EXPERIMENT 6
MESH ANALYSIS AND NODAL ANALYSIS
GROUP 3 LEADER: QUAN, JEWEL MEMBERS: DELA LUNA, NIÑO JHIM ANDREW Masangkay, Luis Paolo Santos, Sveth DATE PERFORMED: FEBRUARY 22 2013 DATE SUBMITTED: MARCH 01 2013
GRADE
Experiment 6: Mesh Analysis and Nodal Analysis Jewel Quan#1, Niño Jhim Andrew B. Dela Luna#2, Sveth Santos#3, Luis Paolo Masangkay#4 #1,3,4
School of IE-EMG, #2School of MME Mapúa Institute of Technology Muralla Street, Intramuros, Manila, Philippines 1
[email protected] [email protected] 3
[email protected] 4
[email protected] 2
Abstract— Abstract- In this experiment we investigated the effects of mesh analysis and nodal analysis on multiple active linear sources in a network. And to verify whether their linear responses at any point are similar to Kirchhoff’s Law. We were given two different circuits one on mesh analysis and another for nodal analysis, we simulated them on the Tina pro software. We were tasked to calculate for voltage and current as well, to see if they are close to simulated values. Keywords— Circuit, Current, Voltage, Power, Mesh, Nodal
I. INTRODUCTION
Fig. 2 Circuit with mesh currents labeled as i1, i2, and i3. The arrows show the direction of the mesh current.
Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three.
Fig. 1 Essential meshes of the planar circuit labeled 1, 2, and 3. R1, R2, R3, 1/sc, and Ls represent the impedance of the resistors, capacitor, and inductor values in the s-domain. Vs and is are the values of the voltage source and the current source respectively.
Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.
A mesh current is a current that loops around the essential mesh and the equations are set solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them.]It is usual practice to have all the mesh currents loop in the same direction. This helps prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise direction. Figure 2 shows the same circuit from Figure 1 with the mesh currents labeled. Solving for mesh currents instead of directly applying Kirchhoff's current law and Kirchhoff's voltage law can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents. Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current. For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the electronic component multiplied by the mesh current in that loop.
If a voltage source is present within the mesh loop, the voltage at the source is either added or subtracted depending on if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is not contained between two meshes, the mesh current will take the positive or negative value of the current source depending on if the mesh current is in the same or opposite direction of the current source.The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.
A dependent source is a current source or voltage source that depends on the voltage or current of another element in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like an independent source. After the mesh equation is formed, a dependent source equation is needed. This equation is generally called a constraint equation. This is an equation that relates the dependent source’s variable to the voltage or current that the source depends on in the circuit. The following is a simple example of a dependent source.
𝑀𝑒𝑠ℎ 1: 𝐼! = 𝐼!
1 𝑀𝑒𝑠ℎ 2: −𝑉! + 𝑅! (𝐼! − 𝐼! ) + 𝐼 − 𝐼! = 0 𝑠𝑐 ! 1 𝑀𝑒𝑠ℎ 3: 𝐼 − 𝐼! + 𝑅! 𝐼! − 𝐼! + 𝐿! 𝐼! = 0 𝑠𝑐 !
𝑀𝑒𝑠ℎ 1: −𝑉! + 𝑅! 𝐼! + 𝑅! 𝐼! − 𝐼! = 0 𝑀𝑒𝑠ℎ 2: 𝑅! 𝐼! + 3𝐼! + 𝑅! 𝐼! − 𝐼! = 0 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒: 𝐼! = 𝐼! − 𝐼!
Once the equations are found, the system of linear equations can be solved by using any technique to solve linear equations. There are two special cases in mesh current: currents containing a supermesh and currents containing dependent sources.
Fig. 5 Kirchhoff’s current law
Fig. 3 Circuit with a supermesh. Supermesh occurs because the current source is in between the essential meshes.
A supermesh occurs when a current source is contained between two essential meshes. The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the current source. This will be an equation where the current source is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a supermesh.
