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2, Network Theorems, and A.C. Bridges, , , , , , , , $1, Electrical Network, , A single closed electrical circuit is called a single loop circuit or simple network. In, sucha circuit, the current in an impedance, and potential difference across it can be calculated, by Ohm’s law or Kirchhoff’s law. On the other hand in a complex circuit, the calculation of, current in an impedance and potential difference across it by applying Kirchhoff’s law is not, simple because then we have to solve several equations. Thus to simplify the calculations in, acomplex network, we apply different Network theorems. It is important to define the some, definitions before going into the explanation of network solutions., , Circuit element : The individual circuit component as inductor, capacitor, resistor or, generator etc. with two terminals by which it may be connected to other electric component, , are called circuit element., , Network : Any electrical circuit containing capacitor, inductance and generators, is known as electrical networks. The circuit which contains no source of e.m.f. is known as, Passive network and the circuit which contains source of e.m.f. is called Active network., Networks are called to be linear network when the current in the network is directly, proportional to the driving voltage while the non linear networks are in which the current is, not directly proportional to the driving voltage. The network whose either of the properties, changes with direction of operation is called the unilateral network. The network whose, properties remain same in either direction of operation is known as bilateral network., , Branch : A branch of network is defined as a group of elements in series (group of, elements along which the current remain constant)., , Mesh or Loop : Aclosed path (circuit) in a network is called mesh or loop., , Node or Junction : Anode is a point or junction where two or more branches meet and, are connected electrically., , Short circuit ; When two points of a circuit are connected together by zero resistance, wire, they are said to be short circuited., , Open circuit : Two points are said to be open circuited when there is no connection, between them,, , As we know that in series combination, the equivalent impedance is equal to the sum, of impedances i.e., , 2g HZ +2, 42, + oy, , While in parallel combination, the reciprocal of the equivalent impedance is equal to, , the sum of reciprocal of all impedances. i.e., , Network Theorems and A.C. Bridges | 45, , , , Scamec wan Cancer,

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jconductor Devices, 46 | RPP : Circuit Fundamentals and Semicon ., , , , we need potential divider and current, , In network analysis, combination of, , divider. The potential l’ can be divided in series, impedences Z, and Z, as (see Fig. 1), , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , VE, _ V4, J = =, " Zj+Z, Leg, WZ, _ ‘Vay, "2° FZ, Lug, Similarly, the current /is divided into parallel combination of I, impedances as (sce Fig. 2) L 7, IZ, IZ i, = = £ 1 25, 1 Z+Z, Leg [#2], = ZT, 242, Leg Fig, 2, , § 2. Network theorems, We can solve complicated network by using some simple theorem. Some of important, theorems are Superposition theorem, Reciprocity theorem, Thevenin theorem and Norton's, theorem., § 2.1. Superposition Theorem, According to this theorem, Jn any linear network containing impedances and more, than one source of e.mJ,, the current flowing in any branch, is equal to the vector sum of the currents that would, separately flow by each source +e. mf. while all the other, sources beirg replaced by their inernal impedances., Consider a network consists of two source of e.m.f., E, and E, and impedances Z, 2) and Z, are connected as, shown in Fig. 3(a). In this case suppose /, and /, are the, , , , , , Currents in mesh (1) and (2). Then by using Kirchhoff's law, mitts, , E,=(Z, +25), + Zl, (I), , Ey = 231, + (Z,+ Z3)I, ou(2), , Solving these equations for 7, and 1, we get, _ (2) +Z,)E, _ Z3E5 (3), Zk, +Z,2Z,4+2,2, ZZ, F2,Z,4 2,2,, (Z, +Z;)E,, i 1+ Z3)E> Z3E AA), , Ala +ZiZy+Z32, 22,4 2,23 + ZZ,, , Now, we suppose the circuit of Fig. 3(b) in which the source of e.m.f. E, is absent., Then the mesh equation will be ;, , Ey =(Z, + Zs)" ZL" ; (5), O° 2h" (Zp + 251," 6), , Scamec wan Cancer,

