Lab_#4_1444_009_Juan_Rubin_1001768492
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Physics Lab (Online Simulation)
AC CIRCUITS
Electricity and Light
Unit 4
TA name: Michael Rooks
Due Date: 03/24/2021
Student Name: Juan Rubin Student ID: 1001768492
Theory:
The capacitive reactance and the inductive reactance of a capacitor and an inductor, respectively, are defined as:
X
C = 1/(
C) and X
L =
L
,
where
= 2
f is the angular frequency in rad/s, and f is the frequency in Hertz (Hz). If the current in the RC circuit is I(t) = I
o
sin(
t)
, then the maximum voltage across each element
and the entire circuit is:
V
Ro = I
o
X
R , V
Co = I
o
X
C , and V
o = I
o
Z ,
where X
R is simply the resistance R of the resistor, and Z = sqrt (
R
2
+X
C
2
) is the impedance of the circuit. The relation between the maximum voltage across each element and the entire circuit is: V
o = sqrt(
V
oR2 +V
oC2
) ….. (1)
For an RL circuit, the voltage across the elements and the entire circuit is:
V
Ro = I
o
X
R , V
Co = I
o
X
L , and V
o = I
o
Z , with the impedance of the circuit being Z = sqrt (
R
2 +X
L
2
)
The relation between the voltages then is: V
o = sqrt (
V
oR
2
+V
oL 2
) …… (2)
1
Physics Lab (Online Simulation)
In the case of a resistor, an inductor, and a capacitor connected in series to an AC source (RLC circuit), as shown on the diagram below, the impedance of the circuit is equal to: Z = sqrt (
R
2
+(X
L
-X
C
)
2
) …… (3) If the current in the circuit is given by I(t) = I
o
sin(
t)
, the voltage across the circuit as a function
of time is then given by V(t) = V
o
sin(
t+
)
, where
is called a phase angle and for the RLC circuit is equal to: tan(
) = (X
L
-X
C
)/(X
R
) …… (4)
Experimentally, the phase angle between the current and the voltage in the circuit, can be determined by measuring the voltage value at the time when the current is zero. If the current I(t)
= I
o
sin(
t)
= 0 at some time t
, then sin(
t) = 0 at that time. Using the trigonometric relation for sin (A + B)
, we can write down for the voltage: V(t) = V
o
( sin(
t) cos(
) + cos(
t) sin(
) ) = V
o sin(
) The phase angle
,
then, can be determined as:
=arccos(V(t)/V
o ) ….. (5)
2
Physics Lab (Online Simulation)
You can refer to the course book on AC Circuits to complete the following questions
Please type all your answers in Blue
This pre-lab is worth 15 points.
1)
AC current through a perfect inductor or capacitor is 90 degrees out of phase with the AC
voltage across the inductor or capacitor.
-1
2)
What is reactance? What is impedance? What are their units?
Reactance is the measurement of inertia running against the flow of current, and impedance is the forms of opposition running against the current, including resistance and reactance. Both parameters’ units are in ohms.
-4
3)
What is the name for the units of an inductor and capacitor and the symbols which represent them?
The unit associated with an inductor is the Henry, denoted by the capital letter “H”; the unit associated with a capacitor is known as the Farad, denoted by the capital letter F.
-4
4)
For an inductor, the current and voltage is out of phase by an angle
. How can this angle be determined theoretically?
The angle can theoretically be determined by having the inductor being out of phase with the voltage. This can be calculated by using the following formula ∅
=
arccos
(
V
t
V
0
)
-2
5)
What are the equations for the time constants, τ, for a RL and a RC circuit?
Time constants equation is T = L/R. R is the resistor value, measured in Ohms, and L is the
determined value of the inductor, measured in Henrys.
The other equation representing the time constants is T = R*C, R is the resistor value,
measured in ohms, and C is the capacitor value, measured in Farads.
-4
3
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Physics Lab (Online Simulation)
This lab uses simulation kit in Phet Labs.
https://phet.colorado.edu/sims/html/circuit-construction-kit-ac/latest/circuit-construction-kit-
ac_en.html
There are 3 choices, open the “Lab”
Activity 1: Voltage across the elements in RC circuit. 1.
Connect a resistor and a capacitor in series and connect them to an AC source. a.
Click on a resistor, a capacitor and an AC source and drag them into the circuit board.
b.
Drag wires and connect the elements in series to form an RC circuit. Two elements are connected when their ends overlap. To disconnect two elements, right-click on the joint and select "split junction".
c.
To change the values for an element, right-click over that element and select change value.
