General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 1 / 22
[International Campus Lab]
DC and AC Circuits
Determine the behavior of resistors, capacitors, and inductors in DC and AC circuits.
Symbols:
: Resistance (or resistor)
: Capacitance (or capacitor)
: Inductance (or inductor)
: Emf of the power source
: Instantaneous terminal voltage or potential difference
: Maximum terminal voltage (or voltage amplitude)
: Instantaneous current
: Maximum current (or current amplitude)
: Instantaneous charge on the capacitor
: Maximum charge on the capacitor
: Frequency
: Angular frequency, 2
: Reactance
: Impedance
1. Direct Current (DC) R-C Circuit
A circuit that has a resistor and a capacitor in series, as
shown in Fig.1, is called an R-C circuit. We idealize the bat-
tery to have a constant emf and zero internal resistance.
We begin with the capacitor initially uncharged (Fig.1a). At
some initial time 0 we close the switch, permitting cur-
rent around the loop to begin charging the capacitor (Fig.1b).
The potential differences across the capacitor and the
resistor are 0 and at 0
respectively. The initial current through the circuit is giv-
en by Ohm’s law: ⁄ ⁄ .
Fig. 1 Charging a capacitor. (a) Just before the switch is closed, the charge is zero. (b) When the switch closed (at 0), the current jumps from zero to / . As time passes, approaches and the
current approaches zero.
Objective
Theory
----------------------------- Reference --------------------------
Young & Freedman, University Physics (14th ed.), Pearson, 2016
26.4 R-C Circuit (p.886~890)
30.4 The R-L Circuit (p.1024~1028)
31.3 The L-R-C Series Circuit (p.1052~1056)
31.5 Resonance in AC Circuits (p.1060~1062)
-----------------------------------------------------------------------------
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 2 / 22
As the capacitor charges, increases and decreases,
corresponding to a decrease in current. The sum of these two
voltage is constant and equal to . After a long time the ca-
pacitor becomes fully charged, the current decreases to zero.
Then the entire battery emf appears across the capacitor and
. Let represent the charge on the capacitor and
the current in the circuit at some time after the switch has
been closed. The instantaneous potential differences and
are
(1)
Using these in Kirchhoff’s loop rule, we find
0 (2)
(3)
As the charge increases, the capacitor charge approach-
es its final value . The current decreases and eventually
becomes zero. When 0, equation (3) gives
(4)
Note that the final charge does not depend on . We
can derive general expressions for as functions of time.
Substituting / into equation (3) yields
1
(5)
Fig. 2 Current and capacitor charge as function of
time for the circuit of Fig.1.
Integrating both sides of equation (5) and solving for and
, we find
1 / 1 / (6)
/ / (7)
Fig.2a and 2b are the graphs of Eqs. (7) and (6) respectively.
After a time equal to , the current in the circuit has de-
creased to 1/ 0.368 of its initial value . At this time, the
capacitor charge has reached 1 1/ 0.632 of its final
value . The product is therefore a measure of
how quickly the capacitor charges. We call the time
constant, or the relaxation time, of the circuit, denoted by :
(8)
When is small, the capacitor charges quickly. If is
small, it’s easier for current to flow, and the capacitor charges
more quickly.
Now after the capacitor has acquired a charge , we re-
move the battery from the circuit and connect point a and c
(Fig.3). The capacitor then discharges through the resistor,
and its charge eventually decreases to zero.
Fig. 3 Discharging a capacitor. (a) Before the switch is closed at the time 0, the capacitor charge is and the current is zero. (b) At time after the switch is closed, the capacitor charge is and the current is . The actual current direction is opposite to the direction shown; is negative. After a long time, and both approach zero.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 3 / 22
In Fig.3 we make the same choice of the positive direction
for current as in Fig.1. Then Kirchhoff’s loop rule gives equa-
tion (3) but 0; that is,
(9)
The current is now negative; this is because positive charge
is leaving the left-hand capacitor plate in Fig.3b, so the
current is in the opposite to that shown in the figure.
At 0, when , the initial current is / . To
find , we rearrange and integrate equation (9), then we find
/ (10)
⁄ / (11)
We graph the current and the charge in Fig.4.
