# PHY 124 Lab 5 - AC circuits

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## Goals

The first purpose of this laboratory is to observe voltages as a function of time in an RC circuit and compare it to its expected time behavior. In the second part the resonance frequency of a series RLC circuit is measured and compared with the expected value.

## Equipment

• 1 AC Signal Generator
• 1 Oscilloscope
• 1 Board with resistors, a capacitor and an inductor
• 7 test leads (wires) with banana plugs (4 red, 3 black)
• 6 alligator clips with banana sockets

Two of the component boards used are shown below, note that you need to be careful as not all boards have the different components in the same place!

In this lab, you use the oscilloscope to study some properties of alternating current (AC) circuits which involve capacitors and inductors. In lab 3 you worked with simpler direct current (DC) components, specifically, resistors. The important difference between the two types of components is that the behaviors of the AC components components depend on the rate of change of the input voltage/current, i.e., the frequency f of the driving signal (SV8, Chap. 21). In this experiment the AC voltage is supplied by the AC signal generator.

## Part I: RC Circuits

Your goal is to measure the capacitive time constant $\tau_C$ in an RC circuit and compare it to the predicted value of $\tau_{C}=RC$.

One way to charge and discharge a series RC circuit (see SV8, Chap. 18.5 for charging and discharging the capacitor) is to use a DC source of electrical potential $V_{0}$ such as a battery, and a switch for connecting and bypassing the battery . This power source is then placed in series with a resistor and capacitor which (for a series RC circuit) are also in series with each other (see Fig 3a below). In this experiment, we will use a different approach; the voltage will be generated using the square wave output from the AC signal generator (see Fig 3b below). As you saw in Lab 2 the signal you would produce manually if you used the battery and switch would be the same as a square wave switching between 0 V and $V_{B}$. One advantage of using the signal generator is that it can switch the voltage on and off much faster and more reproducibly than you could with your hand.

If you think about Fig 3 (a) at time t = 0, we can assume that we begin with no charge difference on the capacitor $C$ (C is measured in Farads) and that the switch is then set to connect the battery to the circuit. The voltage $V_{C}$ across the capacitor is then given by $V_{C} = \frac{Q}{C}$, so that at t = 0 we know $V_{C} = 0$. As the charge $Q$ builds up on the capacitor in time, $V_{C}$ increases until it equals $V_{B}$.

The voltage across the capacitor when the capacitor is charging is given in SV8, Chap. 18.5.

There are two important things to notice about this formula. First when t = 0 the exponential factor becomes 1. You should notice that in the formula above this gives $V_{C} = 0$. Second, notice that if the capacitor starts out with no charge on it then as t approaches infinity the exponential factor goes to 0. This means that if the capacitor is left charging long enough that its voltage will eventually equal to $V_{0}$. In the example above $V_{0}=V_{B}$.

The voltage drop $V_{R}$ across the resistor must satisfy $V_{B}=V_{C}+V_{R}$. Using this formula and equation 5.1, you can eliminate the variable $V_{C}$ and derive what $V_{R}$ is equal to in terms of $V_{B}$ and $e^{-t/\tau_{C}}$ which gives us equation 5.2.

As with $V_{C}$ you can look at $V_{R}$ for the cases with t = 0 and t approaching infinity.

In our experiment, unlike in the example above, the AC generator output voltage changes between and ($-V_{0}$ and $+V_{0}$ (not between 0 and $+V_{0}$ ). The magnitude of the voltage change is thus $2V_{0}$ (not $V_{0}$)!

You can see that for the capacitor voltage in equation 5.1 that the curve approaches $V_{0}$ exponentially (as demonstrated in the formula for capacitor voltage) when it rises from 0 to $V_{0}$. In your lab, it will do the same thing when it rises from $–V_{0}$ to $+V_{0}$. When charging from $+V_{0}$ to –$V_{0}$, the curve approaches –$V_{0}$ exponentially (ie looks like a falling exponential). For this lab you will look at the capacitor charging from $–V_{0}$ to $+V_{0}$.

Also, instead of voltmeters we will be using the oscilloscope to look at the voltage coming from the signal generator, the voltage across the capacitor, and the voltage across the resistor. The diagrams that represent the two experimental configurations, the first to measure $V_{C}$ and the second to measure $V_{R}$ are shown below.

