Sirindhorn International Institute of Technology
Thammasat University
School of Information, Computer and Communication Technology
___________________________________________________________________________
COURSE
: ECS 304 Basic Electrical Engineering Lab
INSTRUCTOR
: Dr. Prapun Suksompong (prapun@siit.tu.ac.th)
WEB SITE
: http://www.siit.tu.ac.th/prapun/ecs304/
EXPERIMENT
: 05 RC Circuit and Resonant RLC Circuit
___________________________________________________________________________
I. OBJECTIVES
1.
2.
To investigate the discharging of a capacitor, and the time constant of an RC circuit.
To determine the resonant frequency and frequency response of a series RLC circuit.
II. BASIC INFORMATION
1.
A function of capacitor in electric circuits is to restore energy by charge charging. Rate
of charging represented by capacitor voltage V(t) can be expressed by the following
equation.
t
V (t )  Vin (1  e  )
where Vin is the voltage applied to the capacitor, and  is the time constant which is
defined, in the case of RC circuits, as
  RC
When the capacitor discharges energy, the voltage across the capacitor will be
decreased according to the following equation.
t
V (t )  V0e 
where V0 is the initial voltage .
When a capacitor of C farads is charged through a resistor of R ohms, the time constant 
in seconds of the charging circuit is  = RC. In one time constant, a capacitor charges to
approximately 63.2 % of the applied voltage across its terminals. If we apply a squarewave signal to an RC circuit, we will get an output signal as shown in Figure 5-1. At the
rising edge of the square-wave, the capacitor will be charged; and at the trailing edge,
the capacitor will be discharged.
In part A of this experiment, the value of  are found using three different methods (see
Figure 5-1-2):
1) Measure t0.37 . Then,   t0.37 .
2) Measure thalf . Then, calculate   thalf ln 2 .
3) Measure R and C. Then, calculate   RC .
2.
Resistors limit the amount of current in dc as well as in ac circuits. In addition to
resistors, reactive components, such as inductors and capacitors, impede currents in ac
circuits.
3.
Inductive reactance of a coil is given by XL = 2f L ohms, where L is the inductance in
henries, f is the frequency of signal in the circuit. The ac voltage across a coil, V L is
equal to the product of the alternating current in the coil and the inductive reactance of
the coil; that is VL = I  jXL where j =  1 .
Voltage
time
Discharging
period
Charging
period
pulse input voltage
voltage across capacitor
Figure 5-1-1: Pulse input voltage and the voltage across a capacitor.
4.
The capacitive reactance is given by XC = 1/(2f C) ohms. It is inversely proportional to
frequency and capacitance.
2
4  V0
2  V0 2
1.47  V0 e
t0.37
thalf
Figure 5-1-2:  measurement.
5. A series RLC circuit, as well as its impedance phasor diagram is shown in Figure 5-2. The
reactance in the series RLC circuit is X = XL  XC, and the impedance is found by using
the formula Z = R + j(XL  XC).
R
V
XL

L
0
R
-90°
C
XC
Figure 5-2: A series RLC circuit and its impedance phasor diagram.
We can understand the behavior of an RLC circuit according to the change of
frequency from the graph in Figure 5-3. This graph illustrates how impedance changes
with frequency. At frequency below fo, XC > XL, and the circuit is capacitive. At the
resonant frequency fo, XC = XL, so the circuit is purely resistive. At frequencies above fo,
XC < XL, and the circuit is thus inductive.
The impedance magnitude is minimum at resonance, and increases in value above and
below the resonant point. At zero frequency, both XC and Z are infinitely large, and XL is
zero. This is because the capacitor looks like an open circuit at 0 Hz, while the inductor
looks like a short. As the frequency increases, XC decreases, and XL increases. Since XC is
larger than XL at frequencies below fo, Z decreases along with XC. At fo, XC = XL and Z =
R. At frequencies above fo, XL becomes increasingly larger than XC, causing Z to increase.
3
Z( )
XC
XL
Z
Z=R
0
fo
f
Figure 5-3: Impedance of a series RLC circuit as a function of frequency.
Im
Vm
R
Vm
2R
0

  
Figure 5-4-1: Frequency Response curve of a series RLC circuit.
