KIRCHHOFF LOWS AGAINST NODE-VOLTAGE ANALYSIS AND MILLMAN'S THEOREM MARCELA NICULAE 1, C. M. NICULAE 2
The present paper aims at introducing a new method for teaching electric circuits in high schools. Using the Kirchhoff's laws to solve electric networks of medium complexity involves a great deal of less appealing mathematical effort. Instead, we propose a more efficient algorithm that combines the node-voltage analysis with Millman's theorem. The method is simple enough, and requires less effort to solve electric circuits of medium complexity. It is this simplicity that improves the student ability to correctly solve the problem.
1. Introduction Circuit analysis normally refers to finding all node voltages and all branch currents for an electrical circuit. In this paper we call this task solving the fundamental problem of the circuit, or, in short, solving the circuit. Several methods are available for linear circuit analysis [1]. The most important are: Branch current method and Mesh current method. Apart from these there are other methods such as: Superposition, Source transformations, and Equivalent circuits. These methods combine one of the main methods with powerful theorems such as: Millman's Theorem [2,3], Superposition Theorem, Thévenin's Theorem and Norton's Theorem (see [4,5] for an historical review about the Equivalent Circuit Concept). Of course, all of these efficient methods are based on Ohm's and Kirchhoff's Laws. The curricula of Romanian high school dedicated to circuit analysis recommends a direct method based on Kirchhoff's laws to be used in applications. Teaching experience showed that this method is time consuming, and less motivating for students. On the other hand, at the academic level, any basic course on electric circuits introduces proper theorems and methods for 1
„Ion Barbu” Theoretical High School, Bucharest, email:
[email protected] University of Bucharest, Faculty of Physics, Atomiştilor 405, Bucharest-Măgurele, C.P. MG-11, 077125, RO. 2
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such an analysis. The experience of teaching, based on a statistical evaluation of students, showed that when students make use of these theorems and methods they respond faster, providing right answers and solutions.
1.1. Prerequisite concepts
The elements of an electrical network are: the nodes, the branches, and the loops. Their definitions are well known (see for example [1]). ● branch is a part of a circuit that contains at least a resistor or a voltage source and connect two nodes; ● node is the junction of two or more branches; ● loop is any closed connection of branches; ● reference node (or reference), is a node in the circuit, with respect to which all other voltages are measured; ● node voltage is measured with respect to reference node; ● Kirchhoff Current Law (also known as Kirchhoff's first law) (KCL). The sum of all currents flowing into a node is zero; ● Kirchhoff Voltage Law (Kirchhoff's second law) (KVL). The sum of all voltages around a loop is zero.
2. The branch current method In electrical engineering, the branch current method (BCM), also known as nodal analysis or node-voltage analysis, is used to determine the voltage (potential difference) between nodes in an electrical circuit. Here, we present a brief description of this method that solves the system of KCL and KVL equations in terms of node voltages.
2.1. BCM steps
The steps of BCM method are: ● Assume directions of currents in each branch – as is presently done in the method based on Kirchhoff's laws. ● Chose one of the circuit nods as reference When choosing the reference node, select the node with the most connections.
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● Write
KCL equations
I I i
53
o
0
(1)
where I i and I 0 are currents that flow into, leave (flow out) the node, respectively. ● Write the Generalized Ohm's Law (GOL) equation for each branch of the circuit VA VB E p En I k Rm where I k is the branch current,
E
R
(2) m
is the branch equivalent resistance,
E n is the equivalent voltage source of the branch, V A and, V B are the voltages at the ends of the k branch. ● Solve each one of the above equations for the branch current – as p
I k V A V B E p E n / Rm
(3)
● Substitute these currents in KCL system (1) ● Solve this system for node voltages ● Substitute these voltages in equations (3) and
calculate the branch currents This method is particularly efficient since for most circuits encountered in practice, the number of independent nodes is smaller than the number b of branches. 3. Millman's Theorem Millman's Theorem, also known as parallel generator theorem, allows for the calculation of a node voltage, in terms of voltages at neighboring nodes [2, 3]. Let X be a node having N neighbors (see fig. 1).