In the Lab 5 pretest you need to calculate the result predicted for $\tau_C$ from $\tau_C = RC$. Since the resistor and capacitor in your circuit may be slightly inaccurate, use the values for R (with a 10% error) and C (with a 10% error) in your setup to calculate the predicted time constant and its error using equation (E.7) in “Uncertainty Errors and Graphs” (“UEG”). You should make a note of the value you get in the Lab 5 pretest as you will be comparing it to the value you measure in the lab.

### Measuring the capacitor voltage

In this part of the lab you will observe the voltage across the capacitor VC (t) with CH1 of the oscilloscope and the AC generator output voltage VAC(t) with Ch2. Begin your set up by creating a series RC circuit following the diagram in Fig. 4a shown above. In your test setup, $C = 10^{-7}$ F. There are several resistors available to be used in series. Their values are shown in Fig. 2. First, connect the resistor and capacitor in series. For this part you will use the 1 kΩ resistor. Make sure that you are using this resistor labeled with this value, which is not in the same place on all the boards. Connect Ch. 2 of the oscilloscope in parallel with the signal generator. Connect the positive terminal of Ch. 1 of the oscilloscope to the point between the resistor and capacitor. Connect the negative terminal of Ch. 1 of the oscilloscope to the negative terminal of the signal generator. Now connect the positive terminal of the signal generator to the end of the resistor that is not connected to the capacitor. After that, connect the negative terminal of the signal generator to the end of the capacitor that is not connected to the resistor. When you are finished your set up should look like Fig. 4a. Make sure the ground connections are made exactly as given in the diagram 4a. It is important that all 3 grounds, ie the function generator ground, the CH1 ground and the CH2 ground are all connected together. The p figure below shows you what Fig 4a should look like in real life. Record the values for the resistance and capacitance on your worksheet.

Once you have connected your circuit use the AC generator output “MAIN OUT LO” with “AMPLITUDE” turned fully clockwise.

Dial a frequency of ~ 500-1000Hz:

• POWER (button) pushed in
• FREQUENCY Hz (big knob): ~ 5-10
• MULT (pushbutton): 100
• VERN (knob): stripe/O oriented straight up (toward O below VERN)
• pushbutton below SQUARE-WAVE shaped figure pushed in

Set the oscilloscope to:

• POWER (red button): pushed in to turn on oscilloscope
• X-Y (button): out (not pushed in)
• COUPLING and SOURCE (switches): AC and CH2
• Both CH1 And CH2 inputs (switches): DC
• Both VAR (red knobs): CAL’D (full clockwise)
• VERT MODE (switch): CH2
• Both VOLT/DIV (knobs): ~0.5 (such that you see the square wave)
• VAR SWEEP (knob): CAL’D (full clockwise)
• TIME/DIV (knob): ~ 0.2 ms (such that you see at least one period of the square wave)
• HOLD OFF (knob): pushed in and turned full counter-clockwise toward MIN
• TRIG LEVEL (knob): pushed in

Adjust the TRIG LEVEL so that you see a stable picture on the oscilloscope.

Now set the oscilloscope to VERT MODE (switch)to DUAL.

You should see both voltage signals $V_{AC}(t)$ and $V_{C}(t)$. Using the VERTICAL POSITION buttons center your voltage signals on the vertical center of the screen. Record the settings for FREQUENCY, VOLT/DIV for CH1 and CH2 and TIME/DIV on your worksheet

Sketch the observed pattern of VAC(t)and VC(t) on your worksheet. Your sketch should try to be as accurate as possible and must include axes scales and labels.

Next you are going to analyze the curve to obtain $\tau_C$. First, look back at Eq. 5.1 describing the capacitor voltage. Notice that when $t = \tau_C$ the equation becomes $V_{C} = V_{0}(1-e^{-1})$. When you put in the value of e, this will give you $V_{C} = 0.63\times V_{0}$. Note however that as you are going from $–V_{0}$ to $V_{0}$ you need to find the point that is $0.63\times 2V_{0}$ above $-V_{0}$. Find this voltage on your curve and then find the time at which is occurs. This time will be your experimental value for $\tau_{C}$ which you should record on your worksheet.

