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University of North Carolina at Charlotte

Department of Electrical Engineering

Laboratory Experimentation Report

Name: Ethan Miller Date: July 16, 2013

Course Number: ECGR 2156 Section: L90

Experiment Title: [5] Magnetic Properties of a Small Transformer, [6] Impedance, [7] AC

Operation of RLC Network

Lab Partner: Waqee Hassan Experiment Number: 5, 6, 7

Objective:

Experiment 5:

The objective of this experiment was to investigate the relationships of magnetic

properties and fields of the transformer. The relationship was found to be the difference between

the current, voltage, and power of the transformer.

Experiment 6:

The objective of this experiment was to validate the existing relationships between

magnitude and phase relationship to AC voltage and current. These relationships were done by

the following (1) resistance load, (2) inductive load and (3) capacitance load.

Experiment 7:

The objective of this experiment was to attain voltages, current and power from the

circuit from a sinusoidal source and evaluate the findings with theoretical calculations.

Equipment List:

Experiment 5:

Items Asset #

Function Generator 000000

1 ohm resistor 000000

Oscilloscope 000000

Experiment 6:

Items Asset #

5.1 And 1 K ohm resistor 000000

Function Generator 000000

Oscilloscope 000000

Capacitor 000000

Inductor 000000

Experiment 7:

Items Asset #

Inductor 000000

1 K ohm resistor 000000

AC supply 000000

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Relevant Theory/Background Information:

Experiment 5:

A machine with different applications that collects from small audio sound to any major

power transmission systems is called a transformer. Transformers have provided systems by a

magnetic coupled electrical energy from primary to secondary winding. This voltage has to

either step up or down to form a ratio between the secondary to primary winding.

In Equation 1, a wire that carries current will produce a magnetic field. Surrounding this

wire is directly proportional to the current and varies inversely with the space of the wire.

(Eqn.1)

Where,

B = magnetic flux intensity or flux density

permeability of the medium

I= current

R = radial space from the wire

From Equation 2 the magnetic field intensity was related to the flux density shown below:

(Eqn.2)

From Equation 3: the integral of magnetic field intensity around a closed path equals the current:

(Eqn.2)

From Equation 4: The integral of magnetic field intensity also equals the number of turns times

the current. F was denoted as magnetomotive force.

(Eqn.3)

From an iron core,

Where,

flux

A = cross-sectional area of core

l = center circumference of core

From rearranging the magnetomotive force.

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Where,

R = magnetic resistance

Building the core with different materials, the permeability of these materials has been

known to be tens of thousands bigger than air. Which means the flux of the transformer was

created from the core.

Experiment 6:

Up until now all of the circuits that have been studied have all been DC circuits. In AC

(alternating circuit) circuit Ohm’s law stays the same but the main difference was in the

relationship of the impedance in a capacitor and the inductor. Impedance is denoted by a letter Z,

a complex number of resistance. The voltage and current are replaced by a complex phasor. Both

of these are representing as a sinusoidal function (current) and a frequency domain (voltage).

The circuit elements for impedance, of an R, L, C (resistors, inductors, and capacitors) are shown

below.

ZR = R

ZL =

ZC =

Inductor has a j part; this j part is the phase angle of 90 degrees because the voltage leads

the current by 90 degrees. Capacitor has a j part; this j part is the phase angle of -90 degrees

because the current leads to the voltage of 90 degrees. Also the magnitude of the inductor is

called the inductive reactance and the capacitor magnitude is called the capacitive reactance.

Experiment 7:

When working on parallel circuits, most of the time it was easier to work with

admittance. The reason for admittance is in parallel circuits the admittance adds up like

impedance. Admittance is just the inverse of impedance shown below.

YR = G = R

YL = -jBL =

YC = jBC =

Where,

G (conductance )

BC = (capacitive Susceptance )

BL = (inductive Susceptance )

Power in an AC circuit, consist of a real or average and imaginary components. This will

be the complex power of the voltage and current phasors. In Equation 4, power is defined by the

voltage times the conjugate of the current. This defined below.

