LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
RLC Circuits
Purpose:
In this lab we will get reacquainted with the oscilloscope, determine the inductance of an
inductor, verify the resonance frequency and find the phase angle, φ, of an LRC circuit.
Equipment:
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
Oscilloscope, CRO or Digital
DMM
LCR meter
Function Generator
Inductor (5H, toroidal)
27 KΩ resistor, 100Ω resistor, or Resistance Box
2 x 10-9 Farad capacitor, or Capacitance Box
Patch Cords, BNC connectors, Alligator Clips
Terminal Board, Wire Jumper Kits
Theory1
When a dc (direct current) voltage source, such as a battery, is applied to an electrical
circuit, the resulting current flows only in one direction, as the polarity (+ and -) of the
voltage remains constant. For an ac (alternating current) circuit voltage source, however,
the polarity of the voltage alternates with time and the direction of the current flow in the
circuit alternates with the same frequency as that of the voltage source.
One of the most commonly used ac voltages is one that varies sinusoidally with
time. This may be generally described by the equation
V (t ) = Vm sin 2πft
Eq. 1
where Vm is the maximum voltage amplitude and f is the frequency of the source. The
angle θ = 2πft = ωt is called the phase angle.
The current in an ac circuit may or may not be in phase with the voltage,
depending on the nature of the components in the circuit. In any case, if the applied
voltage is sinusoidal, the current I will also be sinusoidal, an as a function of time may be
expressed
I (t ) = I m sin(2πft − φ )
Eq. 2
where Im is the maximum amplitude of the current and φ is the phase constant. In Eq. 1,
the phase constant of the voltage has been assumed to be zero for simplicity. In this case,
φ is also the phase difference between the applied voltage and the resulting current.
(More generally, φ = φI – φV and is the angle between Im and Vm, as illustrated in Fig. 1.)
1
The theory section of this lab was shamelessly borrowed from:
Jerry D. Wilson. Physics Laboratory Experiments, 2nd Edition. Lexington MA: D.C. Heath and
Company, 1986. Experiment 40.
1 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
Vm
Im
Vm
Im
π/2ω
-Im
π/ω
3π/2ω
2π/ω
5π/2ω
3π/ω
t
(in terms of
phase angle
= θ/ ω )
φ
-Vm
Phase Constant
t
Figure 1
ac Voltage and current. Here, the voltage leads the current by a constant phase difference φ.
The phase constant φ in Eq. 2 can be either positive or negative. When it is
positive, the maximum value of the current Im is reached at a later time than the
maximum value of the voltage, Vm. In this case, we say that the voltage leads the current
or the current lags behind the voltage by a phase difference φ. Similarly, if φ is negative,
the current leads and the voltage lags.
When there is a capacitive element in the circuit [an RC circuit, Fig 2(a)], the
alternate charging and discharging of the capacitor opposes the current flow. This
opposition is expressed as capacitive reactance, XC, and
XC =
1
2πfC
Eq. 3
where C is the value of capacitance (in farads, F). The unit of XC is ohms.
Similarly, when there is an inductive element in an ac circuit [an RL circuit,
Figure 2(b)], the self-induced counter emf in the induction coil opposes the current. The
inductive reactance, XL, is given by
X L = 2πfL
Eq. 4
where L is the inductance of the coil (in henrys, H). The unit of XL is ohms.
Many ac circuits have both capacitive and inductive reactance elements. The
combined opposition to the current flow of resistive and reactive elements in a series
circuit as shown in Figure 2(c) (a series RLC circuit) is expressed in terms of the
impedance, Z, of the circuit, which is given by
[
Z = R 2 + (X L − X C )
2
]
1
2
2
⎡ 2 ⎛
1 ⎞ ⎤
⎟ ⎥
= ⎢ R + ⎜⎜ 2πfL −
2πfC ⎟⎠ ⎥
⎢⎣
⎝
⎦
2 of 11
1
2
Eq. 5
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
The unit of Z is also in ohms. (R is the total resistance of the circuit. In general, it is
assumed that the resistance of the induction coil is negligible compared with the resistor
element.)
H
H
H
C
L
V
V
L
G
C
R
V
R
R
G
G
(c) RLC circuit
(b) RL circuit
(a) RC circuit
Figure 2
Circuit diagrams for (a) RC, (b) RL and (c) RLC series circuits.
For the RLC series circuit, it can be shown that the phase relation of the voltage
and current is given by
tan φ =
XL − XC
R
Eq. 6
This relationship is often represented in a phasor diagram, in which resistance and
reactances are added like vectors (Figure 3). Note that the angle φ is either positive or
negative, depending on whether the inductive or capacitive reactance is greater. If XL is
greater than XC, φ is positive and the current lags behind the applied voltage in time. The
circuit is then said to be inductive. Similarly, if XC leads XL, φ is negative and the current
leads the voltage. In this case, the circuit is said to be capacitive. [Eq. 6 can be applied
to single reactance circuits as in Figure 2(a) and (b) by letting the appropriate reactance
be zero.]