Figure 5-4-1 shows the frequency response curve of a series RLC circuit. Three
significant points have been marked on the curve. These are 0, the resonant frequency,
1
and 1 and 2. For series RLC circuits, we have 0 
. Points 1 and 2 are
LC
located at 70.7 % of the maximum (at o) on the curve. They are called the half power
points, and the frequency separation between them is called the bandwidth (BW) of the
circuit. For the series RLC circuit, BW is defined as
BW =  2  1 
R
L
For a circuit intended to be frequency selective, the sharpness of the selectivity is a
measure of the circuit quality. The quality of the frequency response is described
quantitatively in terms of the ratio of the resonance frequency to the bandwidth. This
ratio is called the quality factor (Q) of the circuit. Then we have
Q = o/BW
4
For the series RLC circuit, the current in the circuit is maximum at o, as can be seen
in Figure 5-4-1. At resonant frequency, the voltages |VL| and |VC| across L and C,
respectively, are equal. Thus, at o
where
IXL = IXC
I =V
R
VL = VC = VQ
Q = XL/R
= XC/R
1 L
.
R C
= 0 /BW
=
The equation VL = VC = VQ becomes significant for the values of Q > 1. For such values,
VC and VL are greater than the applied voltage V.
Remark: In general, the maximum values for VL and VC will NOT occur at resonant
frequency. (See Figure 5-4-2.)
𝑉𝐿
𝑉𝐶
𝑉𝑅
f
f0
Figure 5-4-2: Voltage values across the resistor  VR  , the capacitor  VC  , and the inductor
 V  in a series RLC circuit.
L
5
III. MATERIALS REQUIRED
- Function generator, multi-meter, and oscilloscope.
- Resistor: 100 , 1 k.
- Inductor: 22 mH.
- Capacitors: 0.01 F, 0.47 F, and 0.1 F.
IV. PROCEDURE
Part A: To investigate the discharging of a capacitor
oscilloscpe
1k
+
4Vp-p
square
wave
0.1 F

Figure 5-5: The RC circuit for investigating the discharging of a capacitor.
1. Connect the circuit of Figure 5-5, where the oscilloscope is in DC mode.
2. Set the output of the function generator to be a square wave with amplitude of 4 Vp-p and
frequency of 500 Hz. Then set the DC offset level to 2 V, such that the square-wave
waveform has its lower peak at 0 V and its higher peak at 4 V.
3. Adjust volts/div and time/div until the waveforms can be seen on the oscilloscope. Then,
draw the voltage waveforms across the function generator (shown by dash line in Figure
5-1), and across the capacitor (shown by solid line) in Table 5-1.
4. Next, find the time constant  by measuring the time elapse t0.37 during the capacitor
discharges energy, while the voltage across the capacitor drops to 0.37 times of the
maximum voltage as shown in Figure 5-6. Then, put the value in Table 5-2.
Figure 5-6: The discharging behavior of a capacitor.
6
Q: Why 0.37 times of the maximum voltage?
A: The voltage value after one time constant is equal to 0.37 times of the maximum
voltage. From the following equation,
t
V (t )  V0e 
At t = ,
V (t )  V0e1
V (t )  0.37V0
Hence, we can find  when the voltage is 0.37 times of the maximum voltage. Note that
V0 in the above equations is equivalent to VPS in Figure 5-6.
5. Measure the time elapse thalf that the voltage across the capacitor drops to half of the
maximum voltage.
6. Calculate  by the following equation, and record the value in Table 5-2.

7.
thalf
ln 2
Calculate the product of R and C, and record the value in Table 5-2.
Part B: Series resonant RLC circuit
Oscilloscope
ChA
ChB
V
R2
100 
L
Sine-wave
generator
C
Figure 5-7: The series RLC circuit.
1.
You are supplied with one inductor of the value 22 mH, and three capacitors with values
of 0.01, 0.47 and 0.1 F. Measure all values of the inductor and capacitors, and
7
calculate the resonant frequency fo = o/2 =
1
Hz for each of the capacitors.
2 LC
Record the calculated resonant frequencies in Table 5-3 under “Calculated resonant
frequency”. Connect the circuit of Figure 5-7 with C = 0.01 F.
