# AC Circuits

Electronics - Devices

Oct 5, 2013 (4 years and 7 months ago)

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AC Circuits

AC Current

peak
-
to
-
peak and rms

Capacitive Reactiance

magnitude and phase

Inductive Reactance

magnitude and phase

AC Circuits

AC Signals and rms Values

There are a number of ways to describe the current or
voltage for a time varying signal:
peak value, peak
-
to
-
peak, and rms
. Write in your own words what each means
and indicate the three values on the sine wave signal
shown below.

V
p

V
pp

V
rms

AC Circuits

What is V average, V
ave
?

How is V
rms

related to V
m
?

We will derive this in a minute.

When you measure an ac current or voltage with a
DMM you are measuring rms values.

AC Circuits

Connect the cables from the Signal/Function Generator and the voltage
input cables to the
2.2 k
W

on the E&M board. The red and
black cables with the red and black plugs are the input to the voltage
sensor. Make sure the two grounds (black cables) are connected to the
same side of the resistor. Start DataStudio and configure the program
to measure voltage using channel A. You will recall that you do this
by clicking and dragging the plug icon to channel A and choosing
Voltage Sensor. For output we will connect both the digital output and
the oscilloscope. Connect both to channel A.

Next select the Signal/Function Generator by clicking the Signal icon
on the lower left side of the DataStudio window. Set the amplitude to
5 volt and the frequency to 100 Hz on the function generator. Select
the sine wave (AC) and click ON.

Measure the peak value and the peak
-
to
-
peak value using the
oscilloscope and measure the rms value using the digital meter.

AC Circuits

Measure the peak value and the peak
-
to
-
peak value using
the oscilloscope and measure the rms value using the
digital meter.

V
m

=

V
pp

=

V
rms

=

What is the ratio of V
rms

to V
p
?

V
rms
/V
p

= 0.707

Is this what you expect? Explain.

Yes, at least we expect the value to be less than one.

AC Circuits

The rms value is found by squaring the signal, integrating
over one period, and then taking the square root. Lets do
it. First square the following signal.

V(t) = V
p
sin(
w
t)

Now integrate this with respect to time from t = 0 to t = T
(the period). Look in Appendix A page A
-
10 for the
integral.

Hint: What is
w
T?

Finally, dividing by T and take

the square root you should get

V
2
(t) = V
p
2
sin
2
(
w
t)

Resistance, Capacitance, and Inductance

When we discuss resistors, capacitors, and inductors in a
circuit there are two important points to remember.

The magnitude relationship between the current in
and voltage across a resistor, capacitor, or inductor.

The phase relationship between the current in and
voltage across a resistor, capacitor, or inductor.

AC Circuits

AC Circuits

Resistance

The simplest case is a resistor in a circuit.

The current and voltage are in phase

The magnitude is V
R
=IR

The current and voltage are in phase in a resistor.

Current and voltage reaches a minimum at time T?

AC Circuits

Capacitive Reactance

When we discuss capacitors in a circuit there are two
important points for a capacitor. One is the phase
relationship between the current in and voltage across a
capacitor. What is the phase relationship? This is the
bold statement on page 855 in your text.

The current in a capacitor leads the voltage by 90
0
.

Voltage reaches a minimum at a later time T+
D
T?

Current reaches a minimum at time T?

AC Circuits

The second point is the magnitude relationship between
current and voltage for a capacitor. What is this
relationship?

This is equation (33
-
5) in your text. To make this look like
Ohms law we define
capacitive reactance
. What is the
definition for capacitive reactance?

What are the units?

ohms

Like

AC Circuits

Let’s check this out. Connect the
2.2 k
W

on the E&M
board in series with the 0.047
m
F capacitor to the output of the
function generator. Set the amplitude to 5 volts, the
frequency to
200 Hz
, and the function to sine wave. Connect the voltage probes
to the Pasco 750 Interface across the resistor. To get the phase
correct connect the black voltage lead to one side of the resistor and
connected the ground lead (black cable) from the ground side of the
signal generator. We will also measure the voltage across the
capacitor by connecting the leads from channel B across the
capacitor. Make sure the positive lead is connected to the positive
side of the capacitor. This is the side connected to the positive lead
from the signal generator. You need to click the plug icon and drag
it to channel B and choose Voltage Sensor. To connect the output to
the oscilloscope do not open a new window but go to the
oscilloscope window and click the second trace icon (Which should
be No Signal) on the right and select Channel B.