S =VI* = P + j Q (Eqn.4)

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Where P is the real or average power in watts and Q is the imaginary power in VAR (Volt

Amps Reactive). Apparent power is the magnitude with the phase angle of P and Q. Real Power

or average power, can also be defined as VrmsIrms cosθ and the imaginary power Q, VrmsIrms sinθ.

Cosθ is defined as the power factor, and sinθ is defined as the reactive factor. The power factor is

designated for either leading or lagging, positive θ is leading and negative θ is lagging. If the

power factor is leading a capacitor is in the circuit and a lagging power factor has an inductor in

the circuit.

Experiment Data/Analysis:

Experiment 5:

In Graph 1, shows the transformer in an x-y plot (hysteresis curve). This graph shows

channel one at 50mV and channel two at 20mV. Both of the channels were found to be the

primary voltage and secondary current. The difference between the primary voltage and the

secondary current was the secondary current divided by the resistance of the circuit. This

secondary current was the current of the second inductor inside the transformer. Also, this

secondary current supplied the power to the 200K ohm resistor and the .025 nano farads

capacitor. The phase angle was determined from this graph, this angle was determined by the

inverse sin of the difference between the primary voltage and secondary current divided by the

total amount of voltage found, shown in Equation 5.

(Eqn.5)

Graph 1: X-Y Plot of Transformer

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Graph 2: X-T Plot Transformer

Graph 2 shows a Y-T (hysteresis curve) of the phase shift between the primary voltage and

secondary current. The time it took to change at these voltages was found from the graph at

2.50ms. This time was the time it took to change from primary voltage to secondary current.

Another calculation was found from the graph, which was the phase angle. The phase angle was

determined by the change in time divided by the period of the graph times by 360 degrees shows

in Equation 6.

(Eqn.6)

Table 1: RMS Values

VRMS IRMS Real Power Imaginary

Power

Complex

Power

Phase angle

4.701 .989 2.633 .066J 2.633+.066J .08185

Recorded Values for the 1 ohm resistor

DC resistance of 5 volt winding Measured Resistance

.478 1.831

Found from the power meter Vrms, Irms and real power are shown in Table 1. The

imaginary power was calculated from the following Equation 7. The complex power was both

the real power and imaginary power put together. The apparent power was also calculated from

the complex power. Apparent was just the magnitude and the angle of the transformer which was

2.6333<.1436.

In Table 1, the DC resistance of 5 volt winding in the one oh resistance was found. This

was compared to the measured resistance of the one ohm resistance. The measured resistance

was the real resistance from the primary inductor of the transformer. The DC resistance of

primary inductor was the imaginary part of the resistance in the one ohm resistor.

(Eqn.7)

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Experiment 6:

FREQ VR IR ZRmeas ZRtheo VC IC ZCmeas θC ZCtheo VL IL ZLmeas ΘL ZLtheo

1000 356 0.36065242 987.1 1000 2.17 0.122353 17.73558 -86.4 15923.57 188 2.625294 71.61102 64.8 62.8

2000 356 0.36065242 987.1 1000 2.15 0.253529 8.480278 -86.4 7961.783 347 2.612157 132.8404 57.6 125.6

3000 356 0.36065242 987.1 1000 2.13 0.375098 5.678515 -90.72 5307.856 506 2.590588 195.3224 54 188.4

4000 356 0.36065242 987.1 1000 2.1 0.489412 4.290865 -92.16 3980.892 673 2.562745 262.609 72 251.2

5000 356 0.36065242 987.1 1000 1.06 0.597451 1.774204 -90 3184.713 825 2.527451 326.4158 72 314

6000 356 0.36065242 987.1 1000 2.02 0.69902 2.889762 -95.04 2653.928 975 2.486078 392.1839 77.76 376.8

7000 356 0.36065242 987.1 1000 1.97 0.79451 2.479516 -100.8 2274.795 1090 2.44 446.7213 75.6 439.6

8000 356 0.36065242 987.1 1000 1.92 0.884706 2.170213 -92.16 1990.446 1220 2.390196 510.4184 74.88 502.4