Substituting equations (6), (7), (10), (11) and ⁄
into equation (1) yields the terminal voltages across the ca-
pacitor and resistor as functions of time:
1 chargingcapacitor (12)
chargingcapacitor (13)
dischargingcapacitor (14)
dischargingcapacitor (15)
Fig. 4 Current and capacitor charge as function of time for the circuit of Fig.3.
2. Direct Current (DC) R-L Circuit
A circuit that includes both a resistor and an inductor, and
possibly a source of emf, is called an R-L circuit, as shown in
Fig.5. The inductor helps to prevent rapid changes in current.
Let be the current at some time after switch S is closed,
and let / be its rate of change at that time. The potential
differences and across the resistor
and the inductor at that time are
(16)
Using these in Kirchhoff’s loop rule, we find
0 (17)
1 ⁄ (18)
Fig. 5 An R-L circuit.
Fig. 6 (a) Graph of versus for growth of current in an
R-L circuit with and emf in series. The final current is ⁄ ; after one time constant , the current is
1 1/ of this value. (b) Graph of versus for decay of current in an R-L circuit. After one time constant , the current is 1/ of its initial value.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 4 / 22
As Fig.6a shows, the instantaneous current first rises rap-
idly, then increases more slowly and approaches the final
value / asymptotically. At a time equals to / , the
current has risen to 1 1/ 0.632 of its final value. The
quantity / is therefore a measure of how quickly the cur-
rent builds toward its final value; this quantity is called the
time constant for the circuit, denoted by :
(19)
Now suppose switch S in Fig.5 has been closed for a while
and the current has reached the value . We open S and
close S at the same time. Then the current through and
decays smoothly, as shown in Fig.6b. The Kirchhoff’s-rule
loop equation is obtained from equation (17) with 0, then
the current varies with time according to
⁄ (20)
The terminal voltages across the capacitor and resistor are
1 ⁄ 0 → (21)
⁄ 0 → (22)
⁄ → 0 (23)
⁄ → 0 (24)
3. Resistor in an AC Circuit
When a sinusoidal current cos flows through , as
shown in Fig.7a, the instantaneous voltage across is
cos (31)
The maximum voltage , the voltage amplitude, is the co-
efficient of the cosine function:
(32)
Hence we can also write
cos (33)
The current and voltage are both proportional to
cos , so the current is in phase with the voltage. Equation
(32) shows that the current and voltage amplitudes are relat-
ed in the same way as in a DC circuit. Fig.7b shows graphs
of and as functions of time. The vertical scales for cur-
rent and voltage are different, so the relative heights of the
two curves are not significant. The corresponding phasor
diagram is given in Fig.7c. Because and are in phase
and have the same frequency, the current and voltage phas-
ors rotate together; they are parallel at each instant. Their
projections on the horizontal axis represent the instantaneous
current and voltage, respectively.
Fig. 7 Resistance connected across an ac source.
Note
Equations (16)-(24) is valid only for an idealized inductor
which has negligible resistance. A real inductor generally
has resistance due to the resistance of a long fine
wire. Thus an inductor can be considered as L-RL combi-
nation in series. Then equations (21)~(24) become
⁄ 1 ⁄ 0 → (25)
⁄ ⁄ 0 → (26)
⁄ ⁄ → 0 (27)
⁄ ⁄ → 0 (28)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 5 / 22
4. Capacitor in an AC Circuit
Next, we connect a capacitor with capacitance to the
source, as in Fig.7a, producing cos through the ca-
pacitor. Integrating ⁄ and using , we get
sin (34)
sin (35)
or cos 90° (36)
The current is proportional to the rate of change of voltage,
so the voltage and current are out of phase by a quarter-cycle,
as in Fig.8b. The peaks of voltage occur a quarter-cycle after
the corresponding current peaks, and we say that the voltage
lags the current by 90°.
Equation (35) shows that the maximum voltage is
(37)
To put this expression in a form similar to equation (32), we
define the capacitive reactance of a capacitor as
1
(38)
Fig. 8 Capacitor connected across an ac source.