Fig. 4 Circuit with dependent source. ix is the current upon which the dependent source depends.
In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents. In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysisusing Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, Ibranch = Vbranch * G, where G (=1/R) is the admittance (conductance) of the resistor. Nodal analysis is possible when all the circuit elements' branch constitutive relations have an admittance representation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, many circuit simulation programs (e.g. SPICE) use nodal analysis as a basis. When elements do not have admittance representations, a more general extension of nodal analysis, modified nodal analysis, can be used.
While simple examples of nodal analysis focus on linear elements, more complex nonlinear networks can also be solved with nodal analysis by using Newton's method to turn the nonlinear problem into a sequence of linear problems. Steps for Nodal Analysis: 1. Note all connected wire segments in the circuit. These are the nodes of nodal analysis. 2. Select one node as the ground reference. The choice does not affect the result and is just a matter of convention. Choosing the node with the most connections can simplify the analysis. 3. Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to assign a variable. 4. For each unknown voltage, form an equation based on Kirchhoff's current law. Basically, add together all currents leaving from the node and mark the sum equal to zero. Finding the current between two nodes is nothing more than "the node you're on, minus the node you're going to, divided by the resistance between the two nodes." 5. If there are voltage sources between two unknown voltages, join the two nodes as a supernode. The currents of the two nodes are combined in a single equation, and a new equation for the voltages is formed. 6. Solve the system of simultaneous equations for each unknown voltage.
Fig. 6 Basic example circuit with one unknown voltage V1.
The only unknown voltage in this circuit is V1. There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node. 1. Current through resistor R1: (V1 - VS) / R1 2. Current through resistor R2: V1 / R2 3. Current through current source IS: -IS
With Kirchhoff's current law, we get:
𝑉1 − 𝑉𝑠 𝑅!
+
𝑉1 − 𝐼𝑆 = 0 𝑅2
This equation can be solved in respect to V1:
𝑉! =
𝑉! +𝐼 𝑅! ! 1
1
𝑅! + 𝑅!
Finally, the unknown voltage can be solved by substituting numerical values for the symbols. Any unknown currents are easy to calculate after all the voltages in the circuit are known.
5 𝑉 + 20 𝑚𝐴 100 Ω 𝑉! = ≈ 4.667 𝑉 1 1 100 Ω + 200 Ω
Fig. 7 In this circuit, VA is between two unknown voltages, and is therefore a supernode.
In this circuit, we initially have two unknown voltages, V1 and V2. The voltage at V3 is already known to be VB because the other terminal of the voltage source is at ground potential. The current going through voltage source VA cannot be directly calculated. Therefore we can not write the current equations for either V1 or V2. However, we know that the same current leaving node V2 must enter node V1. Even though the nodes can not be individually solved, we know that the combined current of these two nodes is zero. This combining of the two nodes is called the supernode technique, and it requires one additional equation: V1 = V2 + VA.
The complete set of equations for this circuit is:
𝑉1 − 𝑉𝐵
+
𝑅! 𝑉! = 𝑉! + 𝑉!
𝑉2 − 𝑉𝐵 𝑅!
+
𝑉! = 0 𝑅!
By substituting V1 to the first equation and solving in respect to V2, we get: 𝑉! =
𝑅! + 𝑅! 𝑅! 𝑉! − 𝑅! 𝑅! 𝑉! 𝑅! + 𝑅! 𝑅! + 𝑅! 𝑅!
II. MATERIALS AND METHODS For this experiment we used a computer unit with a full version of the Tina pro circuit simulator.
then calculated the values obtained manually using nodal analysis and applying the principle for supernodes.