9000 356 0.36065242 987.1 1000 1.87 0.967647 1.932523 -90.72 1769.285 1340 2.337451 573.2741 84.24 565.2

10000 356 0.36065242 987.1 1000 1.79 1.044314 1.714044 -90 1592.357 1440 2.274706 633.0489 90 628

TABLE: 2 Impedance Table

62800

TABLE:3 Ѡ=2πf

6280

12560

18840

25120

31400

37680

43960

50240

56520

ΔtC ΔtL

0.00024 0.00018

0.00012 0.00008

0.000084 0.00005

0.000064 0.00005

0.00005 0.00004

0.000044 0.000036

0.00004 0.00003

0.000032 0.000026

0.000028 0.000026

0.000025 0.000025

TABLE:4 Time

5.1 VC 5.1 VL

0.624 13.389

1.293 13.322

1.913 13.212

2.496 13.07

3.047 12.89

3.565 12.679

4.052 12.444

4.512 12.19

4.935 11.921

5.326 11.601

TABLE:5 Voltage

ZR ZC ZL

1.29 99.88862 14.03029

1.29 99.89349 5.764661

1.29 99.89302 3.674328

1.29 99.89221 4.541811

1.29 99.94429 3.954085

1.29 99.89111 4.082783

1.29 99.891 1.619953

1.29 99.89097 1.596014

1.29 99.89077 1.42853

1.29 99.89236 0.803961

TABLE:6 Resistance Error TABLE:7 Phase Impedance Error

θC θL

12

2.4

0.8

0

28

36

40

20

20

13.6

4

4

0.8

2.4

0

5.6

0

16

16.8

6.4

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Graph 3: Theoretical Resistance

TABLE: 8 Magnitude Vrms Error

VRMS-C-3volts

99.92766667

99.92833333

99.929

99.936

99.93766667 66.5

6499.94033333

VRMS-L-4volts

95.3

91.325

87.35

83.175

79.375

75.625

72.75

69.5

99.93

99.96466667

99.93266667

99.93433333

0

200

400

600

800

1000

1200

1400

0 2000 4000 6000 8000 10000 12000

The

ore

tica

l Re

sist

ance

Frequency

Theoretical Resistance Theoretical Resistance

Theoretical Resistance ECGR 2156-Logic and Networks Lab Experiment #6-Impedance Ethan Miller Partner: Waqee Hassan

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Graph 4: Measured Resistance

Graph 5: Theoretical Inductance Magnitude

0

200

400

600

800

1000

1200

1400

0 2000 4000 6000 8000 10000 12000

Me

asu

red

Re

sist

ance

Frequency

Measured Resistance Measured Resistance

Measured Resistance ECGR 2156-Logic and Networks Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2000 4000 6000 8000 10000 12000

The

ore

tica

l In

du

ctan

ce M

agn

itu

de

V

olt

s

Frequency

Theoretical Inductance Magnitude Theoretical Inductance Magnitude

Theoretical Inductance Magnitude ECGR 2156-Logic and Network Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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Graph 6: Theoretical Inductance Phase

Graph 7: Measured Inductance Magnitude

0

10

20

30

40

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60

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80

90

100

0 2000 4000 6000 8000 10000 12000

The

ore

tica

l In

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ctan

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has

e a

ngl

e

Frequency

Theoretical Inductance Phase Theoretical Inductance Phase

Theoretical inductance Phase ECGR 2156-Logic and Networks Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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Measured Inductance Magnitude Measured Inductance Magnitude

Measured Inductance Magnitude ECGR 2156- Logic and Networks Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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Graph 8: Measured Inductance Phase

Graph 9: Theoretical Capacitance Magnitude

0

10

20

30

40

50

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80

90

100

0 2000 4000 6000 8000 10000 12000 Me

asu

red

Ind

uct

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Ph

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an

gle

Frequency

Measured Inductance Phase Measured Inductance Phase

Measured Inductance Phase ECGR 2156-Logic and Networks Lab Experiment # 6 Impedance Ethan Miller Partner: Waqee Hassan