Then
(39)
The SI unit of is the ohm, the same as for resistance,
because is the ratio of a voltage and a current. is
inversely proportional both to and to . Capacitors tend to
pass high-frequency current and to block low-frequency cur-
rent and DC. A device that preferentially passes signals of
high frequency is called a high-pass filter.
5. Inductor in an AC Circuit
Finally, we connect a pure inductor with inductance and
zero resistance, as in Fig.9a. Although there is no resistance,
there is a potential difference between the inductor terminals
because the current varies with time, giving rise to a self-
induced emf / . Thus the voltage is given by
/ , the negative of the induced emf.
cos sin (40)
or cos 90° (41)
The voltage and current are out of phase by a quarter-cycle.
Since the voltage peaks occur a quarter-cycle earlier than the
current peaks, we say the voltage leads the current by 90°.
Fig. 9 Inductance connected across an ac source.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
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From equation (40), the amplitude is
(42)
We define the inductive reactance of an inductor as
(43)
Using , we can write equation (42) in a form similar to (32)
(44)
The inductive reactance is a description of the self-
induced emf that opposes any change in the current through
the inductor. Thus, according to equation (43), the inductive
reactance and self-induced emf increase with more rapid
variation in current (that is, increasing angular frequency )
and increasing inductance .
Since is proportional to frequency, a high-frequency
voltage applied to the inductor gives only a small current.
Inductors are used in some circuit applications to block high
frequencies while permitting lower frequencies or DC to pass
through. A circuit device that uses an inductor for this purpose
is called a low-pass filter.
Fig. 10 An L-R-C series circuit with an ac source.
5. Alternating Current (AC) L-R-C Series Circuit
To analyze a series circuit containing , , , and an AC
source, we use a phasor diagram, as shown in Fig.10. The
instantaneous total voltage across all three components
is equal to the source voltage at that instant. The phasor rep-
resenting this total voltage is the vector sum of the phasors
for the individual voltage. We use the symbols , , and
for the instantaneous voltage across , , and , and
the symbols , , and for the maximum voltages. We
denote the instantaneous and maximum source voltages by
and . Then, , , , and .
The voltage across a resistor is in phase with the current
and its maximum value is . The phasor in Fig.10b,
in phase with the current phasor , represents the voltage
across the resistor.
The voltage across an inductor leads the current by 90°
and its voltage amplitude is . The phasor in
Fig.10b represents the voltage across the inductor.
The voltage across a capacitor lags the current by 90° and
its voltage amplitude is . The phasor in Fig.10b
represents the voltage across the capacitor.
So the magnitude of the phasor , which represents the
total voltage across the series of elements, is
(47)
We define the impedance of an AC circuit as the ratio of
the voltage amplitude across the circuit to the current ampli-
tude in the circuit. From equation (47), the impedance of the
L-R-C series circuit is
(48)
So we can rewrite equation (47) as
(49)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 7 / 22
Equation (49) has a form similar to , with impedance
in an AC circuit playing the role of resistance in a DC
circuit. Just as direct current tends to flow the path of least
resistance, so alternating current tends to follow the path of
lowest impedance.
We can see the meaning of impedance for a series circuit by
substituting and 1⁄ into equation (48).
1 (50)
Impedance is actually a function of , , , as well as of
the angular frequency . Hence for a given amplitude of
the source voltage applied to the circuit, the amplitude of
the resulting current will be different at different frequencies.
In the phasor diagram shown in Fig.10b, the angle be-
tween the voltage and current phasors is the phase angle of
the source voltage with respects to the current .
tan1⁄
(51)
If the current is cos , then the source voltage is
cos (52)
6. Resonance in AC Circuits
The impedance of the L-R-C series circuit depends on the
frequency, as equation (50) shows. Fig.11 shows graphs of ,
, , , and as functions of . As the frequency in-
creases, increases and decreases, hence there is
always one frequency at which 0. At this frequency,
has its smallest value, equal simply to the resistance
and the maximum value of occurs. This peaking of the
current amplitude at a certain frequency is called resonance.
The angular frequency at which the resonance peak oc-
curs is called the resonance angular frequency.
