suitable filter capacitor, and compare your observation of the voltage across the load resistor, with and without the filter capacitor, with Spice simulations. 4 If they agree, just hand in a graph of the simulation. Otherwise, also sketch your observed oscilloscope trace. Measure the magnitude of the ripple, change the load resistance, and determine whether the ripple changes as expected.

A ``Real'' DC Power Supply


Figure 11: A full wave rectifier with a voltage regulator.
\includegraphics{lab2-bridgec.eps}
You have been given an open power supply constructed as shown in Figure 11. This supply is a full wave rectifier with a filter and a device called a voltage regulator that removes the remaining ripple. This device is an integrated circuit (IC) that contains lots of very small resistors, capacitors, diodes and transistors manufactured on a single silicon wafer. For the moment, we will treat it as a ``black box.'' Its function is to deliver or accept current as needed to maintain a constant output voltage. The 78XX series of voltage regulator maintains an output voltage of XX Volts. Your supply has a 7806, so the DC output voltage should be close to 6 V.
Poke around carefully with the oscilloscope. The filter capacitor is physically the largest capacitor. It also has the largest capacitance. (The other two capacitors, labelled $C_{in}$ and $C_{out}$ are recommended by the manufacturer of the voltage regulator.) Measure the DC and ripple voltages both at the output and at the filter capacitor.

AC Circuit Analysis Exercises


Figure 12: Various filter circuits.
(a) \includegraphics{lab2-filter1.eps} (b) \includegraphics{lab2-filter2.eps}
(c) \includegraphics{lab2-filter3.eps}
For each circuit shown in Figure 12:

  • Apply whichever method you wish (Node Voltage, Mesh Current, ``voltage divider thinking'') to derive expressions for $S = \left\vert \frac{V_{out}}{V_{in}} \right\vert$ and the phase angle.
  • Check your calculations against a Spice simulation.
  • Find the Thevenin equivalent voltage and impedance for the output terminals indicated.
  • Comment on the function of the circuit as a filter converting $V_{in}$ into $V_{out}$. 

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