Lab 3 (Nodal and Mesh Analysis)

EEEB111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY

Experiment 3: Nodal and Mesh Analysis

EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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EXPERIMENT 3

Nodal and Mesh Analysis

Assessed OBE Course Objectives: CO1, CO2, CO3 and CO4

OBJECTIVES The objectives of this laboratory experiment are to simulate a resistive circuit using computer simulation program, LTspice, to build a resistive circuit with DC sources and make node voltage and branch current measurements. Verify nodal and mesh analysis method.

INTRODUCTION Nodal Analysis A node is a point where two or more elements are connected, refer Fig. 3.1.

Figure 3.1: Example of Nodes in a resistive circuit

The voltage at each node is called node voltages. Nodal analysis is an analysis on how to calculate the voltages at each node in the circuit. The procedure can be divided into four (4) basic steps. 1. Assign a reference node (usually the ground node is chosen). Label the other nodes as nonreference nodes, 𝑣1 , 𝑣2 … 𝑣𝑛 with respect to the reference node. These 𝑣1 , 𝑣2 … 𝑣𝑛 are known as node voltages. Refer Fig. 3.2.


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Figure 3.2: Assign reference node and non-reference nodes

2. Apply Kirchhoff’s Current Law (KCL) at each non-reference node, where sum of branch current going into and out of the same node equals zero, 𝑁

∑ 𝐼𝑛 = 0 𝑛=1

where current entering a node is taken as positive (+) and current leaving the same node is taken as negative (-). Refer Fig. 3.3.

Figure 3.3: Apply KCL at non-reference nodes

At Node 1: 𝑣1 = 𝑉1 KCL at Node 2: 𝐼1 − 𝐼2 − 𝐼6 = 0 KCL at Node 3: 𝐼2 − 𝐼3 + 𝐼5 = 0 KCL at Node 4: 𝐼3 − 𝐼4 + 𝐼6 = 0 At Node 5: 𝑣5 = 𝑉2 EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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3. Then substitute the branch current I with node voltage terms, using Ohm’s Law:

𝐼=

𝑣ℎ𝑖𝑔ℎ𝑒𝑟 − 𝑣𝑙𝑜𝑤𝑒𝑟 𝑅

where current flows from a higher potential to a lower potential. At Node 1: 𝑣1 = 𝑉1 KCL at Node 2:

KCL at Node 3:

KCL at Node 4:

(𝑣1 − 𝑣2 ) (𝑣2 − 𝑣3 ) (𝑣2 − 𝑣4 ) − − =0 𝑅1 𝑅2 𝑅6 (𝑣2 − 𝑣3 ) (𝑣3 − 𝑣4 ) (𝑣5 − 𝑣3 ) − + =0 𝑅2 𝑅3 𝑅5

(𝑣3 − 𝑣4 ) (𝑣4 − 0) (𝑣2 − 𝑣4 ) − + =0 𝑅3 𝑅4 𝑅6

At Node 5: 𝑣5 = 𝑉2 4. Determine the unknown node voltages, 𝑣1 , 𝑣2 … 𝑣𝑛 by solving the simultaneous equations in step 3.


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Mesh Analysis Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. A mesh is a loop that does not contain any other loop within it. In the mesh analysis of a circuit with n meshes, the calculation can be divided into the following steps. 1. Assign mesh currents 𝑖1 , 𝑖2 … 𝑖𝑛 to the n meshes. Refer Fig. 3.4.

Figure 3.4: Assign mesh currents

2. Apply Kirchhoff’s Voltage Law (KVL) to each of the n meshes, where sum of voltages in a given mesh is zero. 𝑀

∑ 𝑣𝑚 = 0 𝑚=1

where the sign on each voltage is the polarity of the terminal encountered first as we travel around the mesh. Refer Fig. 3.5.

Figure 3.5: Apply KVL at each mesh

KVL at Mesh 1: −𝑉1 + 𝑣𝑅1 + 𝑣𝑅6 + 𝑣𝑅4 = 0 KVL at Mesh 2: 𝑣𝑅2 + 𝑣𝑅3 − 𝑣𝑅6 = 0 KVL at Mesh 3: −𝑣𝑅4 − 𝑣𝑅3 + 𝑣𝑅5 + 𝑉2 = 0 EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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3. Use Ohm’s law to express the voltages in terms of the mesh currents. KVL at Mesh 1: −𝑉1 + 𝑅1 𝑖1 + 𝑅6 (𝑖1 − 𝑖2 ) + 𝑅4 (𝑖1 − 𝑖3 ) = 0 KVL at Mesh 2: 𝑅2 𝑖2 + 𝑅3 (𝑖2 − 𝑖3 ) − 𝑅6 (−𝑖2 + 𝑖1 ) = 0 KVL at Mesh 3: −𝑅4 (−𝑖3 + 𝑖1 ) − 𝑅3 (−𝑖3 + 𝑖2 ) + 𝑅5 𝑖3 + 𝑉2 = 0 4. Solve the resulting n simultaneously equations to get the mesh currents; 𝑖1 , 𝑖2 and 𝑖3 . Note: 1. Current entering negative potential of element would be a negative current. 2. Branch current is the current that goes through each branch. i.e. element. It is usually (but not always) denoted by a capitalized ‘I’ to differentiate with mesh current ‘i’. If a resistor is not between 2 meshes, its branch current would equal the mesh current. When there are more than two unknown node voltages, the solution of the nodal analysis becomes a bit difficult, and thus computer software assistance would be helpful. In this experiment, we will learn to utilize LTspice for DC analysis of circuits containing resistors and independent voltage sources.

