MAPÚA UNIVERSITY School of EECE Intramuros, Manila Experiment #4 Impedance and Admittance of a Parallel RLC Circuit
Dotimas, Winvel P.
Group No. 2
Date Submitted: February 14, 2018 EE103L-B6 Score Date Performed: February 7, 2018
Engr. Ezperanza E. Chua Instructor
Preliminary Data Sheet
Sample Computation: Table 4.2: Through 300Hz S
= Siemens
Ohm’s Law: (Measured Values) G=
= . = 5.21
βL =
= . = 3.29
βC =
= . = 2.74
|Z| =
= . = 158.7 Ω
|Y| =
= ||
= . = 6.30 or 0.006
Equation Impedance:
G=
= 0.01 = 10 m
βL =
1 1 = 2πfL 2π(300 Hz)(100 mH) = 5.3
βC =
2 = 2(300 )(2.2 ) = 4.1
Pythagorean Theorem: (Impedance Triangle) β = βL – βC = 5.3 mS – 4.1 mS = 1.2 mS |Y| =
√ ( + β ) = √ (10 +1.2 ) = 10.07 mS
Θy = tan-1
( )
= tan-1
(1.2 10 )
= 6.84o
Complex Number: Y =
= + + −
= 10 +j1.2 mS (Rectangular form)
Y = |Y| ∟ Θy = 10.07 mS ∟ 6.84o (Polar form) Z=
= 100 Ω ∟ -6.84o
Procedure: 1. Connect the circuit as shown in Figure 4.1. 2. Set the sine wave function generator to 300 Hz, with constant output amplitude of 4 VRMS as measured on the voltmeter.
3. On Table 4.1, following the results table, record the readings of total current, current flowing in the resistor, inductor and capacitor respectively. This can be done by replacing each link with an ammeter one at a time. What did you observe about the magnitude of the voltage across the resistor, inductor, and capacitor? What is this parameter in the system? •
The voltage across the RLC circuit is the same in magnitude since each component are in parallel making the current the parameter in the system.
4. On another sheet of paper, draw to scale a phasor diagram showing I T, IR, IL, and IC set the voltage as reference. (see next page) What is the phase relationship between I L and IC? How do you combine the resultant value of I L and IC? •
IL and IC are out-of-phase with each other by 180 O and the resultant value could be get by subtracting I C from IL.
5. On Table 4.2, following the results table calculate the required parameters. 6. Switch of the power supply unit.
Interpretation of Results: During the experiment, we gathered data for Table 4.1 using Figure 4.1 found on the lab manual as reference. As a parallel circuit, we knew that the voltage through all the components will be the same as the total voltage from the supply whereas current will differ on each individual component. In Table 4.1, we set the function generator on five (5) different frequencies and obtain a unique voltage reading for each frequency (3.650 V for 300 Hz; 4.041 V for 400 Hz), nevertheless the same throughout the circuit per trial. We also notice the relationship that as the total current increases, the resistor current, IR , also increases, as well as the capacitor current, I C, while the inductor current, IL, decreases. We can expect this as I L and IC are out-of-phase with each other by 180O. Then, we use the values obtain from Table 4.1 to Table 4.2, Ohm’s Law (Measured Values), to get the experimental conductance, G, Susceptanc e, β, and the magnitude of impedance, |Z| and admittance, |Y|. In here, we all get the values using the basic equation of V=IR, where R is reciprocated as 1/R with the unit siemens to have the new equation V=I(1/R) in place so that we can substitute the value we get from Table 4.1 using Ohm’s Law. We can also see that the impedance given (158.7 Ω) is reciprocated as 1/|Z| to get the value of the admittance, |Y|, at 0.006 Siemens or (1/158.7 Ω). In the next part of Table 4.2, Equation Impedance, we now calculate for the susceptance, which is the inverse of the reactance for inductor (X L for 300 Hz = 188.496 Ω and inversing it will get 5.3 Ω -1 which is equal to β L) and capacitor, respectively as well as getting the conductance with the assumption of purely resistive branch which is just the inverse of resistance (1/100Ω = 0.01 siemens). For the impedance triangle, we knew that I L is out-of-phase with IC by 180O, so to get their vertical line we can say that I T=IL-IC, which we also applied to the total susceptance, β, (β =β L- βC), which is based from the triangle drawn above, we can see that their hypotenuse will be {G+ (βL- βC)} which is the admittance, therefore through
Pythagorean Theorem, we can say that the admittance, |Y| = this we can get the angle, Θ, through
√ ( + β ) and through
tan−(G). In the 300 Hz trial, we can see that
the circuit is inductive as X C > XL or βL > βC and the rest as capacitive as β C > βL. For the last part of the experiment, we write the admittance in rectangular form with the conductance being the real part and the susceptance as the imaginary part. Then we convert it to polar form or as a phasor quantity with the angle included through which when inverse we can get its impedance counterpart (in the 300 Hz sample, 10.07 mS ∟ 6.84o = 100
Ω ∟ -6.84o, both acting as inductive).
