Linear and Non-Linear Resistance Load Line Analysis PDF

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EE 336 Lab Report # 02

Linear and Non-Linear Resistance Load Line Analysis Chris Martinez Ben Noel

Performed on September 14, 2010 Submitted on September 21, 2010 Will Gaughan Section 01

Embry-Riddle Aeronautical University Daytona Beach, FL 32114

I. Abstract: This lab served as an introduction to load line analysis and their respective graphical representations on electrical analysis. The two circuit components that are classified in load line analysis are linear and non-linear devices. A linear device is defined as a device in which the input into the device is linearly related to the output of the device. A nonlinear device is defined as when the input of the device is non-linearly related to the output of the device. The devices that we used were a basic resistor, a light bulb, and a light-emitting diode or commonly known as a LED.

II. Introduction/Research/Theory By looking at the voltage-currrent relationship given by Ohm’s Law at its elementary form in comparision to a linear equation (y = mx + b ) gives

V =IR → I=

1 1 ∙V ∴ m=slope= R R

This relationship between the voltage and the current across the resistor is why a resistor is considered a linear device. Figure 1: Linearity (Ohm's Law) In comparison, the light bulb that we used does not have a constant resistance. The resistance of the bulb or more specifically, the bulb’s filament (the metal wire inside the bulb that the electricity passes through; it is also produces the light given off) is dependent on the temperature of the filament. Thus a light bulb is considered a non-linear device.

A LED is also considered a non-linear device. It is also an electric source of light yet operates differently than a standard light bulb. Both are non-linear devices yet the LED has an exponential relationship between the voltage and current.

Load line analysis is a graphical concept that is used in the analysis of non-linear devices in a direct current. Initially, a non-linear device is analyzed in the current, then varying the source voltage then measuring the corresponding voltage across and current through the non-linear device. From those measurements, current through the non-linear device vs. voltage across the device is plotted. Then the device is placed in series with a linear device in the same circuit. The circuit is analyzed using KVL, which results in a load line equation. This load line equation gives a relationship between the source voltage, the voltage across the resistor and the voltage across the non-linear device.

By applying two boundary conditions, such as when the voltage across the non-linear device is zero and when the current through the non-linear device is zero and solving for the corresponding voltage across or current through the non-linear device two sets of voltage across the non-linear device and their corresponding current values are obtained.

The two sets of data are plotted as two points on the graph formed from the analysis of the non-linear device alone and are connected by a straight line called the load line. The intersection of the load line and the original data points plotted to form the operating point. The operating point is the location where the responses from the non-linear device alone are equal to that of the responses from the combination of the linear and non-linear device.

Load Line Analysis 25

Current (mA)

20 15

Non-Linear Device Operating Line

10 5 0 0

5

10

15

20

25

Voltage(non-linear device) (V)

Example Graph 1: Load Line Analysis

The plot above is obtained by measurements taken from the circuit shown in Figure

Applying Kirchhoff’s Voltage Law to the circuit shown in Figure yields the following V s=IR +V NLD

Using the boundary conditions of

I =0 →V NLD =V s

This analysis yields the points

( V)

s ( V s , 0) ∧ 0, R

and

V NLD =0 → I=

Vs R

These points are plotted as squares and connected by the red line. This line is referred to as the Load Line. The location where the original plot intersects the load line is called the operating point. At the operating point, the response from the circuit containing both a nonlinear and linear device is the same as the response from the circuit containing the non-linear device.

III. Circuit Diagrams and Pictures:

Figure 2:

Figure 3:

Figure 4:Load line analysis circuit.

Figure 5:

Figure 6: Load line analysis circuit

IV. Lab Description/Procedure 1. Created circuit shown in figure 2 2. Gathered data to plot Ir vs. Vr 3. Created circuit shown in figure 3 4. Gathered data to plot Ilamp vs. Vlamp 5. Created circuit shown in figure 4 6. Record I vs V 7. Created circuit shown in figure 5 8. Plotted Idiode vs Vdiode 9. Created circuit shown in figure 6 10. Record I vs V

V. Result Analysis A. Resistor Measurements When measured the 820 ohm resistor read the following: R1 = 880 ohms Once we conducted the experiment we got the following results for the current and voltage through and across the resistor: Table #1 Measurement Resistor Only Vs (volts) 0 1 2 3 4 5 6 7 8

Resistor Vr (volts) Ir (mA) 0 0 1 1.32 2.03 2.41 3.06 3.47 4.09 4.58 5.08 5.82 6.11 6.86 7.12 8.09 8.14 9.24

9 10

9.09 10.05

10.36 11.36

12 10

Ir

8 6 Resistor

4 2 0 0

2

4

6

8

10

12

Vr

Figure #7 V vs I for the resistor only The resistance was also estimated from the slope of the line from the different voltage test for the resistor. The slope was calculated between each point in the following manner: I 3−I 2 1 slope ( )= V 3−V 2 R 1 2.41−1.32 slope ( )= 2.03 −1 R