XL
(XL – XC)
φ
XC
XL
Z
R
R
φ
(XL – XC)
Z
XC
(a) Inductive Circuit
(b) Capacitive Circuit
φ=0
XL
Z=R
XC
(c) Resistive Circuit
Figure 3
Phasor diagrams for (a) an inductive circuit, (b) a capacitive circuit, and (c) a resistive circuit.
In a phasor diagram, resistances and reactances are added like vectors.
3 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
Since the voltage and the current are continually changing in an ac circuit, it is
convenient to consider effective or time-averaged values of the voltage and current.
These root-mean-square (rms) values are given by V = Vm 2 and I = I m 2 , where
Vm and Im are the maximum or “peak” voltage and current, respectively. The rms values
of V and I are the values read on most ac voltmeters and ammeters. An Ohm’s law type
relationship holds between these values and the impedance, Z:
V = IZ
Eq. 7
and it follows that
Vm = I m Z
Thus, for a given applied voltage, the smaller the impedance of a circuit, the
greater the current in the circuit ( I = V/Z ). Notice in the reactance term in Eq. 5 that
there is a minus sign and that the individual reactances are reciprocally frequencydependent. As a result, for given L and C values, the total reactance can be zero for a
particular frequency when XL = XC. That is,
2πfL −
1
=0
2πfC
Eq. 8
and solving for f,
fr =
1
2π LC
Eq. 9
where fr is called the resonance frequency. In this condition, the circuit is said to be in
resonance. The impedance is then equal to the resistance in the circuit, Z = R, and the
circuit is resistive [Figure 2(c)].
Since the impedance is a minimum at the resonance condition, the maximum
current flows in the circuit from the voltage source, and maximum power P = I2Z = I2R.
For fixed values of L and C, resonance occurs at the particular resonance frequency fr
given by Eq. 9. However, notice from Eq. 8 that for a given source frequency, resonance
can also be obtained by varying L and/or C in the circuit.
In this experiment, it is desired to measure the phase difference between the applied
voltage and current in ac circuits, and to investigate the resonance condition in a series
(or parallel) RLC circuit using an oscilloscope. Notice from Eq. 6 that for the resonance
condition, XL = XC, the voltage and current are in phase (φ = 0), since tan φ = 0 (and tan 0
= 0).
Suppose that different voltage signals are applied to the horizontal and vertical
inputs of an oscilloscope. If the ratio of the horizontal and vertical frequencies is an
integral or half-integral, then a stationary elliptical pattern such as in Figure 4 is
observed. Assume that the applied voltages have the forms
x = A sin 2πft
y = B sin( 2πft − φ )
4 of 11
Eq. 10
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
where A and B are the amplitudes (i.e. the voltages Vx and Vy). [Compare with Eqs. 1 and
2 that give the voltage V(t) and circuit current I(t), respectively.] Note that the y
intercepts, +b and –b, occur when x = 0, or when 2πft = 0 (since sin 0 = 0). Then from
the second equation, b = B sin(-φ). Hence,
b
sin φ = ±
B
and similarly
sin φ = ±
a
A
Eq. 11
You should be able to prove and understand that if A = B and φ = 90o, the trace would be
a circle. Also, if A = B and φ = 0o, the trace is a straight line.
To measure the phase angle difference of the voltage and current in a circuit, the
voltage signal from the signal generator is applied to the horizontal input. For example,
for the circuits in Figure 2, connections are made to points H and G, where G is the
horizontal ground terminal. The voltage input signal has the form of Eq. 1.
x = A sin 2πft
y = B sin (2πft – φ)
Y
B
b
a
A
X
Ellipse centered
at origin
Figure 4
The elliptical pattern used to measure the phase angle difference
The current signal is applied to the vertical input with connections made to points
V and G, where G is to ground. This is actually a voltage input from across the resistor in
the circuit. However, it is proportional to and in phase with the current through the
resistor (and therefore through the entire series circuit). Hence, the phase angle
difference between the voltage and current can be determined from the shape of the
resulting elliptical oscilloscope pattern, as described above.
Experiment and Analysis:
Part A:
1. With the LCR meter, determine a good value of the resistance of the 27KΩ resistor
(unless you are using the resistance box), the inductor, and capacitor. If these values
are considerably different than the “stated” values, check the batteries of your meter!
5 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
2. Hook up circuit as shown in the sketch, remembering to connect the resistor to the
negative (ground) output of the function generator. Use the nano-farad capacitor and
the 5 Henry inductor if available. Otherwise, using the lowest capacitance value on
the capacitance box.
H
V
L
C
R
G
Figure 5
3. Connect Channel 1 of the oscilloscope across the function generator [positive (red)
lead to point H, negative (ground) lead to point G]. Set the function generator to 1000
Hz and 5 Vpp.