This is the angular frequency at which the inductive and
capacitive reactances are equal, so at resonance,
1
1
√ (53)
The resonance frequency is 2⁄ . This is the fre-
quency at which the greatest current appears in the circuit for
a given source voltage amplitude.
The current at any instant is the same in and . However,
the instantaneous voltages across and always differ in
phase by 180°. At the resonance frequency, , and
and are equal. Then, the total voltage
across the L-C combination in Fig.10a is exactly zero. The
voltage across the resistor is then equal to the source voltage.
So at the resonance frequency the circuit behaves as if the
inductor and capacitor were not there at all.
The phase of the voltage relative to the current is given by
equation (51). Fig.11b shows the variation of with angular
frequency.
If we can vary or of a circuit, we can also vary the res-
onance frequency. This is exactly how a radio or television
receiving set is tuned to receive a particular station.
Fig. 11 How variations in the angular frequency of an ac
circuit affect (a) reactance, resistance and imped-ance, and (b) impedance, current amplitude and phase angle.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 8 / 22
1. List
Item(s) Qty. Description
PC / Capstone software
1 Records, displays and analyzes the data measured by
various sensors.
Interface
1
Data acquisition interface designed for use with various
sensors, including power supplies which provide up to
15 watts of power.
Voltage Sensor
2 Measures terminal voltages.
L-R-C Circuit Board
1 Contains various electronic components including resis-
tors, capacitors, and an inductor.
Patch Cords
(with banana plugs)
3 Carry an electric signal.
Multimeter
1 Measures voltage, current, and resistance.
2. Details
(1) L-R-C Circuit Board
(2) Multimeter
Equipment
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 9 / 22
Experiment 1. DC R-C Circuit
(1) Build an R-C circuit and connect the circuit to the interface.
① Connect the red banana plug patch cord from the red
[OUTPUT1] port of the interface to the bottom jack of the
330μF capacitor on the L-R-C circuit board.
② Connect the black patch cord from the black [OUTPUT1]
port of the interface to the bottom jack of 100Ω resistor.
③ Connect the yellow patch cord across the inductor.
④ Connect one voltage sensor to [Analog input A] on the
interface and attach the leads across the capacitor, making
sure the red cable of the voltage sensor is connected to the
bottom jack of the capacitor.
⑤ Connect the other voltage sensor to [Analog input B] on
the interface and attach the leads across the resistor, making
sure the black cable of the voltage sensor is connected to the
bottom jack of the resistor.
(2) Set up the PASCO Capstone program.
(2-1) Add sensors.
Click the ports which you plugged the sensors or patch
cords into and select [Voltage Sensor] or [Output Voltage
Current Sensor] from the list.
Procedure
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 10 / 22
(2-2) Configure the signal generator.
We will supply a positive 4V square wave signal through
the circuit. Set the parameters of [Signal Generator Output 1]
as below.
[Waveform] : Square
[Frequency] : 1Hz 1s
[Amplitude] : 2V 2V~2V
[Voltage Offset] : 2V 2V 0V~4V
[Auto] (starts or stops the generator automatically.)
(2-3) Adjust the sample rates of the measurements.
Select [5.00kHz] for all sensors in the [Controls] palette.
(2-4) Set the auto recording conditions.
Click [Recording Conditions] in the [Controls] palette.
Measurement will be started manually and continued only
for 3 seconds. Thus, set the parameters of [Stop Condition]
only, as below.
[Condition Type] : TimeBased
[Record Time] : 3.000s
(2-5) Define an equation.
We will add a graph of total voltage across the capacitor and
the resister to verify Kirchhoff’s loop rule. Click [Calculator] in
the [Tools] palette and enter any name (for example, Vtotal)
and equal sign ‘=’ in a new line of [Calculations]. Now you will
enter the sum of the measurements of two voltage sensors. If
you type ‘[‘, a pop-up list opens. Complete the equation by
selecting [Voltage, Ch A (V)] and [Voltage, Ch B (A)] from the
list. Enter the [Units].
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 11 / 22
(2-6) Create graph displays.
Click and drag the [Graph] icon from the [Displays] palette
into the workbook page. You need 4 graph displays.
Select [Time s ] for the -axis of all graphs.
