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are n resistances R1, R2, , Rn connected in series across a direct voltage E By Kirchhoff s law,
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FIGURE 1610
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CHAPTER SIXTEEN
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(1635a) (1635b) (1635c)
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(1635d )
Equations (1635c) and (1635d ) state that the series circuit (Fig 1610a) may be replaced by a simple circuit comprising one resistance R (Fig 1610b) whose value is equal to the sum of the series resistances
15 Parallel Circuits with Several Resistances In the circuit in Fig 1611a, there are n resistances in parallel across the same voltage E By Ohm s law,
E I R1 2
E I R2 n
E Rn
(1636)
Applying Kirchhoff s current law to the junction a, I I1 E E R where 1 R 1 R1 1 R2 1 Rn 1 R1 I2 1 R2 In 1 Rn (1637a) (1637b) (1637c) (1637d )
Equations (1636) give the currents in the various branches of the parallel circuit Equation (1637a) states that the total current I supplied by the emf source is the sum of these currents Equations (1637c) and (1637d ) state that the parallel circuit (Fig 1611a) may be replaced by a simple circuit (Fig 16llb) comprising one resistance whose value is expressed by Eq (1637d )
16 Series-Parallel Circuits (Networks) The determination of the currents in a
network of resistances (Fig 1612) is accomplished by the application of Kirchhoff s laws Thus, applying Kirchhoff s emf law to the loops 1a1b12 and a1b1b2a2,
FIGURE 1611
ELECTRICAL ENGINEERING
FIGURE 1612
I1R1 I1R1
I3R3 I2R2
(1638a) (1638b)
Applying Kirchhoff s current law to the junction a, I3 I1 I2 (1639)
Equations (1638) and (1639) give three independent relations among the currents I1, I2, and I3 from which these currents can he determined Thus write these equations as follows: R1I1 R1I1 0I2 R2I2 I1 Using determinants (see Sec 13), let R1 R1 1 E 0 0 I1 R1 E R1 0 1 0 I2 R1 R1 1 I3 0 E R2 0 1 0 R2 0 1 R1R2 0 R2 1 0 R2 1 R3 0 1 R3 0 1 R1R2 (R1R2 R1R3 R2R3) (1641) I2 R3I3 0I3 I3 E 0 0 (1640a) (1640b) (1640c)
R2E R1R2
R2R3
(1642a)
ER1 R1R3
R2R3
(1642b)
E(R2 R1) R1R2 R1R3 R2R3
(1642c)
CHAPTER SIXTEEN
17 Power and Energy in Direct-Current Circuits Power in a dc circuit is
de ned as follows: P EI (1643)
where E is the emf impressed on the circuit, and I is the total current furnished by that emf If E is expressed in volts and I in amperes, P is in watts The power in any branch k of the circuit is de ned through Joule s law as follows: Pk I 2Rk k (1644)
where Pk is the power in branch k, Ik is the current owing through branch k, and, Rk is the resistance of branch k The energy supplied to a dc circuit in an interval of time of length t is W Pt EIt (1645)
If P is in watts and t is in seconds, W is expressed in watt-seconds or joules The energy supplied to each branch in an interval t is Wk Pk t
2 I k Rkt
(1646)
But, by the principle of conservation of energy, the total energy supplied must be equal to the sum of the energies supplied to all the branches Hence, if a circuit has n branches,
W or
2 I k Rkt
(1647a)
I 2Rk k
(1647b)
Equation (1647a) is the law of conservation of energy in a dc circuit, and Eq (1647b) is the law of conservation of power in a dc circuit
SINGLE-PHASE ALTERNATING-CURRENT CIRCUITS
18 Sine Functions
De nition A sine function of x is of the form e Em sin (x ) (1648)
Graph A graph of the sine function given in Eq (1648) is shown in Fig 1613, wherein Em is the maximum value or amplitude of the sine function, e0 is the initial value of the sine function or its value corresponding to x 0, and is the initial angle of the sine function
ELECTRICAL ENGINEERING
FIGURE 1613
Periodicity Sine functions are periodic and have a period of 2 In other words, a sine function repeats itself every 2 radians, or e where n is any integer Cycle The set of values which a sine function assumes in the course of a period constitute a cycle The term cycle is often used in ac circuits and is denoted by the symbol
19 Average and Effective Values Frequent reference is made in the text to the average and effective values of a sine function These are now de ned
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