2.
Turn on the sine-wave generator.
3.
Turn on the oscilloscope, and adjust it to view the sine wave output of the generator.
4.
Increase the output of the generator until the scope indicates an 8 Vp-p voltage.
Maintain this voltage throughout this experiment.
5.
Observe the rms value of VR across the resistor as the frequency of the generator is
varied. Record the resonant frequency fo at which VR is maximum. Measure fo and
record the value in the “0.01F” row of Table 5-3.
Caution: As you adjust the frequency f, the voltage across the generator output will
change. Readjust it back to 8 Vp-p.
6.
Replace the 0.01 F with a 0.47 F capacitor. Check the generator output voltage to
verify that it is 8 Vp-p; adjust the voltage if necessary. Find fo and record the value in
the “0.47 F” row of Table 5-3.
7.
Replace the 0.47F with a 0.1F capacitor. Check the generator output voltage and
adjust to maintain 8 Vp-p. Find f0 and record the value in the “0.1F” row of Table 5-3.
8.
Check the generator output voltage, and adjust the value to maintain at 8 Vp-p. With C =
0.1F, vary and record the frequency according to Table 5-4. Record the corresponding
values of VR, VL, and VC in Table 5-4, where VL and VC are the voltage across L and C,
respectively.
8
Table 5-1. Square wave input voltage and voltage across capacitor
volts/div = _______________, time/div = _________________.
Table 5-2. Time constant of an RC circuit
 from t0.37
 from thalf / ln 2
Table 5-3. Resonant frequency of a series RLC circuit
Inductor L, mH
Capacitor C, F
TA Signature:
________________________
 from RC
TA Signature:
________________________
Resonant Frequency fO, Hz
Calculated
22
0.01
22
0.47
22
0.1
Measured
TA Signature:
________________________
9
Table 5-4. Frequency response of a series RLC circuit
Frequency
deviation
Frequency
f, Hz
Voltage across R,
VR, V
Voltage across L,
VL, V
Voltage across C,
VC, V
fO – 1000
fO – 500
fO – 200
fO – 100
fO
fO + 100
fO + 200
fO + 500
fO + 1000
TA Signature:
________________________
10
QUESTIONS
True or False
1. _______ In a circuit containing impedance, the current and voltage are 90 degrees out of
phase.
2. _______ The value of reactance in a series RLC circuit can exceed the value of
impedance.
3. _______ The value of resistance in a series RLC circuit can exceed the value of
impedance.
4. _______ In a series RLC circuit, the capacitor voltage can be greater than the source
voltage.
5. _______ A series RLC circuit operating above its resonant frequency is inductive.
6. _______ There is only one combination of L and C for each resonant frequency.
Fill in the blanks.
7.
At resonance, the power factor of a circuit is ____________.
8.
Impedance is a combination of _______________ and _______________.
9.
Give the definition of bandwidth.
Bandwidth is ____________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
10. The series RLC circuit in Figure 5-2 has an inductance of 100 H, a capacitance of 200
pF, and a resistance of 10 . Determine its resonant frequency and bandwidth. Also
determine XC, XL, IL, and VL at 1.0 MHz if the source voltage V = 1.0 Vpeak.
Resonant frequency = _______________
Bandwidth = _________________
At 1.0 MHz
XC = ______________
XL = ______________
IL = ______________
VL = ______________
11
11. The series RLC circuit in Figure 5-2 has a capacitance of 25 F, and a resistance of 10
. We then adjust the inductance until VR is maximum. Determine VR, VL, and VC at 
= 2000 rad/sec if the source voltage V = 50 Vpeak.
VR = ______________
VL = ______________
VC = ______________
12. What are the characteristics of series resonant circuits?
13. Suppose you set the voltage across the output of the signal generator at 2Vrms. You then
connect the generator output across a 100Ω resistor. Now, you measure the voltage across the
generator again but you get a value which is significantly less than 2 Vrms. Why?
14. How can you to adjust the DC offset of the signal generator?
15. Is changing the DC offset on the signal generator the same as changing the vertical
position of the trace in oscilloscope? Explain.
16. What are the differences between the DC and AC mode of the oscilloscope?
12