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10-1

### 10. AC Circuits*

Objective: To learn how to analyze current and voltage relationships in alternating current (a.c.) circuits. You will use the method of phasors, or the vector addition of rotating vectors that represent the voltages or currents. This method does not require calculus.

Reading assignment Before you come to the lab study the sections on alternating current, capacitor circuits, inductors and inductor circuits,

Read the following sections. (Section numbers may be slightly different depending on the edition of your textbook: Check the section titles.)

Knight, Jones and Field : 26.1 Alternating Current, capacitor circuits 26.4, inductors and inductor circuits 26.5, oscillation circuits 26.6

Serway and Vuille (212): 21.1 Resistors in an AC Circuit , 21.2: Capacitors in an AC Circuit, 21.3 Inductors in an AC Circuit, 21.4 The RLC Series Circuit, 21.5: Power in an AC Circuit, 21.6 Resonance in a Series RLC Circuit

Serway and Jewett (252): 33.1 AC Sources , 33.2 Resistors in an AC Circuit, 33.3: Inductors in an AC Circuit, 33.4 Capacitors in an AC Circuit, 33.5 The RLC Series Circuit, 33.6 Power in an AC Circuit, 33.7 Resonance in a Series RLC Circuit

Pre-lab assignment:

1. The vector shown is rotating counter-clockwise around the circle shown with an angular frequency of

. At time

, the angle is

### 

. Write a formula for the x and y components of the vector (a “phasor”) at general time t. The length of the vector is Io.

Ix =

Iy =

The learning objectives are the following:

1. To learn how alternating currents or voltages can be represented as the y-components of rotating vectors, or phasors.

2. To understand the amplitude and phase relationships amongst the different components (resistors, capacitors, inductors) of a.c. circuits.

3. To learn effective use of a typical oscilloscope.

4. To understand the phenomenon of resonance in series RLC circuits. (This material is presented in your textbook iunder alternating current circuits.)

### *© William A Schwalm 2012

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10-2 2. (Phys 252 only) At the right is a schematic circuit di 3.

###  

s

I t

The idea behind a current phasor is that either Ix or Iy can be used to represent an alternating current as

the phasor rotates, since these components vary “sinusoidally.”

2. (Phys 252 only) At the right is a schematic circuit diagram showing a resistor, capacitor, inductor and alternating voltage source.

(a) Label these with R, C, L and a.c.

(b) At a given instant t in time, suppose is the current flowing out of the source, as

shown. Let , and be the currents flowing through R, L and C at this instant. Write the equations that hold between each pair currents. Explain why.

(c) Suppose the capacitor starts out with zero charge. As the current flows in the direction shown, what happens to the right and to the left capacitor plate? (Two separate answers.)

(d) As we have learned, the voltage drop across the coil, that is to say across the inductor L, is proportional to the rate of change of the current.

.

What is the relation between the current through the resistor and the voltage drop across the resistor? Give an equation.

(e) The voltage across the capacitor is related to the charge Q on either of its plates by the definition of capacitance. Thus give a formula for the voltage drop across the capacitor.

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(f) Write an equation (one equation) relating the voltage

s

##  

of the alternating voltage source to the three voltage drops across the three other circuit components.

time source current time A B C voltage B C A = = = Io B

Phasor for source current Phasors for voltage drops

A

C

A ?

B ?

C ?

3. (Phys 212 and 252) In the following figure you see the source current as a function of time. The other curves represent voltages across the resistor, capacitor and inductor, in some order. Based on your answers above, identify which curve represents which voltage drop. Then explain briefly each choice. (Don’t just say that current leads voltage across the “grob,” (whatever) because you also have to explain why that is, based on how grob works.)

4. Below left is a current phasor, a vector rotating counter-clockwise with angular frequency such that its vertical component (y component) gives the source current . At the right are three voltage phasors marked A, B, and C. These relate to voltages. Based on your responses above, identify which of the three phasors corresponds to the voltage drop across each of R, L and C ? (The letters A, B, C are not necessarily going to correspond to the curves in the previous question.) Explain your choices. For instance, which voltage phasor should line up with the current phasor because the voltage drop is proportional to current?

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10-4 

C

R

### = 0.

(Yes) 

Exploring equipment: Read the Appendix on the use of oscilloscopes

The hold introduces extra time delay between sweeps by missing some sweeps. Pulling the hold knob you can chop the input between CH1 and CH2. That means the input (not triggering) is sampled by rapidly alternating back and forth between the two inputs. This is handy for very low sweep rates and is used in place of alternate sweeps in this case.