Change the unit of time to [ms] to read more precise values
of time.
Select following measurement for the -axis of each graph.
Output Voltage, Ch 01 : input signal through the circuit
Voltage, Ch A : terminal voltage across the capacitor
Voltage, Ch B : terminal voltage across the resistor
Vtotal : sum of two terminal voltages. See the step (2-5).
(3) Begin recording measurements.
Click the [Record] button at the left end of the [Controls] pal-
ette to begin collecting data.
The position of each graph depends on your configuration.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 12 / 22
(4) Analyze the graphs.
Adjust the scale of the graph. You can read off the coordi-
nates of the data using [Show coordinate…] icon.
(5) Analyze your results.
(5-1) Current through the R-C circuit
The graph of [Voltage, Ch B (A)], which is the terminal volt-
age across the resistor, represents through the
circuit, since is proportional to .
If you want to plot the graph of , you can use [Calculator].
Enter the equation ⁄ 100Ω in a new line of
[Calculations] and add one more graph of .
The graph of [Voltage, Ch A (A)], which is the terminal volt-
age / across the capacitor, represents on the
capacitor, since is proportional to . (You can also add
the graph of as explained above.)
Attach the graphs to the table below and compare them with
Fig.2 and Fig.4 of the [Theory].
Charging Discharging
or
or
(5-2) Kirchhoff’s loop rule
The graph of [Output Voltage, Ch 01] shows the emf
of the power source.
The graphs of [Voltage, Ch A (V)] and [Voltage, Ch B (A)]
show / and respectively.
Measure the voltages from the graphs for any 5 points of
time and verify Kirchhoff’s loop rule.
0 (2)
/ ⁄
The graph of [Vtotal] shows the sum of and . Com-
pare it with the graph of and verify the rule is always
valid.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 13 / 22
(5-3) Capacitance of the capacitor
Equations (12) and (14) represent the terminal voltage of the
charging or discharging capacitor.
1 / chargingcapacitor (12)
/ dischargingcapacitor (14)
If you know , you can find using your result and the
equations above.
After a time equal to , the capacitor charge has risen to
1 1/ 0.632 of its final value, i.e. the terminal voltage
across the capacitor reaches 63.2% of the input voltage
. Find the time from the graph, and then calculate the
capacitance of the capacitor using following equations.
1 1/ 1 / or ⁄
Similarly, find the time at which the charge of discharg-
ing capacitor reaches 1/ 36.8% of the initial value. Calcu-
late using following equations.
1/ / or ⁄ .
charging discharging
result
average
marked
(6) Repeat the experiment with 100μF capacitor.
Using 100μF and 100Ω, repeat the steps (3)-(5).
(7) Compare the results.
Compare the graphs of the terminal voltages across 330μF
and 100μF capacitors.
[…viewing of multiple runs | Select …] or [▼] activates view-
ing multiple runs together.
Attach the graphs and answer the following questions.
Q
Describe the difference of the graphs.
What is the physical meaning of the difference?
Explain your result using the time constant
A
Note
Most resistors and capacitors have a tolerance rating
expressed as a percentage. The tolerance value is the
extent to which the actual value is allowed to vary from its
nominal value. The tolerance is usually indicated by col-
ored stripes or codes on the surface. The most common
tolerance variation is 1~10% for resistors and 5~20% for
capacitors.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 14 / 22
Experiment 2. DC R-L Circuits
(1) Build an R-L circuit and connect the circuit to the interface.
① Connect the red banana plug patch cord from the red
[OUTPUT1] port of the interface to the right jack of the
8.2mH inductor on the L-R-C circuit board.
② Connect the black patch cord from the black [OUTPUT1]
port of the interface to the bottom jack of 100Ω resistor.
③ Connect one voltage sensor to [Analog input A] on the
interface and attach the leads across the inductor, making
sure the red cable of the voltage sensor is connected to the
right jack of the inductor.
④ Connect the other voltage sensor to [Analog input B] on
the interface and attach the leads across the resistor, making
sure the black cable of the voltage sensor is connected to the
bottom jack of the resistor.
⑤ Insert the iron core inside the inductor.