### Measuring the resistor voltage

To measure $V_{R}$ you need to exchange the positions of the resistor and capacitor and connect Channel 1 of the oscilloscope parallel to the resistor so that you can monitor $V_{R}$. Your set up should look like as Fig. 4b. Again, make sure the ground connections are made exactly as given in the diagram 4b. It is important that all 3 grounds, ie the function generator ground, the CH1 ground and the CH2 ground are all connected together

Observe the voltages $V_{AC}(t)$ and $V_{R}(t)$ and sketch them on your worksheet.

You will notice that $V_{R}(t)$ approaches zero (and NOT $+V_{0}$ or $–V_{0}$ ). This follows from $V_{0}=V_{C}+V_{R}$, since $V_{C}$ approaches $V_{0}$, $V_{R}$ approaches zero. You should also notice that the initial $V_{R}$ is ($2V_{0}$) above 0, since the voltage change is ($2V_{0}$) and the initial current is $2V_{0}/R$.

You are now going to obtain a measurement of $\tau_C$ from the voltage across the resistor. As before, adjust the TIME/DIV and VOLT/DIV settings so that you have one decaying exponential curve on the screen. Sketch this curve on your worksheet.

At $t =\tau_{C}$ , $V_{R}= e^{-1}\times 2V_{0}$, which gives $V_{R}=0.37\times 2V_{0}$. Find the point on your curve where this value occurs for $V_{R}$. Then find the time associated with this voltage. This will be your measured time constant, $\tau_{C}$ from this part of the experiment. Make sure you estimate the error for your value. Record the value and error on your worksheet.

## Part II. Resonant AC circuits

Your goal is to measure the resonance frequency of an in series RLC circuit and compare it to the predicted frequency you find by setting the reactance of the circuit to zero, i.e. minimizing the impedance and hence maximizing the current for a given voltage.

The following exercise is in the MapleTA Lab 5 Pretest and the resulting equation is important for this part of the lab.

You want to find the resonance frequency $f_{0}$, which is the frequency which minimizes the impedance $Z$ (see SV8, Chap. 21.4 and 21.6)) of the in series RLC circuit by causing (XL - XC), the difference between the inductive reactancee XL and the capacitive reactance XC, to be zero (see SV8, Chap. 21.6, Eq. [21.18]). From SV8, Eq. [21.19], enter the equation for reactance (use $f_{0}$ to denote the frequency in your equation). Then solve for $f_{0}$, the resonance frequency.

Calculate the resonance frequency $f_{0}$ from the equation and get its value, using the values of the inductance L and the capacitance C in your setup (see Fig 2). As before, since you cannot count on these values being completely accurate, propagate the errors of L and C ( assume 10% for both) into the error for $f_{0}$ using expressions (E.7) and (E.8) in “UEG”.

Make a note of the values you get, you will need them to compare to your measured values.

### Procedure for measuring the resonance frequency

As the resonance frequency $f_{0}$ maximizes the current $I = V/Z$ it also maximizes the voltage across the resistor $R$ which is $V_{R} = IR$ . To find $f_{0}$ you will vary the frequency of the AC generator until you observe maximum voltage $V_{R}$.

Wire the circuit as given in the diagram below. Begin with a series “loop” with the signal generator, the resistor, the capacitor, and the inductor Make sure you connect the low voltage side of the resistor to the ground (black) of the generator and the oscilloscope. Make sure you wire the capacitor, inductor and the resistor in series exactly as shown in the diagram. Use the 100Ω resistor (see Fig 2 above) for this part. You will be observing the voltage across the resistor so attach Channel 1 of the oscilloscope parallel to the resistor. You will not need to look at the voltage from the signal generator, so disconnect it from Channel 2.

Switch the AC generator from square wave to a sine wave by pushing in the button labeled with a sine curve. Set the generator frequency to ~3000 Hz (push in the 100 MULT button and turn the FREQUENCY Hz knob to ~30), and turn the small black knob (to the left of the big black knob) so that the stripe and O on it are straight up if the knob isn't already in this position. Set the VOLTS/DIV to ~0.2 V and set the TIME/DIV at ~0.1 ms. Set the scope VERT MODE and the COUPLING SOURCE to CH1. You should see a couple of periods of the sine curve. Enter the set values and the value of the inductance (see Fig.2 ) on your worksheet.