Fig. 1. Circuit with a common node X connected to N branches
The Millman's Theorem states that the voltage of the node X is given by
V V X k Rk
1 / Rk
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(4)
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That is, the voltage at X is the sum of branch voltages, multiplied by their associated admittances, divided by the total admittance. By total admittance we mean the inverse of the equivalent resistance of the N resistors connected in parallel. At the high school level, the admittance, Y, is defined as the inverse of electrical resistance, 1 / R . 4. Our method
The steps of our method are: ● Simplify,
if possible, the circuit branches. For instance, if a branch has more than one source voltage, merge them into a single one. Do the same for branch resistances (see definitions associated with system 2). ● Chose one of the nodes in the circuit as reference node. See step 2 of BCM method. ● Apply Millman's theorem to each independent node of the circuit. ● Solve the above system in terms of node voltages. ● Assume directions of currents in each branch. ● Write Ohm's law for each branch and solve it for branch current. The simplification mentioned as a first step is possible because it is always possible to find an equivalent branch which has a single voltage source in series with a resistor. In fact, this is the main result of the Thévenin's theorem [1, 4]. At high school level, Thévenin's theorem for linear electrical networks can be stated as follows: Any combination of voltage sources and resistors with two terminals is electrically equivalent to a single voltage source V, and a single series resistor R. 5. The test problem
Consider the circuit shown in fig. 2.
2k
10V
1.99k
4k
4V
1k
0.99k
10
10
Fig. 2. The test problem
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Problem
Solve this circuit (i.e. find node voltages and branch currents). For the purpose of comparison, this problem will be solved using: Kirchhoff's laws method, the branch current method, and our method. 5.1. Kirchhoff’s laws method
The test circuit has 5 branches (see fig. 3) I1 E1
R1
R5
A I5
B I4
I3
I
R3
R2
II
r1
R4
C
III
I2 E2
r2
Fig. 3. For each branch of the test circuit we assumed a direction for the current. We consider three loops and assumed a direction for each one
Notice that this circuit has 6 loops and 7 nodes, but only 3 loops and 2 nodes are independent. When A and B are chosen as independent nodes, Kirchhoff's current laws for these nodes may be written as I1 I 3 I 5 , I4 I2 I5.
A B
(5)
Kirchhoff's voltage law applied to loops I, II, and III gives E1 r1 R1 I 1 R3 I 3 ,
(I)
R 3 I 3 R5 I 5 R 4 I 4 ,
(II)
E 2 r2 R2 I 2 R4 I 4 ,
(III)
(6)
The solution of this system reads
I k k / , for k = 1, 2, …, 5,
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(7)
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where 1 R4E1R5 R3R4E2 + R3R4E1 + R4E1r2 + + R4E1R2 +r2R5E1 + R2R5E1 + E1R3r2 + E1R3R2, 2 E2R5R1 + r1R3E2 + R1R3E2 + E2R3R5 + + E2R5r1 + R3R4E2 + r1R4E2 + R1R4E2 – R3R4E1 3 r1R4E2 + R1R4E2 + R4E1R5 + R4E1r2 + R4E1R2 + r2R5E1 + R2R5E1 4 E1R3r2 + E2R5R1 + r1R3E2 + R1R3E2 + E2R3R5 + E2R5r1 + E1R3R2 5 R3R4E2 + r1R4E2 + R1R4E2 R3R4E1 – E1R3r2 E1R3R2 and R3R4R5 + R3R4r2 + R3R4R2 + R3R5r2 + R3R5R2 + + r1R4R5 + r1R4R3 + r1R4r2 + r1R4R2 + r1R5r2 + r1R5R2 + + r1R3r2 + r1R3R2 + R1R4R5 + R1R4R3 + R1R4r2 + R1R4R2 + + R1R5r2 + R1R5R2 + R1R3r2 + R1R3R2 Look at this solution and imagine the typical student reaction. Imagine also the teacher effort in deriving this solution. What you feel, the student feels too. This is why such a solution is less probable to become the object of teaching in high schools. This kind of solution is usually called a literal solution. Deriving such a solution is time consuming and has less teaching results. The students become quickly bored in watching the teacher effort. It is naturally they think like that: „What a boring lecture! I hope the teacher assumes that us will never be able to do such a task. Maybe during this boring lecture I can do some other interesting things”. As the reader probably knows: „If something is boring, it is very difficult to do”. Therefore, during physics lectures, boring moments like this should be avoided as much as possible. Also, if possible, the literal solution should be avoided. In practice, we proceed with numbers from the very beginning. Numbers can be reduced, simplified, and finally, the mathematical effort is considerably reduced. Even so, solving a system derived directly from Kirchhoff's laws is time consuming. Our problem has been conceived to have a nice numerical solution: I1 = 4mA, I2 = 1mA, I3 = 3mA, I4 =2mA and I5 = 1mA.
