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Series-Parallel Circuits
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Series-parallel circuits, also called networks, consist of individual groups of series and parallel resistors. Such circuits, as long as they consist only of individual groups of series and parallel resistances, can always be reduced to a single equivalent resistance. Consider, as an example, the series-parallel circuit shown in Fig. 31, in which we wish to nd the battery current I. It is given that the battery voltage is constant 45 volts, and the resistance values are in ohms.
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Solution First, by eq. (35), the two parallel 10-ohm resistors can be replaced with an equivalent single 5-ohm resistor, and the three parallel 12-ohm resistors can be replaced with a single 4-ohm resistor. When this is done, Fig. 31 becomes the simple series circuit shown in Fig. 32 (where 45 V is read as 45 volts ).
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CHAPTER 2 Electric Current. Ohm s Law
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Since the circuit of Fig. 32 is now a purely series circuit (section 2.5) we have that the original circuit of Fig. 31 reduces to a single equivalent resistance of 15 ohms, as shown in Fig. 33. Hence, by Ohm s law, eq. (11), the battery current I is equal to I V=RT 45=15 3 amperes; answer: Problem 20 A battery of constant 36 volts is connected to the series-parallel circuit in Fig. 34. Resistance values are in ohms.
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In the gure, let it be required to nd: (a) (b) (c) battery current I, power output of battery, current in 6-ohm resistor.
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Problem 21 A battery of constant 24 volts is applied to the series-parallel network shown below. Resistance values in ohms. Find the battery current. (Answer: 2.97497 amps)
CHAPTER 2 Electric Current. Ohm s Law
Problem 22 Given the series-parallel network as follows, resistance values in ohms, nd:
(a) (b)
potential of point x with respect to ground, potential of point y with respect to ground.
Problem 23 What value of resistance must be connected in parallel with a 36-ohm resistor if the parallel combination is to be equivalent to a single 20-ohm resistor
* The term ground is often used to denote a common reference point or reference line in a network. Such a ground may or may not be connected to an actual earth ground.
Determinants and Simultaneous Equations
3.1 Introduction to Determinants
The purpose of this chapter is to prepare us for future work in the writing and solution of network equations. We ll nd that network analysis produces systems of simultaneous equations, and such systems are most conveniently handled by making use of what are called determinants. The study of determinants is not basically di cult, but it will call for close attention to details on your part. The results, however, will be well worth the time and e ort you put into it. Let us begin with some de nitions, as follows. A determinant is a square array of numbers, or letters used to represent numbers, placed between two vertical bars. The numbers or letters are arranged in horizontal rows and vertical columns. An example of what a determinant looks like is shown in Fig. 35. Notice that the rows are numbered from the top down, and the columns from left to right.
Fig. 35
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CHAPTER 3 Determinants and Equations
Each number or letter in a determinant is called an element of the determinant. Since a determinant is always a square array of elements, the number of rows is always the same as the number of columns. Note that all rows and columns have the same number of elements. A determinant is classi ed according to the number of rows (or columns) it has. The determinant in Fig. 35 is thus a fourth-order determinant, because it has four rows (and also, of course, four columns). A determinant has a value equal to a single number. For instance, later on we ll be able to show that for the determinant of Fig. 35, D 312. The location of an element in a determinant will always be speci ed by giving FIRST the number of the ROW and THEN the number of the COLUMN it is located in. For example, the location of the element 4 in Fig. 35 would be given as (4,1), meaning it is located at the intersection of the fourth row and the rst column. As another example, the location of the element 6 would be speci ed as (3,4), meaning at the intersection of the third row and the fourth column. Many of our discussions will be easier to follow if we represent the elements of a determinant by a letter with subscripts. The system of notation used is illustrated by the fourth-order determinant of Fig. 36. This gure illustrates how subscripts are used to identify the location of the elements in a determinant.
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