Put a checkmark in “values” to turn it on. How does A/C voltage differ from DC voltage? At first glance it appears that while DC voltage flows in one direction, AC voltage
tends to flow in different directions.
Change the component values to those in Table 1, then paste a screen shot of your
circuit below. 4
Physics Lab (Online Simulation)
Table 1
R(
)
C (F)
E
o
(V)
f (Hz)
4.0
0.06
12
1.0
2.
Perform the experiment:
a.
Measuring the maximum voltage using the voltmeter. i.
Use the voltmeter to measure the maximum voltage across the A/C voltage source. Note* Use the pause button to pause the simulation. What value did you obtain?
11.8V
b.
Measuring the maximum voltage on the Voltage Chart. i.
Use the Voltage Chart to measure the amplitude of the voltage on the chart
(it's half the height from a trough to a crest). What value did you obtain?
12V
5
Physics Lab (Online Simulation)
c.
Taking measurements. Measure the maximum voltage across the resistor, V
R
, the capacitor, V
C
, and across both elements, V
RC
. Note* you can use the Voltage
chart in combination with the voltmeter to judge when to pause the simulation to obtain a max value. Complete Table 2.
Table 2
Frequency (Hz)
V
R
(V)
V
C
(V)
V
RC
(V)
V
RC
2 (V)
V
R
2
+ V
C
2 (V)
% difference
1 Hz
9.75
6.6
11.5
132.25
138.62
4.5%
Results: Is the sum of the square of the maximum voltages across R and C, V
R
2
+ V
C
2
, equal to the square of voltage across both elements, V
RC
2 in agreement with Eq.1?
When analyzing the data outputted by both methods of calculations it appears that they
agree with Eq.1 explained by their proximity in value.
Activity 2: Voltage across the elements in RL circuit 1.
Connect a resistor and an inductor in series and connect them to an AC source. a.
Click on a resistor, inductor and AC source and drag them into the circuit board.
b.
Drag wires and connect the elements in series to form an RL circuit. Two elements are connected when their ends overlap. To disconnect two elements, right-click on the joint and select "split junction".
c.
To change the values for an element, right-click over that element and select change value. 6
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Physics Lab (Online Simulation)
Change the component values to those in Table 3, turn on “values” by placing a checkmark, then paste a screen shot of your circuit below. Table 3
R (
)
L (H)
Eo (V)
f (Hz)
50
10
10
1.0
Perform the experiment:
Taking measurements. Measure the maximum voltage across the resistor, V
R
, the inductor, V
L
, and across both elements, V
RL
. Complete Table 4.
Table 4
Frequency (Hz)
V
R
(V)
V
L
(V)
V
RL
(V)
V
RL
2 (V)
V
R
2
+ VL
2 (V)
% difference
1 Hz
6.32
7.45
10
100
95.44
4.6%
Results: Is the sum of the square of the maximum voltages across R and L, V
R
2
+V
L
2
, equal to 7
Physics Lab (Online Simulation)
the square of voltage across both elements, V
RL
2 in accordance with Eq.2?
When analyzing the data outputted by both methods of calculations it appears that they agree with Eq.2 explained by their proximity in value.
Activity 3: Impedance of an RLC circuit. 1.
Set up the RLC circuit.
a.
Click on a resistor, an inductor, a capacitor, and an AC source and drag them into the circuit board.
b.
Select an ammeter and voltmeter from the Tools Menu on the right. Connect the Ammeter in series to the RLC circuit.
c.
Drag wires and connect the elements in series to form an RLC circuit with an ammeter and AC source.
Change the component values to those in Table 5, turn on “values” by placing a checkmark, then paste a screen shot of your circuit below.
8
Physics Lab (Online Simulation)
Table 5
R (
)
L (H)
Eo (V)
f (Hz)
C (F)
10
10
10
0.1
0.1
2.
Perform the experiment:
a.
Measure the maximum voltage across the entire RLC circuit, V
o
.
b.
Measure the maximum current in the circuit, I
o
9
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Physics Lab (Online Simulation)
c.
Measure the value for the voltage when the current in the circuit is zero, V
I=0 3.
Data and Results.
a.
For the experimental values of the impedance and the phase angle, use the relation
Z=V
o
/I
o
, and Eq.5, respectively.
b.
For the theoretical value of the impedance and the phase angle, use Eqs.3-4. c.
Complete Table 6.
Table 6
10
Physics Lab (Online Simulation)
Vo (V)
Io (A)
V
I=0
(V)
Z
exp
(
)
Z
th
(
)
% differenc
e in Z
exp
in (deg.)
th
in (deg.)
% difference
in 10
0.73
7.3
13.88
14.2
2.25%
81
84.1
3.69%
11
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