Resistive, Capacitive and Inductive Series AC Circuits

Objectives: To study electrical resonance in AC circuits, to observe the effects of capacitors and inductors in AC circuits, to analyze AC series RLC circuits.

Equipment: 1- Instek GOS-620 oscilloscope with power cord and probe, 1- BK 2831D function generator with power cord and output cable, 1- BK 4017A multimeter with power cord and test leads, 1- Pasco Cl-6512 RLC circuit board, 2- banana clip leads, 2- alligator clip leads (Figure 1).

Figure 1

Discussion: You previously examined the phenomena of resonance in your investigation of the speed of sound in air in last semesters lab. All electrical and mechanical systems have natural modes of vibration. Last semester you matched these modes in a column of air to the vibrations of a tuning fork to send the column of air into resonance. When a periodic force, such as a series of pulses or taps, is applied to a system, it vibrates with a frequency equal to that of the force. This type of vibration is knows as a forced oscillation. Normally the response to a forced oscillation is quite small, but if the frequency of forced oscillations approaches the natural or resonant frequency of the system, the amplitude of the oscillations grows quickly. Resonance occurs whenever the two frequencies match.

In theory, the amplitude of vibrations at resonance should approach infinity. In mechanical systems friction limits the growth of the amplitude of the oscillations, in electrical systems, resistance plays the same role. In this exercise you will study resonance in alternating current RLC (resistive, inductive, capacitive) circuits.

Analog radios use resonance to tune the receiving circuitry to the broadcast frequency of the station being tuned in. Each radio station broadcasts at a precise carrier frequency. When your receiver is in resonance at this frequency you are in tune with that station. In most circumstances this is accomplished by changing the capacitance of the receiving circuit at a fixed inductance.

AC RLC Circuits

AC circuits contain continuously varying currents and voltages that oscillate sinusoidally with time.

Consider a purely resistive AC circuit:

The applied voltage oscillates with some angular frequency , it's instantaneous value may be described as:

v=V maxcos (t )=V maxcos ()

Note that the current through the resistor is:

i= vR=V maxR

cos ( t )

Since both the current and voltage both oscillate as functions of the same cosine term they are in phase, i.e. they rise and fall in sync.

Next, consider a purely capacitive AC circuit.

The instantaneous charge, q, on the capacitor is:

q=Cv=C V max cos t

Now, let's look at the instantaneous current in the circuit, which is controlled by the capacitor:

i=dqdt=CV max sin t

In this case, the instantaneous voltage:

v=V maxcos t

varies from the current by a factor of -sin. The significance of this, in terms of a phase difference, is that the current peaks ahead of the voltage by a factor of 90 degrees.

Ignoring the minus sign, note that:

I max=CV max 1=V maxR

R 1C

The term 1/C is know as the capacitive reactance, Xc , of the circuit. The units of Xc are ohms.

Capacitive reactance acts like resistance in this circuit.

Finally, consider a purely inductive circuit.

Note that the inductor has no DC resistance. Recall that the instantaneous voltage across the inductor is:

v=L didt=V maxcos t

To determine the instantaneous current:

di=V maxL

cos t dt

di=V maxL cos t dt

i=V maxL

1 sin tC

i=V maxL

sin tC

We use the ics to evaluate C: At t = 0, i = 0 and C = 0 so:

i=V maxL

sin t

In this case the current peaks behind the voltage. In an inductor, the voltage peaks 90 ahead of the current.

The term L is know as the inductive reactance, XL , of the circuit. The units of XL are ohms. Inductive reactance acts like resistance in this circuit.

PROCEDURE

Resonance in AC Circuits

Figure 2 shows the equipment necessary to explore electrical resonance in a circuit. Connect the supplied BNC cable to the output terminal (on the lower right) of the function generator. Connect the red positive lead at the other end of the same cable to the bottom of the 10 resistor on the RCI circuit board, and the black negative lead to the bottom of the 100F capacitor. This arrangement creates a series circuit containing a 10 resistor, an 8.2 mH inductor and a 100 mF capacitor, all powered by the signal from the function generator.

It will be necessary for you to use an oscilloscope to analyze the signals in this circuit. You should, by now, be familiar with the operation of the Instek GOS-620 Oscilloscope from last weeks lab (http://www2.cose.isu.edu/~hackmart/spl2osc.pdf). You may want to spend some time reviewing this material before proceeding.