Slowly turn the big FREQUENCY Hz knob to vary the frequency of the generator smoothly between ~1000 and ~10000 Hz and observe the magnitude of VR. Set the frequency of the signal generator to the setting where you think the maximum VR occurs. Record this frequency as your first estimate for the resonance frequency on your worksheet. Leave the frequency at this setting and measure the period of the wave you see on the oscilloscope; then convert this to frequency and record the value on your worksheet. The ratio r of the frequency determined with the oscilloscope divided by the dialed FREQUENCY of the generator will be used as a correction factor when extracting $f_{0}$ from your graph below.

Now dial FREQUENCIES in ~ 5 steps of ~ 500 Hz below the maximum and record the dialed frequencies and your measured values for $V_{R}$ on your worksheet. Repeat for ~ 5 steps of ~ 1000 Hz above the maximum. (Note that you may have to readjust the TRIGGER LEVEL knob closer to “12 o’clock” for a stable picture as your signal becomes smaller. You also may have to change the TIME/DIV in order to see a full period clearly).

Enter the your dialed frequencies, your measured values of $V_{R}$ and the ratio r in to the table below and click submit. You will get a plot of $V_{R}$ vs frequency where the frequency has been corrected using the r factor you entered. Estimate the resonance frequency from the graph and compare it with the value you calculated in the preparation assignment.

Ratio of actual frequency to dialed frequency
Frequency and Voltage Values
f 1  Hz   VR V
f 2  Hz   VR V
f 3  Hz   VR V
f 4  Hz   VR V
f 5  Hz   VR V
f 6  Hz   VR V
f 7  Hz   VR V
f 8  Hz   VR V
f 9  Hz   VR V
f 10  Hz   VR 10  V

### A simple calibration of the scope time base

This part is not mandatory, but you may find it useful in this and other experiments.

Signals that “leak” from the 60 Hz power lines in the room, not to mention the fluorescent lights and many electronic instruments in the room, provide a useful source for calibrating the time base (TIME/DIV) of your oscilloscope. Do the following procedure that your Lab TA will probably demonstrate to the whole class.

Take a red test lead and plug one of its banana-plug ends into the red banana socket input to CH1 of the oscilloscope. With your time base set to 5 ms per DIV, the COUPLING set to LINE, and the SOURCE set to CH1, increase the sensitivity (decrease the VOLTS/DIV) of CH1 until you see an oscillatory signal on the screen. It will greatly increase the magnitude of the “stray” signal if you hold the banana plug on the “free” end of the red test lead with your hand and also put that hand on the top of the table. Reason: You and the table top become part of an “antenna” picking up this stray signal. You may have to adjust the TRIG LEVEL to get the signal to be stable, and you will have to adjust the VOLTS/DIV and POS for CH1 to get the signal placed correctly on the screen.

The powerline frequency is 60 Hz. What period does that correspond to? Calculate it. If the signal you're picking up with your “antenna” is at 60 Hz, the period of that signal should correspond to the period you just calculated. Count the number of divisions (whole number plus fraction) that correspond to one period on the screen, convert that to time by the TIME/DIV setting, and compare that time to what you expect for the period of 60 Hz. Since the power company works hard to keep the powerline frequency at 60 Hz, you have a nice “time-base calibration source” from its period.

Beware: Because the AC power throughout the building is “three-phase”, there are actually three different 60 Hz signals at the same time, and their relative amplitudes depends just where your “antenna” is. For this reason, it's unlikely that your stray signal will look like an ideal sinewave. What you hope is that there is some distinctive feature on your stray signal that does repeat at 60 Hz – some little “spike” or something. Use that feature as your marker for 60 Hz.

Use this to see how accurate is your oscilloscope time base.

It should be pretty accurate – within a few percent. Turn the VAR SWEEP knob counter-clockwise, away from its CAL'D position. What does this do to the apparent measured period for the 60 Hz signal? Now you should understand why that knob should be kept at its full-clockwise, CAL'D (for “calibrated”) position!