(2) Set up the PASCO Capstone program.
Follow the steps of the previous experiment, EXCEPT NEXT.
(Before you modify configuration, you should save your data
of the R-C experiment with a different file name!)
(2-1) Configure the signal generator.
[Waveform] : Square
[Frequency] : 0.01s
[Amplitude] : 2V 2V~2V
[Voltage Offset] : 2V 2V 0V~4V
[Auto] (starts or stops the generator automatically.)
If the input voltage alternates before the terminal voltage
approaches a certain value, set [Frequency] at 50Hz.
(2-2) Adjust the sample rates of the measurements.
Select [ . ] for all sensors in the [Controls] palette.
If you set the sample rates too high, the computer system
may stop with an error message. If you have any problem
proceeding the experiment with 50kHz, decrease the sample
rates. However, remember that too low rate precludes the
accurate analysis of the result.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
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(2-3) Set the auto recording conditions.
Click [Recording Conditions] in the [Controls] palette.
The measurements will be started manually and continued
for only 0.03 seconds. Set the parameters of [Stop Condition]
as below. (The computer system may stop if the measuring
time is too long.)
[Condition Type] : TimeBased
[Record Time] : .
(3) Begin recording measurements.
The position of each graph depends on your configuration.
(If the input voltage alternates before the terminal voltage
approaches a certain value, set [Frequency] at 50Hz. See
step (2-1))
(4) Analyze the graphs.
(5) Analyze your results.
(5-1) [Optional] Internal resistance of an inductor
The graphs show that the terminal voltages across the in-
ductor and the resistor do not approach the values as we
predicted.
These result from the internal resistance of the inductor.
For more accurate observations, we should consider .
Measure the asymptotic values of and and calculate
using ⁄ ⁄ .
Note
The terminal voltage across the inductor theoretically
decreases from the input voltage, 4V , however, your
graph will probably not, since the interface cannot detect
the sudden change of voltage.
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(5-2) Current through the R-L circuit
The graph of [Voltage, Ch B (A)], which is the terminal volt-
age across the resistor, represents through the
circuit, since is proportional to .
If you want to plot the graph of , use [Calculator] as ex-
plained in previous experiment.
Attach the graphs and compare them with the Fig.6.
0 → → 0
or
(5-3) Kirchhoff’s loop rule
Follow the instructions of the previous experiment.
(5-4) Inductance of the inductor
1 ⁄ 0 → (21)
⁄ 0 → (22)
⁄ → 0 (23)
⁄ → 0 (24)
If you know , you can find using your result and the
equations above.
At a time equal to ⁄ , the current has risen to 1 1/ ,
or 63.2%, of its final value, i.e. the terminal voltage across
the resistor reaches 63.2% of the input voltage. Find the time
from the graph, and calculate the inductance of the
inductor using following equations. (If you cannot find the
exact value of time, calculate the time by supposing two ad-
jacent points are in a linear relationship.)
1 1/ 1 ⁄ or
Similarly, find the time at which the current across the
resistor is decreased to 1/ 36.8% of the initial value. Cal-
culate using following equations.
1/ ⁄ or
0 → → 0
result
average
marked
(6) Repeat measurements with different conditions.
Repeat the steps (3)-(5) with the combinations of 33Ω
and 8.2mH, and 10Ω and 8.2mH.
Repeat the experiment without iron core inside the inductor.
Compare the results and explain why.
1 1/ ⁄
⁄ 1 ⁄
or
Note
For more accurate calculation, we should consider the
internal resistance of the inductor.
⁄ 1 ⁄ 0 → (25)
⁄ ⁄ 0 → (26)
⁄ ⁄ → 0 (27)
⁄ ⁄ → 0 (28)
In this case, the equation in step (5-4) becomes
You should also consider that the experimental value of
the terminal voltage across the resistor is lower than the
predicted value of an idealized circuit.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
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Experiment 3. Resistor in an AC Circuit
(1) Connect 10Ω to the interface signal output.
(2) Set up the PASCO Capstone program.
(2-1) Add an [Output Voltage Current Sensor].
(2-2) Configure [Signal generator].
