Circuits Review Part 1: Resistive Circuits, Linear Scale Ohmmeter

Purpose

The goals of this laboratory are to verify Ohms Law, Kirchhoffs Voltage Law, and Kirchhoffs Current Law. In addition, the operational ranges for voltages and currents as related to component tolerances will be explored.

Theoretical Background

Ohms Law

Ohms Law states that the voltage across a linear resistor is directly proportional to the current flowing through the resistor. Mathematically, the voltage, current, and resistance are related by

V = I*R

Kirchhoffs Voltage Law (KVL)

Kirchhoffs Voltage Law states the algebraic sum of the voltages around any closed path or loop in a circuit must be equal to zero. That is for the circuit below:

Vs = V1 + V2

Figure 1.1

Kirchhoffs Current Law (KCL)

Kirchhoffs Current Law states that the algebraic sum of the currents entering a node must be equal to the algebraic sum of the currents leaving the node.

Mathematically:

I1 = I2 + I3

Figure 1.2

Theoretical Analysis

Referring to Figure 1.3, calculate the circuits Branch Voltages, Branch Currents, Node Voltages, and Loop Currents with the following assumptions. (Note: Mesh Analysis may be useful here.)

1. First assume all the resistors are exact (i.e. a 10 ohm resistor is exactly 10 ohms).
2. Then assume all resistors are 10% above the exact values.
3. Then assume all resistors are 10% below the exact values.

Record all your data in tabular format similar to Table 1.1. Let the exact values of source voltage and resistors be defined by the following values:

Vs = 5.0V

R₁ = R₂ = R₃ =100 R₄ = R₅ =1 K R₆ = R₇ =2.2 K

Figure 1.3

Simulation

Build the circuit Figure 1.3using the Electronics Workbench. Then place virtual Volt and Ammeters in the circuit to find all required voltages and currents that you found in the theoretical analysis section. Print out the schematic showing all the readings before moving on to the next section.

Laboratory Procedure

Part A

Using the Analog/Digital trainer, build the circuit and re-measure all Branch Voltages, Branch Currents, Node Voltages, and Loop Currents using a digital multimeter.

Record all the specified measurements in a table similar to Table 1.1.

TABLE 1.1

Branch Voltages	Branch Currents	Node Voltages	Loop Currents
VR1 =	iR1 =	V1 =	I1 =
VR2 =	iR2 =	V2 =	I2 =
VR3 =	iR3 =	V3 =	I3 =
VR4 =	iR4 =	V4 =
VR5 =	iR5 =
VR6 =	iR6 =
VR7 =	iR7 =

1. Verify each of the following using your experimental results. Account for component tolerances in the calculations if necessary.

1. Ohms Law
2. Kirchhoffs Voltage Law
3. Kirchhoffs Current Law

2. Compare your experimental results for voltages and currents with the results from a theoretical analysis. Do your experimental values fall within operational ranges? Explain in detail.

Part B

Build the circuit shown in Figure 1.4.

Figure 1.4

Set your source voltage and resistance values as follows; let V_s = 5.0 Volts and R_s = 100 . It does not matter if these values are exact as long as they remain consistent throughout the experiment.

1. Measure and record the values of load voltages and currents for various values of R_L (load resistance) as specified in Table 1.2. Use a variable resistor to vary the load resistance.

Table 1.2

RL (Ohms)	VL (V)	IL (mA)
10
20
40
70
100
150
200
300
500
1000

1. Using your data and a spreadsheet program plot V_L vs. I_L. Determine and report the values of x and y intercepts, and the slope of the line.

Part 2: Norton and Thevenin Equivalent Circuits

Purpose

The goals of this laboratory are to demonstrate the equivalence between a multiple resistive network and its Thevenin or Norton equivalent circuits. The concepts of load line and maximum power transfer will also be introduced.

Theoretical Background

Thevenin and Nortons Theorems can be employed at any load terminal. The advantage of applying these theorems is that a complex circuit can easily be reduced to a simpler one. This is very effective in simplifying the process of determining the current, voltage and power at the load.

Figure 1.5

Thevenin Theorem

Thevenins theorem states that an entire network can be replaced, exclusive of the load, by an equivalent circuit that contains only two components. They consist of an independent voltage source in series with a resistor in such a way that the current-voltage relationship at the load remains unchanged.

Method for finding a Thevenins equivalent circuit.

Refer to Figure 1.5

1. Disconnect the load R_L to open the terminal A-B. Find the Voltage at the terminal. This is called open circuit voltage V_oc. V_oc is the independent voltage source for the

Thevenins equivalent circuit.

2. Next, short-circuit the terminal A-B by adding a wire.
3. Find the current at the terminal. This is called short circuit current I_sc.
4. Calculate the Thevenin equivalent resistance R_Th, where R_Th = V_oc/I_sc.
5. Draw the Thevenin equivalent circuit using V_oc in series with R_Th, as demonstrated in Figure 1.6.

Figure 1.6

Nortons Theorem

Nortons theorem is similar to Thevenins theorem with the exception that the equivalent circuit is an independent current source in parallel with a resistor.

Method for finding a Norton equivalent circuit (Refer to Figure 1.5):

1. To begin use the same steps 1-4 that you applied in the Thevenin method. But, you will draw the Nortons equivalent circuit using Isc in parallel with RTh. (See Figure 1.7)

Figure 1.7

Maximum power transfer to the load

Once you have simplified circuit using Thevenins Theorem, you can easily find the voltage and the current at the load. One of the main applications of this technique is to find the maximum power transfer to the load. Referring to figure 1.6:

P_L=V_oI_o or = I_o²R_L

For max condition R_L=R_Th

So P_max=V_oc²/(4R_Th) Watt

Theoretical Analysis

Referring back to Figure 1.5

Let V_s = 5.0 V, R₁ = R₂ = R₃ = 1K Ohm.

1. Disconnect R_L so that the terminal A-B is open. Calculate the open circuit voltage V_oc.
2. Short-circuit the terminal A-B. Calculate the short circuit current I_sc.
   1. Calculate Thevenins equivalent resistance R Recall RTh = Voc/Isc. (You can also find R_Th by short circuiting V_s and calculating resistance between terminal A-B.). What is the R_L for maximum power transfer to the load? Find P_max?
   2. Repeat step 4 using the Norton equivalent circuit.
   3. Draw the Thevenin and Nortons equivalent circuits. Clearly mark all the values (use values from steps a-c) with correct polarity.

Simulation

Draw the circuit in Figure 1.5 using Electronics Workbench. Repeat steps 1 and 2 (a, b, c) of Theoretical Analysis using virtual Amperage & Volt meters to determine the currents and voltages. Print out your schematic diagram before moving on to the next section.

Laboratory Procedure

1. Build a circuit as shown in the Figure 1.5. Letting: V_s = 5.0 V, R₁ = R₂ = R₃ = 1K Ohm.
2. Measure Voc, Isc, and calculate RTh at the terminal A-B. Draw Thevenin and Nortons equivalent circuit.
3. Build a circuit using your Thevenins equivalent circuit from step 2, then chose RL= 1000, 1500, 2000 and calculate the P_L for these cases and discuss your results.