AC Circuits

Measure the peak voltage across the resistor and the
capacitor. Use the resistance to calculate the peak
current. Record the values in the table below. Change
the frequency and repeat the measurements.

frequency voltage voltage current capacitive

(resistor) (capacitor) reactance

Hz Vp (V) Vp (V) Ip (mA) X
C

(
W
)

___
_20
__ __________ __________ __________ __________

__
2000
__ __________ __________ __________ __________

AC Circuits

Measure the peak voltage across the resistor and the
capacitor. Use the resistance to calculate the peak
current. Record the values in the table below. Change the
frequency and repeat the measurements.

f

1/f

V(cap)

V(res)

I

XC

50

0.02

5

0.16

7.27273E
-
05

68750

100

0.01

5

0.32

0.000145455

34375

300

0.003333

4.9

0.93

0.000422727

11591.4

600

0.001667

4.6

1.8

0.000818182

5622.222

1000

0.001

4.2

2.6

0.001181818

3553.846

1500

0.000667

3.6

3.4

0.001545455

2329.412

2000

0.0005

3

3.9

0.001772727

1692.308

3000

0.000333

2.4

4.3

0.001954545

1227.907

AC Circuits

Use DataStudio or open Excel and plot the capacitive
reactance as a function of frequency. Does the curve
look like what you would expect from the definition of
capacitive reactance? Explain.

Yes!

So it should
look like a 1/x
curve

AC Circuits

Use the definition to figure out how to plot the data so
that you get a straight line. Replot the data and do a
linear fit. From the slope calculate the capacitance.

So slope
equals 1/2

C

AC Circuits

Next we will measure the phase shift between the voltage
across the capacitor and the current through the capacitor.
Set the frequency to 2000 Hz. Make sure both positive
voltage leads are connected to the side connected to the
positive side of the signal generator. The two traces
should look like Figure 33
-
4.

AC Circuits

Measure the phase difference between the current and
voltage. Do this by measuring the period which is 360
0

and the time shift between the two signal. Then

q

= 360
0

t/T where t is the time shift and T is the period.

Change the frequency and see if the phase difference
between the two signals changes.

T

t

AC Circuits

Inductive Reactance

When we discuss inductors in a circuit there are two
important points for an inductor. One is the phase
relationship between the current in and voltage across an
inductor. What is the phase relationship? This is the
bold statement at the bottom of page 836 in your text.

The voltage across an inductor leads the current by 90
0
.

Voltage reaches a maximum at time T?

Current reaches a maximum at a later time T+
D
T?

The second point is the magnitude relationship between
current and voltage for a inductor. What is this
relationship?

This is equation (33
-
7) in your text. To make this look
like Ohms law we define
inductive reactance
. What is
the definition for inductive reactance?

What are the units?

AC Circuits

ohms

On the axis below show what you expect for the inductive
reactance as a function of frequency.

AC Circuits

X (ohms)

L

f (1/sec)

Replace the capacitor with the inductor in the circuit.
Using the digital meter measure the voltage across the
inductor and resistor as we did for the capacitor.

frequency voltage voltage current capacitive

(resistor) (capacitor) reactance

Hz Vp (V) Vp (V) Ip (mA) R (
W
)

___
_20
__ __________ __________ __________ __________

__
2000
__ __________ __________ __________ __________

AC Circuits

AC Circuits

Measure the peak voltage across the resistor and the
inductor. Use the resistance to calculate the peak current.
Record the values in the table below. Change the
frequency and repeat the measurements.

f

1/f

V(res)

V(ind)

I

XL

50

0.02

4.8

0.5

0.0022

229.17

300

0.0033

4

1.26

0.0018

693

600

0.0017

3.9

1.55

0.0018

874.36

1000

0.001

3.7

2

0.0017

1189.2

2000

0.0005

3.2

3

0.0015

2062.5

3000

0.0003

2.7

3.7

0.0012

3014.8

4000

0.0003

2.2

4.1

0.001

4100

5000

0.0002

1.8

4.3

0.0008

5255.6

AC Circuits

Plot the inductive reactance as a function of frequency.
Does the graph agree with what you know about inductive
reactance?

Yes!

So t he slope is 2

L.

The inductance is

AC Circuits

Measure the phase shift between the current and voltage
across the inductor. Your two traces should look like
Figure 33
-
6.

AC Circuits

Summary of AC circuit equations

This is everything you need to know from today

AC Circuits

Summary of AC circuit concepts

This is a little more than you want to know!

AC Circuits

Phase shift and Phasors