PRE-LAB ASSIGNMENT 1. Study on Nodal and Mesh Analysis, pages 82 – 99 in Circuit Analysis textbook. 2. Download and become familiar with the LTspice software available at http://lms.uniten.edu.my/moodle/mod/resource/view.php?id=119306 . The software is also available at the lab’s computers. 3. Refer to Appendix 2 for instructions and example on how to do the computer analysis for Experiment 3 using LTspice. 4. Complete the Part A computational analysis of this lab as Pre-Lab Assignment. i.e. BEFORE coming to lab.


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Part A: Circuit Analysis with LTspice

Refer to Appendix 2 for instructions and an example on how to do the computer analysis using LTspice for a resistive circuit. a. Table 3.1 lists the designated resistors values, R1, R2, R3, R4, R5 and R6 according to workbench number. Use these nominal values for your computational analysis. Table 3.1: Designated resistor values for LTspice netlist

Workbench Number #01, #07, #13

#02, #08

Resistor

R1 R2 R3 R4 R5 R6

#03, #09, #14

#04, #10

#05, #11

#06, #12, #15

Resistor Values ( Ω ) 1.0 k

2.2 k

2.0 k

3.3 k

4.7 k

6.8 k

6.8 k

1.0 k

2.2 k

2.0 k

3.3 k

4.7 k

4.7 k

6.8 k

1.0 k

2.2 k

2.0 k

3.3 k

3.3 k

4.7 k

6.8 k

1.0 k

2.2 k

2.0 k

2.0 k

3.3 k

4.7 k

6.8 k

1.0 k

2.2 k

2.2 k

2.0 k

3.3 k

4.7 k

6.8 k

1.0 k

b. Simulate the circuit shown in Figure 3.6 using LTspice to determine the indicated node voltages and branch currents for: R6 = 0 Ω (short circuit), R6 designated value and R6 →  Ω (open circuit). Set V1 = 16V and V2 = 8V.

Figure 3.6: Circuit for LTspice netlist computation

Note: 1. When R6 = 0 Ω (short circuit), there will be no node 4 as the node 2 voltage would be the same at node 4 voltage as well, since all the current will pass through. 2. When R6 →  Ω (open circuit), you need to remove R6 from the circuit, since no current will flow through R6. EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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c. Record the results obtained from LTspice in Table 3.2: Table 3.2: Computer Analysis using LTspice netlist - Results R6 ()

0

Computational Branch Currents (mA)

Computational Node Voltages (V)

𝒗𝟏

𝒗𝟐

𝒗𝟑

𝒗𝟒

𝒗𝟓

𝑰𝟏

𝑰𝟐

𝑰𝟒

𝑰𝟓

Not applicable

R6 

d. Attach copies of the LTspice netlists and the results obtained for ALL three (3) different values of R6, with this experiment’s lab report.


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UNIVERSITI TENAGA NASIONAL Department of Electronics and Communication Engineering College of Engineering Semester: I / II / Special Academic Year: 20 ….. / 20 ….. COURSE CODE: EEEB111 LAB INSTRUCTOR: TITLE: Nodal and Mesh Analysis

EXPERIMENT NO.: DATE:

3 TIME:

OBJECTIVES: The objectives of this laboratory experiment are to simulate a resistive circuit using computer simulation program, LTspice, to build a resistive circuit with DC sources and make node voltage and branch current measurements. Verify nodal and mesh analysis method. PRE-LAB: Part A : Circuit Analysis with LTspice Marks will be given in LTspice assignment’s marks distribution below.

