AC Circuits

1
AC Circuits

• AC Current
• peak-to-peak and rms
• Capacitive Reactiance
• magnitude and phase
• Inductive Reactance
• magnitude and phase

2
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.

3
AC Circuits

• What is V average, Vave?
• How is Vrms related to Vm?
• We will derive this in a minute.
• When you measure an ac current or voltage with a
  DMM you are measuring rms values.

4
AC Circuits

• Connect the cables from the Signal/Function
  Generator and the voltage input cables to the 2.2
  kW resistor on the EM 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.

5
AC Circuits

• Measure the peak value and the peak-to-peak value
  using the oscilloscope and measure the rms value
  using the digital meter.
• Vm Vpp Vrms
• What is the ratio of Vrms to Vp?
• Vrms/Vp 0.707
• Is this what you expect? Explain.

Yes, at least we expect the value to be less than
one.
6
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) Vpsin(wt)
• 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 wT?
• Finally, dividing by T and take
• the square root you should get

V2(t) Vp2sin2(wt)
7
AC Circuits

• 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.

8
AC Circuits

• Resistance
• The simplest case is a resistor in a circuit.
• The current and voltage are in phase
• The magnitude is VRIR

The current and voltage are in phase in a
resistor.
Current and voltage reaches a minimum at time T
9
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
900.
10
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
11
AC Circuits

• Lets check this out. Connect the 2.2 kW
  resistor on the EM board in series with the
  0.047 mF 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.

12
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) XC (W)
• ____20__ __________ __________ __________
  __________
• __2000__ __________ __________ __________
  __________

13
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.

14
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.

15
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.

16
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.

17
AC Circuits

• Measure the phase difference between the current
  and voltage. Do this by measuring the period
  which is 3600 and the time shift between the two
  signal. Then
• q 3600 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.

18
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 900.
19
AC Circuits

• 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?

ohms
20
AC Circuits

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

21
AC Circuits

• 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__ __________ __________ __________
  __________

22
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.

23
AC Circuits

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

24
AC Circuits

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

25
AC Circuits
Summary of AC circuit equations
This is everything you need to know from today
26
AC Circuits
Summary of AC circuit concepts
This is a little more than you want to know!
27
AC Circuits

• Phase shift and Phasors