III. RESULTS AND DISCUSSION Tables I and II shows the results gathered during the experiment using the simulated values using Tina Pro and by solving the given circuits using Mesh Analysis and Nodal Analysis. TABLE I Mesh Currents Simulated Values
For the first part of the experiment we drew the mesh circuit diagram and ran it with our desired values. Calculated Values
I1
I2
I3
62.3
34.43
-85.25
I1
I2
I3
62.295 34.426 -85.25
Voltages V1
V2
V3
6.23 4.59
8.36
V1
V3
V2
V4
V5
13.77 25.41
V4
V5
6.23 4.59 8.361 13.77 25.41
MESH ANALYSIS
The measured values in table 1 was generated by using the program Tina Pro as stated by the manual. The calculated values were solved using Mesh Analysis.
Fig. 8 Circuit for the mesh analysis
The software is gave us the mesh currents I1, I2 and I3 along with the voltages V1 V2 V3 V4 and V5 across their resistors. After that, we computed these values manually using mesh analysis. In the second part we drew and simulated the nodal circuit diagram in the Tina pro.
Fig. 9 Circuit for the nodal analysis
We obtained the node voltages V1 V2 and V3, and the currents I1 I2 and I3. We were careful on the polarities of different variables because making an error on that would mean that could jeopardize the entirety of our experiment. We
Fig. 10 Manual solution for the mesh analysis
TABLE II
NODAL ANALYSIS Mesh Currents Simulated Values
Calculated Values
Voltages
V1
V2
V3
I1
I2
I3
-4.49
5.51
15
-44.87
27.56
-50
V1
V2
V3
I1
I2
I3
-4.487
5.513
-15
-44.87
27.565
-50
Like the first table, the measured values in table 2 was generated by using the program Tina Pro as stated at the manual. The calculated values are solved using Nodal Analysis.
We learned that mesh analysis method is a more convenient tool for analyzing planar multiple loop circuit with multiple active sources. Of course, the source must be a voltage source. If in case there is a current source, it must be transformed to an equivalent voltage source. Consequently, we verified that the similar principle applied in Kirchhoff’s Voltage law is applied in Mesh analysis. We consider the fact that the summation of voltage drop is equal to the summation of voltage rise within a mesh loop. A major current or a mesh current is assumed to be flowing across each mesh loop, from which the values of other sub currents are calculated. Furthermore, the directions of all the mesh current in a circuit are all uniform, that is, all clockwise or all counterclockwise. For paths and elements shared by two mesh loops, the actual current flowing is equal to the difference between the two opposing mesh currents. On the other hand, we have also succeeded in investigating the efficacy of nodal analysis method on multiple active linear sources in a network. A node is a point at which two or more elements have a common connection. We learned that for a nodal analysis to be applicable, the circuit must be planar or two dimensional, and the active sources must be in the form of current source. If in case there is a voltage source, it must be converted to an equivalent current source. Moreover, we verified that similar Kirchhoff’s Current law (KCL) is applied in nodal analysis. The point of concentration is at nodes, where the primary equation is obtained. A ground is set as a reference voltage source designated as the 0V point to help in the analysis of the circuit. Each current in the primary equation is expressed in terms of the potential difference and the resistance. That is: I!"#!$%"& =
I!"#$%&' à I! = I! + I! à
!" !"
=
!" !"
+
!" !"
We have also learned about the special case, the supernode. This is used when between two nodes there is only an active source present. This pair of nodes along with the active source is treated as a supernode. As a node, we can say that the current entering the supernode is equal to the current leaving the supernode.
Fig. 11 Manual solution for the nodal analysis
IV. CONCLUSION In general, we have succeeded in achieving the objectives of the experiment. We were able to investigate the efficacy of mesh analysis on multiple active linear sources in a network. Mesh is a loop which does not contain any other loops within.
ACKNOWLEDGMENT We would like to thank our loving families and friends who are always ready to help us. Also, gratitude is in order for our laboratory teacher and the laboratory assistant. We would also like to extend our gratitude to our lecture professor who taught us the mesh and nodal analysis. REFERENCES [1] [2]
http://en.wikipedia.org/wiki/ Mesh_analysis http://en.wikipedia.org/wiki/ Nodal_analysis