4. Connect Channel 2 of the oscilloscope across the resistor [positive (red) lead to point
V, negative (ground) lead to point G]. Display the voltage across the resistor. Note
that by changing the frequency of the function generator, you can change the
amplitude of the voltage. Why?
5. Display both channels of the oscilloscope at the same time, and measure the time
phase shift ∆t between the current (actually the voltage across the resistor), and the
voltage across the function generator.
6. Convert this to an angular phase shift, φ, from the relation:
2π∆t
φ=
T
Where ∆t is the horizontal displacement between the two peaks (Channel 1 and
Channel 2), and where T is the period of the sin wave generator (1/f). Make sure that
you have a good value for the period (i.e. check the setting on the generator vs. the
oscilloscope reading).
7. Compare the value of φ from Step 5 to the value expected from theory (see Part C
below).
8. Repeat Steps 5-7 for a frequency of 1500 hertz.
Part B: Resonant frequency
1. Replace the 27KΩresistor with a 100 Ω resistor and in order to make the resonant
peak sharper (Why will this happen?). Also be sure to use a 2 nano-farad capacitor
(Use the LCR meter to get a good value) to increase the resonance frequency.
6 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
2. Using the best value for L, calculate the resonant frequency.
3. With the oscilloscope connected across the resistor, vary the frequency of the
function generator and determine the highest voltage across the resistor. Record this
value (don't worry about being too precise).
4. Replace the oscilloscope across the resistor with the DMM ,and use it to indicate the
voltage across the resistor (essentially, you a measuring the current across the resistor
since I = V/R and R is constant.)
5. Vary the frequency, in increments of 2-5% of your resonance frequency f (in Hz),
from about 75% below your value for resonant frequency to about 75% above. Try to
get a total of about 20 points. Record values of VR, and f. When you are finished
with this section, plot VR vs. f and determine the fractional half width of the resonance
curve: ∆ω/ω. Compare this value to the theoretical value ∆ω ω = Z 3C L . Note
that Z equals the total resistance in the circuit…this includes the resistance of the
capacitor and inductor (be sure to measure them!).
Part C: Phase Angle, The Old-Fashioned Way
1. Set f = 1000 Hz. Return the 27 KΩ resistor to the circuit. Use the same capacitance
as in Part A.
2. Set Channel 1 and Channel 2 of the oscilloscope to ground, and be sure that both
horizontal lines are along the x-axis (i.e. the vertical deflection for both channels
when grounded should equal zero!).
3. Reset the voltage levels to DC, and set the time/div knob to "X-Y mode.". Obtain an
elliptical pattern on the screen of appropriate size for measurement by adjusting the
Horizontal and Vertical Gains. Adjust the Intensity and Focus controls to obtain a
sharp pattern.
4. Make the measurements on the shape of the elliptical trace required to determine the
phase difference angle φ. (Measure 2b and 2B or 2a and 2A for convenience, sin b/B
= sin 2b/2B). Use Figure 4 as a reference, if needed.
5. Compare φ as measured in Part A and Part C, Step 4 to each other and to the
theoretical value:
X − XC
φ = tan −1 L
.
R
NOTE: Be sure to compare radians with radians, or degrees with degrees!
6. If time permits, repeat Step C (1) for inductor and capacitor (instead of measuring the
voltage across the resistor). Sketch a graph of vL(t), vC(t) and ε(t).
7 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
Results:
Write at least one paragraph describing the following:
1. what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
2. your results, and what you learned from them
3. Think of at least one other experiment might you perform to verify these results
4. Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
8 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. If you do not turn in this page with your instructor’s signature with
your lab report, you will receive a 5% point reduction on your lab grade. How you divide
clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
9 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
DATA TABLES
Part A:
Best value of resistance: ____________
Best value of capacitance: ____________
Best value of inductance: ____________
f = 1000 Hz
Time phase shift ∆t: ____________
Experimental Phase Angle, φ: ____________
2π∆t
: ____________
Theoretical Phase Angle, φth =
T
f = 1500 Hz
Time phase shift ∆t: ____________
Experimental Phase Angle, φ: ____________
2π∆t
Theoretical Phase Angle, φth =
: ___________
T
Part B:
Resonant Frequency, fr: ____________
CRO: Highest Voltage across Resistor: ____________
DMM: Highest Voltage across Resistor: ____________
f (Hz)
VR (V)
Experimental Half Width, ∆ω/ω: ____________
Uncertainty in Experimental Half Width: ____________
Theoretical Half Width, ∆ω/ω: ____________
Agreement Between Experiment and Theory? ________
10 of 11
LPC Physics 2
RLC Circuits
©
2003 Las Positas College, Physics Department Staff
Part C:
Experimental Phase Angle, φexp: ____________
Uncertainty, δφexp: ____________
⎛ X − XC ⎞
Theoretical Phase Angle, φ th = tan −1 ⎜ L
⎟ : ____________
R
⎝
⎠
Uncertainty, δφth: ____________
Phase Angle from Part A: ____________
Agreement between values of Phase Angle?: ____________
11 of 11