Connect the oscilloscope probe to the CH1 (X) vertical connector on the oscilloscope input section and then connect the tip of the probe between the resistor and inductor on the RCI circuit board, as shown. This completes the setup (Figure 2). Be sure to have your lab instructor exam your setup before energizing anything.

Next, draw an electrical schematic of this arrangement in your lab notebook, which should look something like the schematic in Figure 3. Pay special attention to the two grounds shown in this circuit. How are they articulated in the circuit you constructed on the table?

Figure 2

Figure 3

Set the function generator to the 1000Hz (1K) range, turn it on, and decrease the output frequency to around 150 Hz using the coarse and fine frequency settings. On the oscilloscope, the magnification (10X) should be engaged on the probe. Set the time per division should to 10ms to begin. The mode settings are channel 1 and alternating current. Set the volts per division to 0.1V, with the fine adjustment knob completely clockwise. Turn on the oscilloscope and verify a sine wave is present on the scope. Adjustments may be necessary to the horizontal and vertical sweeps in order to produce a stable wave.

Next, vary the frequency produced by the function generator both up and down to determine which direction decreases the amplitude (vertical height) of the wave pattern. It may be necessary to make minor changes in the scope settings to study the pattern when these frequency changes are made.

Remember that an oscilloscope measures voltage as a function of time. In this circuit:

V max= Imax Z

where Z is the overall impedance (resistance) of the circuit taking into account the resistance, and the capacitive and inductive reactances:

Z=R2 X LX C 2At resonance XL = XC and Z has its minimum value. With this in mind, what extreme value of the

amplitude should indicate that the resonant frequency has been found, maximum or minimum? The resonant frequency of a RLC series circuit is given by:

f = 12LC

where L is the value of the inductor and C is the value of the capacitor. What happens to the sine wave when you select a frequency range on the function generator higher or lower than that containing the resonant frequency? Does your recorded value match the predicted value?

Disconnect your circuit and turn everything off before proceeding to the next step.

A Series AC RLC Circuit

We will consider only a circuit with a resistor, a capacitor and an inductor in series, energized with an alternating EMF. Do you think that the order of these components matters in a series circuit? Why or why not? What must be the same everywhere at any particular instant in such a circuit?

In a purely resistive AC circuit, the resistor acts in exactly the same manner as in a DC circuit. Though the voltage across and current through the resistor oscillate harmonically, Ohm's law still holds at any particular instant in time. As we noted before, the instantaneous voltage may be written:

v=V maxcos t =V maxcos

while the current may be written:

i= vR=V maxR

cos t

The voltage and current increase and decrease in step in a purely resistive circuit and are said to be in phase. The maximum voltage across the resistor in a resistive circuit may be easily measured with an oscilloscope and is denoted Vmax. The maximum current in the circuit is simply Vmax/R. Multimeters read RMS (Root Mean Square) values. A RMS value is a statistical average of the signal: voltage, current, etc. It is defined as:

Signalmax2

=Signal rms

The RMS voltage across the resistor in a series RLC circuit is V rms=V max /2 , the RMS current, using Ohm's Law, is I rms=V rms /R .

Capacitors, by contrast, behave differently in AC and DC circuits. In an AC circuit the capacitor impedes (resists) the changing current much like a resistor does in a DC circuit. This occurs due to the fact that the current in the circuit is at a maximum when the capacitor first begins to charge and the potential between the plates is zero. The voltage and current, which are in phase for a resistor, are 90 out of phase for a capacitor. The current leads the voltage by 90 in a purely capacitive AC circuit. The values for both voltage and current increase and decrease harmonically, but with a phase difference of 90.

The peak or maximum value of the current in a capacitive AC circuit is:

I max=CV max=V maxX C

where Xc, defined previously, is known as the capacitive reactance of the circuit. The RMS current is given by a similar expression, with Vrms replacing Vmax. The instantaneous voltage drop across the capacitor is:

vC=V max sin ( t)= Imax X C sin( t )

Recall that the instantaneous current everywhere in the circuit is given by cos ( t) . Here the voltage across the capacitor is given by a sin term, which indicates that the voltage lags the current by 90.