Conclusion: In conclusion, we proved that RLC circuits on AC in parallel behave the same way as DC does which the current for each component is different from one another and that the total supply voltage is the same as the voltage for each component. Instead, we investigated the impedance and admittance when being driven by a sinusoidal alternating current and its overall effect for the circuit. As we have taken in consideration DC circuits, the first part of the experiment after taking experimental voltage and current values tackles Ohm’s Law in which voltage is equal to the product of current and resistance. However, being in AC with the presence of phasor current and phasor voltage, we must say that the resistance will now be called as impedance, as it impedes the flow of the current which is true for a series connection. However, for a parallel connection, we must say that the impedance must be reciprocated which is now called as admittance, as it admits the flow of the current. The same is true for the conductance , G, as it is the opposite of resistance, thereby producing G=I/V . We also must take in account the pure reactance, X , in an AC circuit but being in parallel we must now consider its inverse form known as susceptance, β. All values taken as an inversed form will have a unit for siemens. After doing the experimental part of getting the susceptance, impedance and admittance through Ohm’s Law and the measured values, we now compare it with the equation impedance which now includes phasors. With the help of an impedance triangle, we know that the inductor current leads the voltage by 90 O whereas the capacitor current lags voltage by 90 O, concluding that the inductor current and
capacitor current is out-of-phase by 180O. As we know, the susceptance value is just a reciprocated form of the reactance value and with that in mind, we can conclude that the total susceptance, β, is βL-βC. With this representing as the vertical value of the triangle and the conductance, being purely resistive, represented as a horizontal value, we can now conclude that through Pythagorean’s Theorem that the magnitude of the admittance is the hypotenuse and is equal to the square root of the sum of the square value of conductance and the square value of susceptance. We can now also get the angle theta from this through
tan− (G).
Having completed all the necessary values, we can now write the admittance in rectangular form with the conductance representing the real part and the susceptance, much like the reactance, representing the imaginary part. Through here, we can get its equivalent polar form and getting the reciprocated value of the admittance, we can now get its equivalent value of impedance. To conclude, the characteristic of the circuit will always depend on the magnitude of inductive reactance and capacitive reactance in AC.
Final Data Sheet Table 4.1:
Frequency (Hz)
Total Voltage (V T)
Total Current (IT)
Resistor Current
Inductor Current
V RMS
mARMS
300
3.650
400
Capacitor Current (IC)
(IR )
(IL)
mARMS
mARMS
23
19
12
10
4.041
25
22
10.5
15
500
3.992
25
22
8
20
600
4.1
27
23
7
21
700
40
28
23
6
23
mARMS
Table 3.2:
Formula βL = βC = |Z| = = |Y| = || G =
Ohm’s Law
(Measured Values)
1 βL = 2πfL
Equation Impedance
700
5.21 mS
5.44 mS
5.51 mS
5.61 mS
5.75 mS
3.29 mS
2.60 mS
2 mS
1.71 mS
1.5 mS
2.74 mS
3.71 mS
5.01 mS
5.12 mS
5.75 mS
158.7 Ω
161.64 Ω
159.68 Ω
151.85 Ω
142.86 Ω
0.006
S
S
0.006 S
0.006
0.01
0.01 S
S
S
0.006
0.01
S
S
0.006
0.01
S
S
5.3 mS
3.98 mS
3.18 mS
2.65 mS
2.27 mS
4.1 mS
5.53 mS
6.9 mS
8.29 mS
9.7 mS
1.2 mS 10.07 mS
-1.5 mS
-3.72 mS
10.11 mS
10.67 mS
6.84O
-8.81O
-20.41O
-29.42O
-36.61O
= + + −
10+j1.2 mS
10-j1.55 mS
10-j3.72 mS
10-j5.64 mS
10-j7.43 mS
Y = |Y| ∟ Θy
10.07 ∟6.84O
10.12 ∟-8.81O
10.67 ∟-20.41O
11.48 ∟-29.42O
12.46∟36.61O
94 ∟20.41O
87 ∟29.42O
80 ∟36.61O
βC =
Pythagorean Theorem (Impedance Triangle)
300
0.01
G=
Frequency (Hz) 400 500 600
2
β = βL – βC |Y| =
√ + β Θy =
tan−() Y =
Complex Number
Z=
-5.64 mS -7.43 mS 12.46 11.48 mS mS
100 ∟-6.84O
99 ∟8.81O