MARKS:

/12 /12 /6

3 correct Netlist 3 correct Simulation Results Table 3.2

TOTAL: /30

INSTRUCTOR’S COMMENTS: EXPERIMENTAL RESULTS: Part B : Nodal Analysis Table 3.3 V1 measured Table 3.4 Part C : Mesh Analysis Table 3.5

/3 /2 /9 /6

POST-LAB: Part B : Nodal Analysis Table 3.6 Part C : Mesh Analysis Table 3.7 Table 3.8

/3 /3 /2\

CONCLUSIONS:

/2 TOTAL: /30

INSTRUCTOR’S COMMENTS: STUDENT NAME:

STUDENT ID:

GROUP MEMBER:

STUDENT ID:


SECTION:

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EQUIPMENT 1. Resistors: 1 kΩ, 2 kΩ, 2.2 kΩ, 3.3 kΩ, 4.7kΩ and 6.8 kΩ 2. Digital Multimeter (DMM) 3. DC Power Supply 4. DMM Probes x 2nos. 5. Crocodile Clips Connectors x 2 nos. 6. Protoboard 7. Wire 22 AWG x 1 no. PROCEDURES This laboratory experiment is to develop a familiarity with computer techniques as applied to DC circuit. A circuit to be investigated needs to be constructed and examined in detail in the laboratory session. The simulated values obtained earlier from LTspice simulation will be the reference to verify the measured results obtained experimentally. Part B: Nodal Analysis

a. Measure the resistances of designated resistors with the DMM and record the values in Table 3.3. Use the same resistor values as in Part A: Computer Analysis. Table 3.3: Measured values of designated resistors Resistors

Nominal Value (k)

Measured Value (k)

Error (%)

R1 R2 R3 R4 R5 R6

b. Build the circuit shown in Figure 3.7.

Figure 3.7: Circuit for node voltage measurements, value of R6 EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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c. Set the voltage sources V1 = 16 V and V2 = 8 V, using the DMM for setting accuracy. Record below: V1 measured = ____________ V

V2 measured = ____________ V

d. Using DMM, measure the node voltages; 𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 and 𝑣5 with respect to reference node. (i.e. the ground i.e. black DMM probe must be at the reference node).

e. Record the measured values in Table 3.4. f. Repeat and record in Table 3.4, all the node voltages measurements for when the resistor R6 is zero (0) i.e. short circuit, as per Figure 3.8.

Figure 3.8: Circuit for node voltage measurements, R6 = 0

g. Repeat and record in Table 3.4, all the node voltages measurements when the resistor R6 approaches infinity () i.e. open circuit, using the DMM, as per Figure 3.9.

Figure 3.9: Circuit for node voltage measurements,R6 -> 


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Table 3.4: Experimental Results – Nodal Analysis R6 ()

Measured Node Voltages (V)

𝒗𝟏

0

𝒗2

𝒗𝟑

𝑣4

𝑣5

Not applicable

R6



Part C: Mesh Analysis a. Using the same circuit construction (i.e. maintain the voltage sources and resistor values) as per Part B, measure the branch currents 𝐼1 , 𝐼2 and 𝐼4 as shown in the following Figures 3.10, 3.11 and 3.12. b. Record the measured results in the subsequent Table 3.5.

Figure 3.10: Circuit for branch current measurements, value of R6


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Figure 3.11: Circuit for branch current measurements, R6 = 0

Figure 3.12: Circuit for branch current measurements, R6 -> 

Table 3.5: Experimental Results – Mesh Analysis R6 ()

Measured Branch Currents (mA)

𝑰𝟏

𝑰𝟐

𝑰𝟒

0 R6 

Note: 1. Branch current 𝐼1 = mesh current 𝑖1 2. Branch current 𝐼2 = mesh current 𝑖2 3. Branch current 𝐼4 = mesh current 𝑖1 - mesh current 𝑖3 EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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POST LAB ASSIGNMENT: Part B: Nodal Analysis 1. Summarize the percentage error for node voltages between the nominal computational values using LTspice (Table 3.2) and nodal analysis experimental measured values (Table 3.4) in Table 3.6. Table 3.6: Summary of % Error – Node Voltages % Error for Node Voltages

R6 ()

𝒗𝟏

𝒗𝟐

𝒗𝟑

𝒗𝟒

𝒗𝟓

Not applicable

0 R6  Note: % error should be very small, in region of <5%.

Part C: Mesh Analysis 1. From Table 3.5, determine the mesh currents 𝑖1 , 𝑖2 and 𝑖3 of Figures 3.10, 3.11 and 3.12. Complete the required values of Table 3.7. Table 3.7: Measured Mesh Currents R6 ()

Measured mesh currents (mA)

𝒊𝟏

𝒊𝟐

𝒊𝟑

0 R6 

Not applicable

2. In Table 3.8, compare the computational value of branch current 𝐼5 with the determined value of 𝑖3 . Table 3.8: Summary of % Error – branch current 𝐼5 and mesh current 𝑖3 R6 ()

% Error for computational 𝑰𝟓 and

𝑰𝟓

(of Table 3.2)

measured 𝒊𝟑 𝒊𝟑

(of Table 3.7)

% Error

0 R6  Note: % error should be very small, in region of <5%. EEEB 111 ELECTRICAL/ELECTRONICS MEASUREMENT LABORATORY - UNITEN

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CONCLUSIONS: List TWO (2) main understandings that you have gained from this experiment. (i)

(ii)


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Lab 3 (Nodal and Mesh Analysis)

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