0

0.5

1

1.5

2

2.5

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3.5

0 2000 4000 6000 8000 10000 12000

The

ore

tica

l Cap

acit

ance

Mag

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Vo

lts

Frequency

Theoretical Capacitance Magnitude Theoretical Capacitance Magnitude

Theoretical Capacitance Magnitude ECGR 2156-Logic and Networks Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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Graph 10: Theoretical Capacitance Phase

Graph 11: Measured Capacitance Magnitude

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 2000 4000 6000 8000 10000 12000

The

ore

tica

l Cap

acit

ance

Ph

ase

an

gle

Frequency

Theoretical Capacitance Phase Theoretical Capacitance Phase

Theoretical Capacitance Phase ECGR 2156-Logic and Networks Lab Experiment #6 impedance Ethan Miller Partner: Waqee Hassan

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

0 2000 4000 6000 8000 10000 12000

Me

asu

red

Cap

acit

ance

Mag

nit

ud

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V

Frequency

Measured Capacitance Magnitude Measured Capacitance Magnitude

Measured Capacitance Magnitude ECGR 2156-Logic and Networks lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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Graph 12: Measured Capacitance Phase

In Table 2, the impedance for the resistor stayed the same value of resistance, voltage and

current throughout the whole circuit with the different frequencies. The reason for this was when

the current with through the resistor, this created a voltage drop across the resistor. From this

component in an AC circuit acted the same as in a DC circuit. For the inductance component, the

voltage was measured from the oscilloscope and the current through the inductance was

measured by taken the voltage across the 5.1 ohm, then by ohm’s law the current was found

through the circuit. As the current went through the inductor, this created a resistance, but this

resistance was nothing like the resistance found in a normal resistance of a resistor. Shown in

Table 5 are the voltages across the 5.1 ohm with the capacitor and inductor. The inductor

resistance was found to be a complex resistance because when a current went through this

inductor, a differential equation was form as shown in Equation 8. Equation 8 simplified the

impedance for the inductor, this then became the j was the complex angle for the

inductance. The capacitor was found in the same way as the inductor. The only difference was

the differential equation and the complex angle as shown in Equation 9 was different. This

complex angle was –j. J in either case was an angle of 90 degrees. The impedance for a capacitor

was

Each of this components was under a different frequency to achieve different level of

impedance. Shown in Table 3 shows the frequency converted to omega. The phase angle was

calculated for each of the reactive elements (capacitor and inductor). The difference between the

voltage and current wavelength on the oscilloscope was the time it took for the current to reach

the voltage. On a purely inductor the current lagged Shown in Equation 10, shows how the

phase angle was calculated, f was the frequency and delta t was the change in time.

(Eqn.8)

(Eqn.9)

(Eqn.10)

-103

-101

-99

-97

-95

-93

-91

-89

-87

-85

-83

-81 0 2000 4000 6000 8000 10000 12000

Me

asu

red

Cap

acit

ance

Ph

ase

an

gle

Frequency

Measured Capacitance Phase Measured Capacitance Phase

Measured Capacitance Phase ECGR 2156-Logic and Networks Lab Experiment #6 Impedance Ethan Miller Partner: Waqee Hassan

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In Tables 6, 7, and 8 show the percent error from this experiment. Table 6 shows the error

resistance in the capacitor and inductor. Table 7 shows the phase impedance error and Table 8

shows the magnitude error for both capacitor and inductance. From these tables there was an

error in the measurements taken. The measurements were found to be in a real lab of degrees

varied form the current in an inductor and the voltage in a capacitor.

Graphs were made to show the difference between the theoretical and measured values. In

Graph 3 and 4 of the 1 ohm resistance showed the measured valued of the resistance was a little

less than what the theoretical resistance. Graphs 5, 6, 7, and 8 show the theoretical and measured

phase and magnitude values of the inductance. The magnitude voltage was set to 4 volts but as

the frequency went up the magnitude voltage also went up in a linearly fashion. It appeared the 4

volts did not matter at all because the higher the frequency the higher the magnitude voltage was.

