ELECTRICAL ENGINEERING

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that circuit The direction of ow of the current is indicated by an arrow (see Figs 166 to 168) pointing from point 1 to point 2 of the characteristics Kirchhoff s EMF Law The sum of the potential drops around any loop is equal to the sum of the emf s in that loop, provided the notations are as in the preceding subsection Any current which ows in a direction opposite to that speci ed in the preceding subsection causes a negative drop Moreover, in any one loop, oppositely directed currents cause drops of opposite signs Kirchhoff s emf law for some of the loops in Figs 166 to 168 is expressed in Eqs (1624) to (1629): Fig 166a: e12 Fig 166b: e12 L1 di12 dt L2 L2 di12 dt L3 L3 di12 dt L4) L4 di12 dt (1625a) (1625b) i12 R1 i12 R2 i12 R3 i12 R4 i12(R1 R2 R3 R4) (1624)

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di12 (L1 dt Fig 166c: e12 q C1 q C2 i12 dt Fig 166d : e12 i12R1 i12R2 (R1 Fig 167a: Loop 1a1b12: Loop a1a2b2b1a1: Loop a2a3b3b2a2: Loop a3a4b4b3a3: L1 di12 dt L2 di12 dt (L1 q C3 1 C1

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(1626a) (1626b)

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dt C1 i12 dt C2 1 C1 1 C2 i12 dt (1627a) (1627b)

R2)i12

di12 dt

e12 0 0 0

i1R1 i1R1 i2R2 i3R3 i2R2 i3R3 i4R4

(1628a) (1628b) (1628c) (1628d )

CHAPTER SIXTEEN

Fig 168: Loop 1a1b1b22: e12 Loop a1b1b2a2: 0 i1R1 L1 di1 dt i2R2 L2 di2 dt (1629b) i1R1 L1 di1 dt R3i12 L3 di12 dt i12 dt C3 (1629a)

Kirchhoff s Current Law The sum of currents entering a junction is equal to the sum of currents leaving that junction Note: Entrance to a junction is denoted by an arrow directed toward the junction; exit is indicated by an arrow directed away from that junction Kirchhoff s current law for the junction a, Figs 167 and 168, is expressed in Eqs (1630) Fig 167a to c Fig 167d : Fig 168: i12 i12 i12 i1 i1 i1 i2 i2 i2 i3 i3 i4 i4 i5 i6 (1630a) (1630b) (1630c)

10 Use of Kirchhoff s Laws Kirchhoff s emf and current laws are used to de-

termine the current owing in each branch of a given circuit when the emf and characteristics of that circuit are speci ed It can be shown that in order to solve for the currents, the number of Kirchhoff s emf equations must be equal to the number of simple loops in a circuit, while the number of Kirchhoff s current equations must be equal to the number of junctions less one

DIRECT-CURRENT CIRCUITS

11 Circuit with One Resistance Consider a circuit (Fig 161) in which direct

voltage E is impressed on a resistance R In order to determine the current I, use is made of Ohm s law Thus E IR, and I E R (1631)

12 Circuit with One Inductance The circuit in Fig 163 consists of a direct

voltage E impressed on an inductance L Equation (1610) is used to determine the current i: E Solving Eq (1632) for i, L di dt (1632)

ELECTRICAL ENGINEERING

1 L E t L

E dt k

(1633a) (1633b)

Equation (1633b) shows that the current i increases inde nitely when a direct voltage E is impressed on an inductance Indeed, an inductance acts virtually as a short circuit for a constant direct emf

13 Circuit with One Capacitance The circuit in Fig 169 consists of a direct

voltage E impressed on a capacitance C Here, by Eq (1617a), q i i CE dq dt 0 C dE dt (1634a) (1634b) (1634c)

Equation (1634c) states that the steady current in Fig 169 is zero In other words, a condenser acts virtually as an open circuit when a direct emf is impressed on it Since an inductance acts as a short circuit and a condenser acts as an open circuit whenever a direct emf is impressed on them, they will be considered as nonadmissible characteristics in this section The section will be devoted, therefore, to series, parallel, and series-parallel circuits of resistances only