[Waveform] : Sine
[Frequency] : 100Hz 0.01s
[Amplitude] : 5V 5V~5V
[Auto]
(2-3) Create a scope display.
Click and drag the [Scope] icon from the [Displays] palette
into the workbook page.
(2-4) Add an independent -axis on the right of the graph.
(2-5) Define the axes.
-axis: Time(s)
-axis(left): Output Voltage (V)
-axis(right): Output Current (A)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
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(2-6) Select [Fast Monitor Mode] in the controls palette.
[Fast Monitor Mode] displays data without recording. Check
that [Recode] button is changed to [Monitor] button.
(3) Start monitoring data.
(3-1) Click [Monitor] in the controls palette.
(3-2) Click [Activate … trigger] in the toolbar to horizontally
align repetitions of the signal.
Color in legend matches plot color.
(4) Analyze your results.
Measure the phase difference between cos
and cos . Compare your result with Fig.7 in [Theory].
Experiment 4. Capacitor in an AC Circuit
Connect 100μF to the interface signal output.
[Waveform] : Sine
[Frequency] : 100Hz 0.01s
[Amplitude] : 5V 5V~5V
Measure the phase difference between cos
and cos . Compare your result with Fig.8 in [Theory].
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
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Experiment 5. Inductor in an AC Circuit
Connect 8.2mH to the interface signal output.
[Waveform] : Sine
[Frequency] : 0.002s
[Amplitude] : 5V 5V~5V
Measure the phase difference between cos
and cos . Compare your result with Fig.9 in [Theory].
Experiment 6. AC L-R-C Series Circuit
(1) Build an L-R-C series circuit.
Connect 8.2mH, 10Ω, and 100μF in series.
(2) Set up the PASCO Capstone program.
Follow the setups of the experiment 3~5. Also, the follow-
ing additional setup is required.
(2-1) Create a table display.
Click and drag the [Table] icon from the [Displays] palette
into the workbook page. Select [User-Entered Data] and
enter Frequency Hz and Curret A on each column.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
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(2-2) Create a graph.
-axis: Frequency (Hz) (User-Entered Data)
-axis: Current (A) (User-Entered Data)
You now have [Scope], [Table] and [Graph] displays on the
workbook page.
(2-3) Change the unit of [Current] to [mA] in the [Scope] dis-
play for more accurate measurements
(3) Start monitoring data.
Set [Frequency] of [Signal Generator] at .
Click [Monitor] and then click [Stop].
Measure the amplitude of the alternating current using
[Show data coordinates …].
Enter the frequency and the current amplitude on the table.
(5) Repeat measurement, increasing the frequency.
Increase the frequency in 10Hz increments until 200Hz
and continue to enter values of frequency and current,
In the range of slight variation in current, Increase the fre-
quency in 5Hz or 2Hz increments
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
Lab Office (Int’l Campus)
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(6) Analyze your result.
The graph shows the current through the circuit as a func-
tion of frequency.
Find the resonance frequency of the circuit from the graph
and compare your measured value with the theoretical value.
(It is recommended to calculate the theoretical value using
the measured and in the experiment 1 and 2. The actu-
al value of the components may differ from the nominal value.)
result
theory 1 2 √⁄
(7) Observe the variation of the phase difference.
Compare the phase differences between and in the
condition of , , and .
Attach the results on your report and answer the following
questions.
Q
At the resonance frequency;
1. Which graph among the figure 7-9 shows the phase
relationship between and ?
2. Find the impedance of the circuit using equation (50),
1⁄ .
3. How is the behavior of the combination of the capacitor
and the inductor in the circuit? Explain using a phase dia-
gram as in Fig.10 of theory.
A
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
DC and AC CircuitsVer.20160901
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Your TA will inform you of the guidelines for writing the laboratory report during the lecture.
Please put your equipment in order as shown below.
□ Delete your data files and empty the trash can from the lab computer.
□ Turn off the Computer and the Interface.
□ Return the Iron Core of the inductor to its storage location on the L-R-C circuit board.
□ Handle the L-R-C Circuit Board with care. The plastic case is fragile.
□ Turn off the Multimeter.
Result & Discussion
End of Lab Checklist
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