1 slope ( )=1.029 R

Then the inverse of the slopes were taken and averaged. Then the percent error between the theoretical value from the slope and the actual value was calculated with the following formula: R1:

|R actual−Rtheoretical|∗100 %

%error=

Rtheoretical

|880 −903|

%error=

880

∗100 %

%error=2.21 %

B. Light Bulb When conducting the experiment with the lamp alone we obtained the following results: Table #2 Light Bulb

Vs (volts) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Light Bulb Vr (volts) 0 1.01 2.02 3.08 4.06 5.13 6.16 7.08 8.17 9.12 10.11 11.14 12.16 13.19 14.16 15.09 15.92 16.84 17.68 18.71 19.94

Ir (mA) 0 3.2 4.73 6.08 7.31 8.34 9.33 10.31 11.13 12.08 12.91 13.6 14.44 15.26 15.94 16.57 17.18 17.77 18.33 18.94 19.61

The results were consistent with the non-linear theory explained earlier in this report. This is illustrated by the figure below: 25 20

I

15 10 Load Line 5 0 0

5

10

15

20

25

V

Figure #8 Load-line for Light bulb As listed in the procedure, KVL was then used with a resistor thrown into the circuit to develop an equation for a load line analysis. The calculations are below: Over the loop:

∑ V =0 −V s + IR+ V NDL=0 When I =0 V NDL=V s V NDL =20

When I=

V NDL = 0

VS R

I =22.7 mA

From this calculation we obtain the points (0,22.7) and (20,0) and plotted the load line above in Figure #8. The operating point is (9.12,12.08).

C. LED When conducting the experiment with the LED alone we obtained the following results: Table #3 LED LED Vr

Vs 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Ir

0 0 0.1 0 0.2 0 0.3 0 0.4 0 0.5 0 0.6 0 0.7 0 0.8 0 0.9 0 1 0 1.1 0 1.2 0 1.3 0 1.4 0 1.5 0 1.6 0.12 1.7 1.01 1.8 2.33 1.9 5.37 2 10.9 2.1 18.6 2.2 Infinite

The results were consistent with the non-linear theory explained earlier in this report. This is illustrated by the figure below: 20 18 16 14

I

12 10 LED Load Line

8 6 4 2 0 0

0.5

1

1.5

2

2.5

3

V

Figure #9 Load-line for LED As listed in the procedure, KVL was then used with a resistor thrown into the circuit to develop an equation for a load line analysis. The calculations are below: Over the loop:

∑ V =0 −V s + IR+ V NDL=0 When I =0 V NDL=V s V NDL =2.5

When I=

VS R

V NDL = 0

I =2.84 mA

From this calculation we obtain the points (0,2.84) and (2.5,0) and plotted the load line above in Figure #9. The operating point is (1.7,1.01).

VI. Discussion and Conclusion One of the most important things to learn is the proof in the experiment of a resistor truly being a linear device. As shown Figure #1 resistors do have a linear behavior and ohm’s law is correct. The slope of a current vs voltage curve can be used to determine the resistance with a good degree of accuracy in accordance with our percent error of only 2.21 percent. In the case of our experiment we discovered just how important running the voltmeter or ammeter in parallel and series respectively is. We ended up getting bad readings and had to get help from the TA only to realize the diagram was very specific for a reason. It had been addressed in the last lab, but it was fully engrained in this one. In respects to the behavior of the LED and the light bulb we definitely saw a nonlinear pattern existent in the relationship between the voltage and the current. This indicates that we have to use a method other than ohm’s law to calculate resistance, voltage, or current when doing theoretical calculations. The theory presented for the load line method proved to work well. We were able to find the operation point for both setups of LED and light bulb.

VII. Statement I think that our lab instructor did a good not great job in explaining the lab before we got started, yet did an excellent job in explaining concepts or lab instructions to my

partner and I that were not clear to either of us at the time. I liked the casual atmosphere that he created so that it was easy for us to ask him questions.

VIII. Appendices 1. Do the characteristic curves of each non linear component agree with the theoretical behavior predicted? Yes we got a linear curve for the resistor, a nonlinear curve for the light bulb, whose slope decreased with voltage, and a nonlinear curve for the LED, whose slope increased with voltage.

2. The circuit above contains a non-linear element. The voltage across this element (VA) is equal to the current through the element (I) squared. Write KVL and solve for VA. What size resistor will result in the current I = 2A?

−V s + IR+ VA=0

−12+ I (1)+ I 2=0 2

I + I −12=0

( I +4 ) (I −3)=0 I =3 A

If I = 2A what is R? −V s + IR+ VA=0 2

−12+2 R+ 2 =0

R=4 ohms...