The phase angle also, went up as the frequency went up, but it came to a halt as it got closer to

Graphs 9, 10, 11 and 12 shows the theoretical and measured phase and magnitude values of

the capacitor. The magnitude was set to 3 volts but as the frequency went up the magnitude went

down in a linearly fashion. From the observation of the graph, it appeared the 3 volts did not

matter at all because the higher the frequency was the lower the magnitude was. The phase angle

also went down as the frequency went up. Majority of the points on Graph 12 was around , which is roughly where it’s supposed to be.

Experiment 7:

Table: 9 Voltage, Current and Real Power readings

Figure # Voltage rms Current rms Real Power

3 40.07 28.2 .815

4 28.88 28.20 .1067

5 24.98 28.22 .7052

3 40.19 41.22 1.080

6 39.94 26.6 1.064

7 40.10 31.4 .887

Figure 1 RL and RC circuit diagrams.

From Figure 1, the resistor inductor and capacitor values were measured, these values were

L-2.65 henrys

RL(internal resistance of the inductor)-122 ohms

Rp-1.489K ohms and C was 1.780µ farads.

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Rs’ was found by subtracting the Rs and RL value. The value of Rs’ had to be less than Rs.

Rs’ was 878 ohms. An Ac supply of 40 volts was connected in series with the power meter and

the RL circuit in Figure 1.

The circuit was connected in Figure 1, with the power meter connected in series for the RL

circuit. The voltage, current and real power values were measured across the resistor and

inductor. These values are shown in Table 9, which were the total measured values of both Rs’

and the inductor. In Table 9, the inductor and Rs’ voltage, current and real power was measured

with the power meter. For the RC circuit the same process was done to measure the voltage,

current and real power.

During this experiment a mistake was done on the power meter. After taking the

measurements from the whole RL circuit, compared to the measures values of the partial RL

circuit did not came out to be total power conserved. The power meter showed the current that

was well above the total current for the circuit, as well as the real power. Once this was found,

the experiment was tested again to see if it gave the same results, which it did. Then there was a

need to get another power meter. Once there was a new power meter the total power was

conserved.

In an ideal capacitor, the power would be zero, which means that was no energy loss due to

heat, but in a true capacitor there was a loss of energy due to the heat. From this experiment the

energy loss due to heat all depends on the construction of the capacitor. But the loss of this

energy was very small. In an ideal inductor, the total power was zero assuming that it has zero

winding resistance. From a real inductor total power was not zero and there was some loss of

energy in the circuit. Overall both the capacitor and the inductor do loss some energy in AC

circuits. Other observations include, that the capacitor holds the value of the AC source voltage

and the inductor holds the current from the AC source.

List of Attachments:

Original Data sheet

Conclusion:

Experiment 5:

In conclusion, Graphs 1 and 2 showed the primary voltage and the secondary current.

From these graphs there was a relationship between the primary voltage and secondary current,

this relationship was determine by the windings inside the transformer. The primary current and

voltage created a magnetic flux which was induce to the secondary voltage and current. The

phase angle was also determined from the oscilloscope, this determined how much the current

shifted by.

Experiment 6:

In conclusion, the experiment was to confirm the relationship between the impedance,

magnitude and phase angle of a resistor, inductor and a capacitor at different frequencies. The

resistor acted the same way as in a DC circuit, resistor had the same resistance. The inductor did

not have the same resistance as the frequency went up. The more the frequency went up the

larger the impedance was, the inductor reaches infinity. The capacitor did not have the resistance

as the frequency went up. The more the frequency went up the smaller the impedance was,

capacitor reaches zero.

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Experiment 7:

In conclusion, the measured voltages, currents and real power of both inductor and

capacitor from an AC source differ from each element. The capacitor held the voltage across the

AC source and the inductor held the current across the AC source. From the Table 9, the

capacitor had the current leading and the inductor had the voltage leading, this was determined

from there phase angles. The capacitor held more real power than the inductor did.

References: [5] Lab handout “Magnetic Properties of a Small Transformer”

[6] Lab handout “Impedance”

[7] Lab handout “AC Operation of RLC Networks”

This report was submitted in compliance with the UNCC Code of Student Academic Integrity

(1997-99 UNCC Catalog, p 336) ____ECM___.