Equipment: The a.c. voltage source or signal generator

(a) The same current flows at each instant through each circuit element. This means the current through the capacitor Ic is the same at each instant as the current IR through the

resistor.

(b) The sum at each instant of the voltage drops around the circuit is zero. Thus if Vc is the

voltage drop across the capacitor, VR

EMF provided at the same instant by the signal generator, then at each instant in time

But—and here’s the tricky part—

(c) The maximum voltages across each circuit element do not add up. This is because these maximum voltages occur at different times during the cycle.

The photo on the left shows the signal generator attached to a resistor and capacitor. The signal generator is in the rear and there is a resistor, capacitor and an inductor (not part of the circuit) each mounted on a Plexiglas box. The wires complete the circuit, so that during half the cycle current flows out one terminal of the signal generator, through the resistor, into the capacitor (charging the capacitor) and back into the generator. Then on the other half of the cycle it flows the other way. Therefore notice from the way the circuit is set up that .

r 0 - VC0 - VR0

### 

0. (No! )

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The fact that the voltages across the different kinds of circuit elements do not reach maximum at the same time (i.e. they are out of phase with one another) and thus don’t add up is the reason you have to use an oscilloscope rather than a volt meter and is also the reason we will analyze the circuits using phasors.

### Problem 1

Comparing voltages across the capacitor and the resistor in a series RC circuit.

The Control Data Corporation needs you to analyze a bunch of control circuits for an aircraft. In the midst of this, your supervisor is asking your group to demonstrate, using the oscilloscope, the relationship between the voltage across a capacitor C and a resistor R in a particular series RC circuit. You are supposed to show this to your supervisor as a display on the oscilloscope screen by having the CH1 trace show the resistor voltage drop

R

##  

### t

and the CH2 trace show the voltage drop

C

##  

### t

across the capacitor. You will have to figure out how to do it, although there is one tricky point.

In-class response question: The group should draw on the white board a circuit diagram for a

series RC circuit with a signal generator. Reproduce it here also. Then, answer the following both on the board and below:

(a) Between which two points would you connect the oscilloscope input leads for CH1 in order to measure

R

##  

### t

? Indicate on the diagram.

(b) Between which points would you connect the leads of CH2 in order to measure

C

##  

### t

?

Now here’s the tricky part.

(a) You want to have the top of the oscilloscope screen represent a more positive voltage. (b) You also want to measure the voltages in the same direction relative to the instantaneous current flow (that is, both clockwise or both counter clockwise in the circuit.) However: (c) You cannot ground more than one point in a circuit or you have a short circuit.

The problem comes about because one side of each input channel is already grounded.

Notice the actual CH1 and CH2 inputs are BNC connectors. The outer sleeve of a BNC is a metal cylinder welded to the oscilloscope frame. This side of the input usually corresponds to the common on a voltmeter and to the negative or downward direction on the screen. Thus when the other input lead (central terminal of the BNC) is more positive that the sleeve, the trace on the screen goes up, and when it is more negative than the sleeve the trace goes down.

Therefore you have to connect the sleeve side (grounded side) of each input to the same point in the circuit.

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In-class group response: Go back to your circuit diagram and figure out which point you would

like to connect the grounded sides of the CH1 and CH2 inputs (same point). Indicate this point on the diagram. Then explain why this makes CH2 measure the capacitor voltage upside down! (Dang.) Thus it seems you will have to see

R

and

C

##  

### t

on the screen. This is not you assignment. Explain here and on the white board. A diagram showing both the RC circuit and the oscilloscope connections will be appropriate.

Ok, here is a possible solution to the problem: If you look carefully on the control panel you may find a control (often a push-pull thing) that will reverse the sign of the display for CH2. Thus it will flip the CH2 trace on the screen vertically. This should allow you to show what your supervisor needs to see. Try to find such a control.

In-class response: Draw a picture of how you expect the two traces on the oscilloscope screen to

look. Indicate clearly which trace is which.

In-class response: Draw a phasor diagram showing

voltage drops across the resistor and across the capacitor as phasors of length

R0 and

### V

C0

respectively. Draw them at a time when the current phasor points along the x axis.