Inductors also behave differently in AC and DC circuits, with inductors acting like resistors in AC circuits. Since the voltage across an inductor is proportional to the change in current, the voltage attains its maximum value when the current is changing most rapidly. Since i vs t is harmonic, the maximum rate of change (maximum slope) occurs when the curve goes through zero, i.e., when the current has an instantaneous value of zero. The current and the voltage are therefore 90 out of phase for an inductor. But unlike a capacitor the current lags the voltage by 90 in a purely inductive circuit. The peak or maximum value of the current in an inductive AC circuit is:

I max=V max(L)

=V maxX L

where XL , defined previously, is known as the inductive reactance of the circuit. The RMS current is given by a similar expression with Vrms replacing Vmax. The instantaneous voltage drop across the inductor is:

v L=V max sin( t)=I max X Lsin( t)

Recall, again, that the instantaneous current everywhere in the circuit is given by cos ( t) . Here the voltage across the capacitor is given by a -sin term, which indicates that the voltage now leads the current by 90.

The peak voltages for a series RLC circuit are given by:

V R=I max RV L=I max X LV C=I max X C

Since the peak voltage across each of these elements occurs at a different time, these values are of little use without some factor that takes into account the phase differences between these quantities. This factor is known as the impedance Z of the circuit. The sum of the voltage drops across the circuit may be written:

V M=(V R2+ (V LV C)2)V M=I M (R2+ (X LX C)2)V M=I M Z

By using impedance, we may write a form of Ohm's law that takes into account the time varying nature of AC circuits: V M=I M Z . The phase angle between the current and the voltage for the entire circuit is:

tan= X LX C

R

When XL > XC (typically at high frequencies), the phase angle is positive, meaning that the current lags behind the voltage. When XL < XC , the phase angle is negative and the current leads the voltage. When XL = XC , the phase angle is zero. In this case the impedance matches the resistance and the current is at its peak value. The frequency for which XL = XC is the resonant frequency of the circuit, given by:

L= 1C

2= 1LC

12LC

= f .

PROCEDURE

RLC AC Circuits

Set the function generator to produce a 60 Hz signal with the output level knob all the way clockwise. Measure the voltage output of the function generator with the oscilloscope and verify the 60 Hz frequency. Once you have measured the output voltage of the function generator, reduce it from whatever it is to 5 volts. What is the RMS output of the function generator at this level? What is the angular frequency of the signal?

Next use the multimeter to measure the output voltage of the function generator (make sure that the range is set to accommodate the voltage you measured with the oscilloscope). How do these readings compare? What is the relationship between the multimeter and oscilloscope measurements?

Using the 33 resistor this time, connect the function generator leads to the circuit board otherwise as before. Draw a schematic diagram of this circuit in your lab notebook. Use the

multimeter (set to the appropriate AC voltage range) to measure the individual voltage drops across the resistor, inductor, and capacitor. Then record the voltage drop across the entire circuit. How do these two values compare?

What is the relationship between the peak and RMS currents for each of the three components? Could this result have been predicted? Since the components are in series, the AC current at all points in the circuit has the same amplitude and phase. The voltage across each element will have different amplitudes and phases. State the relationship between peak, RMS, and instantaneous voltage and current values for each component in this circuit. Sketch each of these in your lab notebook.

Compute the impedance and phase angle for this circuit. Is the circuit predominantly inductive or capacitive? Compute resonant frequency of this circuit. Try the same series of measurements for at least two other combinations of series resistors, inductors and capacitors on your printed circuit board.

Figure 4.

Note: the output signal swing of the function generator is limited to 10 volts open circuited or 5 volts into 50 when connected to the board.

Exercises

1. Explain the phase difference between voltage and current in a capacitor. Why does this phase difference occur? Plot v and i vs t to show the phase relationship.

2. Explain the phase difference between voltage and current in an inductor. Why does this phase difference occur? Plot v and i vs t to show the phase relationship.

3. In a series RLC circuit, what determines whether the inductive or capacitive behavior dominates?

4. Why does the amplitude of oscillations become smaller on an oscilloscope when a series RLC circuit is in resonance?

5. An RLC circuit is used in a radio to tune into a an FM station broadcasting at 99.7 mHz. The resistance in the circuit is 12 and the inductance is 1.40mH. What capacitance should be used?

6. A series RLC circuit has the following values: L = 20mH, C = 100nF, R = 20, and Vapp = 100 volts. Find the resonant frequency of this circuit, the amplitude of the current at the resonant frequency, and the amplitude of the voltage across the inductor at resonance.