(8)
Of course, as you probably noticed, the problem has a nice numerical solution.
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Suppose the student finishes his homework and he/she fails in finding the right answer. What a common student thinks in this particular moment looks like this: "I spent a lot of time to understand the method, I did my best in applying it, I wrote, and solved the system. Now, there must be a mistake, but where, I have no idea.
5.2. The branch current method
The steps in applying this method are: The directions of currents are specified in fig. 3. Since node C connects the most branches, it will be used as reference node. KCL system is given by (5). Before writing the equations for currents, we must use proper units for measuring the currents and the resistors. Here, we used mA, V, and k. This way, the involved numbers have less digits. After minor calculations (for instance, 1.99 + 0.01 = 2 of equivalent resistance in branch 2) the equations of branch currents in terms of node voltages are derived
I 1 10 V A / 1 , I 2 4 VB / 2 ,
I 3 VA / 2 ,
(9)
I 4 VB / 1 ,
I 5 V A VB / 4 . After substituting these currents in (5) the following system is derived 10 V A V A V A VB , 1 2 4 VB 4 VB V A VB . 1 2 4 The solution of the above system is V A 6V and VB 2V .
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After substituting these voltages in (9), we get our “nice” solution (8) will be discovered. Undoubtedly, such a method is better “tolerated” by students. This is a great relief both for teacher and students. But we can do much better, as our method proves.
5.3. Our method
Simplifications The equivalent resistances of branches 1 and 2 were computed and placed near the nodes A and B, respectively (see fig. 4). These locations are appropriate for the needs of Millman's theorem. A I5 4k
D I1 1k
2k I2 F
I4 1k
2k
I3 10V
B
4V
C Fig. 4. Following the requirements of Millman's theorem, two nodes, D and F, were labeled. With respect to node C, their voltages are 10V and 4V, respectively
Node C has been chosen as reference (see the reason in BCM)
To apply the Millman's theorem, two nodes, D and F, have been labeled. With respect to the reference node, C, the voltages of these nodes are, 10V and 4V, respectively. When applied to nodes A and B, the Millman's theorem yields to the following system 10 VB 1 4 , VA 1 1 1 1 2 4
4 VA 2 4 , VB 1 1 1 2 1 4
Nice look, don't you think? It is easy to see that V A 6V and VB 2V is the solution of the above system.
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This point forward, the branch currents are calculated in a straightforward manner, using the Ohm's law (see fig. 5). 10V I1 1k
6V I5 4k
2k I2 4V
I4 1k
2k
I3 10V
2V
4V
C
Fig. 5. Last step. Deriving branch currents using Ohm's law
Indeed, at this step, the students can derive numerical values for branch currents using the Ohm's law only. For instance, I 1 10 6V / 1k 4 mA . This way, we come across the “nice” solution. We hope that this example suffices to emphasize the power of our method. 6. Conclusions
Our method is particularly efficient because it transforms a complex problem into a set of small sub-problems, each of which requiring less effort and time. Also, our method allows for increasing the time allocated to major concepts in the domain, instead of looking for the solution of a large system of equations. It also contributes to developing the algorithmic thinking of students, while preparing them for jobs in many engineering fields. With much less effort involved, even problem solving can become funny. Actually, all the methods and theorems involved in circuit analysis are very old by now. After such a long time passed from their discovery, reconsidering them is a necessity as the references [4, 5] prove.
REFERENCES [1] G. RIZZONI, Principles and Applications of Electrical Engineering, 4th edition, McGraw Hill, 2002. [2] J. MILLMAN, A Useful Network Theorem, Proc. Inst. of Radio Engineers, 28: 413, 1940. [3] V. NERGUIZIAN, M. RAFAF, C. NERGUIZIAN, „Electric and Electronic Circuit Analysis with Millman Theorem”, 9th WSEAS International Conference on CIRCUITS, July 11-13, 2005, pp 497-091. [4] D. H. JOHNSON, Origins of the Equivalent Circuit Concept: The Voltage-Source Equivalent, Proc. IEEE, 91: 636-640, 2003. [5] D. H. JOHNSON, Origins of the Equivalent Circuit Concept: The Current-Source Equivalent, Proc. IEEE, 91: 817-821, 2003.
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