Measurement plan: Describe very briefly how you will set up the circuit and create the display the supervisor wants. Recall that an RC circuit has a time constant, which is the product RC of resistance and capacitance. Plan to observe the voltage signals in three cases, (a) when the angular frequency

### 

applied by the source is less than 1/RC, (b) when it equals 1/RC and (c) when it is greater than 1/RC. Record your plan here.

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Implementation: Carry out your plan. Record the results as graphs of what you actually observe in each case. Include the actual numbers on the scales taking into account the time base and voltage per division. Graph at least two cycles for each. Be sure your supervisor sees the display for at least one of the three cases.

Analysis: Write a summary of what you learned. In particular, describe the phase relation between the two voltage drops. In each of three experiments, did the capacitor or the resistor voltage reach a maximum first, and was this what you expected? Can you say how much (what fraction of a cycle) one voltage leads the other by? Why or why not? Also, how do the voltage maxima (the amplitudes) compare in each of the three cases? Can you explain this?

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### Problem 2

Now your group is asked to perform a similar study and to provide a similar demonstration for a series RLC. You are to demonstrate to your supervisor using the oscilloscope how the voltages across the R, the inductor L and the capacitor C vary in a series RLC circuit as a function of the applied angular frequency. (Recall the pre-lab exercises.) Further, you should compare to predictions that you make using phasor pictures.

Work assignment: You need first to be able to show on the screen:

(a) both the resistor voltage drop

R

##  

### t

in CH1 and the capacitor voltage drop

C

##  

### t

in CH2. (You have to rewire between a) and b))

(b) both the inductor voltage drop

L

##  

in CH1 and the capacitor voltage drop

C

##  

### t

in CH2. (Rewire again. This time it’s a little different.)

Measure the phase angle from the current to the source voltage in the cases (a) when

, (b) when

, (c) when

### 5. / LC

.

Finally you need to produce a nice, professional-looking graph showing the ratio of the current amplitude Io divided by the source voltage amplitude Vo as a function of

, where

### 

.

That is, make a graph of o

o

### 

.

The supervisor wants this written up, of course, so you will hand it in as part of your report, as follows.

You should be aware from your reading that the length of the voltage phasors representing voltage drops across each of the circuit elements are

Of course the phasor, which are rotating vectors, do not all point in the same direction.

In-class response: (a) The voltage phasor for which one of the circuit elements (R, L or C) is

parallel to the current phasor at every instant in time? (b) Why?

(c) Also, how are the directions of the other two voltage phasors related to the direction of the current phasor? Draw a picture of the three voltage phasors in this situation, representing phasors as arrows with proper labels, at an instant when the current phasor points directly along

the positive x axis.

0 0 0 0 0 0

R L C

### 

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In-class response: The source voltage at any instant is the y component (in some textbooks, the x components are used) of the vector sum of the voltage phasors. Referring to your drawing and

to the formulas for the phasor lengths, what are the components of the total voltage phasor at the instant shown in your figure?

Vox = Voy =

In-class response: Thus what is the magnitude of the source voltage phasor as a function of

frequency? (This gives the amplitude of the source voltage.)

In-class response: What is the “phase angle” between the source voltage and the source

current? (Remember, the current phasor is along the x axis at the moment, so this asks the angle between the axis and the total voltage phasor. It’s easy to get its tangent.)

Measurement plan: Your group should agree on a measurement plan for performing the assigned work. You will turn in the graph as part of your report. Include a brief outline of the measurement plan here, devoting at least some space to each point. Be sure in particular to tell us how you will wire the circuits in each case.

Implementation: Carry out your plan. Record the necessary data and ancillary observations here and on additional sheets. Before leaving lab, you should also have a rough sketch of the curve representing data from your resonance experiment.

Analysis: You have worked out a prediction of what to expect in each case using phasors. Thus, for each of the points above, the analysis may consist of comparing what you would expect to see (which of course you must state, basing it on the phasors) with what you actually did see. As usual, you should comment especially on things that didn’t go as expected. In particular, your analysis should include some discussion of the form of the graph you have constructed, and how this relates to the phenomenon of resonance as described in the text. What is the resonance frequency? What happens to the current there? etc.

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Conclusion: Usually the last five minutes are devoted to discussing the lab activity and how the activities related to the learning objectives. Give here a brief discussion of how the objectives were addressed, noting anything of particular interest. Include the main points of the class discussion.

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