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Network Theorems and A.C. Bridges | 47, where f,fand /,/are the currents in mesh 1 and 2, respectively due to source By only,, , By solving these equations, we pet, , , , and h Ze, £257, +24, wo), , Further we consider the circuit of F, , , , Then the mesh equation will be,, 0= (2, t Z5)1, 1 Zz; 1," 9), Ey Zh" (24 25)" (10), , where I, "and /," are the currents in mesh | and 2, , , , , , , , respectively due to source £, only., On solving, we get, , , , , , , , , , Fig. 3 (ec), e(L1), , , , , , itm tee, 2 iby + ZZ;, From equations (3), (4), (7), (8), (11) and (12), we have, hah", , = ie "4 i, ", , Thus, the superposition theorem is proved. Thus, the current J, is equal to sum of 1’, and J, "i.e. the current flowing in empedance Z, when sources £, and & 5 are present is equal, to the sum of current flowing in impedance Z, whew only one source is resent in the circuit., While the other source is replaced by their internal impedance,, , § 2.2. Reciprocity Theorem, , According to this theorem, in any linear network, if'a source of c.m.f. introduced in a, mesh (first) produces a current in any other mesh (second), then the same source £ introduced, in second mesh will produce the same current in the first mesh., , Let a linear network [shown in Fig. 4(a)] has a 2{a}+—>-+2 LS, , Source of e.m.f. E, which is in mesh |, , In this circuit, the mesh equation, will be, : 23, E = (Z, + ZyMy + Zyl, Qn) El, , 3h r (2, a I), Solving these equation for current in Z, i.e. J,, , (12), , , , , , , , , , , , , , , , , , , , , , , , , , , , Fig. 4 (a), [y=, Let us now consider the network shown in Fig. —} 2, 4(b), the source of emf E is now placed in mesh 2., In this circuit, the mesh equation will be 1 [2s] ly Qe, 0=(Z4,4+4)h+ 23h, E= Zyl, +(Z.+Z)h, afi +( Fig. 4 (b), , Solving these equations for current in Z tel, , Scamec wan Cancer,

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48 | RPP: Circuit Fundamentals and Semiconductor Devices, , -Z,E, cre, Lo Z\Z + 27234732, , Which is same as the current L previously in mesh 2. / . ld in ai, , Thus the reciprocity theorem is provied. A network in which this theorem hold in said, to be bilateral. Network which has resistance, inductor or capacitor as circuit clement are, bilateral. When electron tube or other control devices are present in the circuit, the network, may not be bilateral and theorem may not be apply., , § 2.3. Thevenin's Theorem, , The essence of this theorem is to simplify the complicated circuits containing several, generators and impedances by an equivalent circuit, which has the same characteristics as, the original., , According to this theorem, Any two terminal linear network containing energy, sources (generators) and impedances can be replaced with an equivalent circuit consisting, of a voltage source E' in series with an impedance Z’. The value of E' is the open circuit, voltage between the terminals of the network and Z' is the impedance measured between, the terminals with all energy sources being replaced by their internal resistance., , Consider a network containing energy source and impedance as shown in F ig. 5(a)., Let 4 and B be two output terminals of this circuit and between these terminals the load, impedance Z , 's connected. Fig. 3(b) represents the Thevenin’s equivalent network., , [, , 4 4 A Zz, , A, TOW 2, , E Nn Zs (Nn Zz, => ES) (4 4., , , , , , , , —o, (a) # ow ?, , Fig.é, , Let, J; and /) are the currents flowing in active and, , we want to calculate the value of E’ and Z’ from Fig., to a circuit as shown in Fig. 5(b)., , passive mesh tespectively. Now,, 5(a) and then show that it is quivalent, , Applying Kirchhoff’s voltage law in both meshes, we get, A+ 23) 2,1, =E me, ~ 21, + (Z, +2Z;5+Z) I, =0 (2), From equation (2), we have, Z2+Z3+Z,, = (Za +Z+Z)) I,, 4, Putting this value in equation (1), we get, (2, + Zs) (Zp + Z3+Z,), fase 2-2 61,, l “3, , , , 43, , Scamec wan Cancer,

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Network, Theorem, sand A.C. Brid,, , lges | 49, , = ZZ sa, of +Z,Z. 5, 3 FZ Z +L 42 +252 72, t, sao oe, 2(Z,+2Z3)+2)2,+Z, (Z, +24), 1, = PEM +25) _, , L, , , , ZZ, Zy+ 143 z, ZtZ, , Calculation of E "This is the voltage between, , , , Aand B when Z, is removed. This is shown in Fig (6)., , 2, ia §, ee ee ee) Or Es] |, &, , , , 301 242;, EZ,, , , , , , , , , , , , , , , , , , , , , , , , , , Voltage drop across Z, i.e. E ‘= (4), , ° Z+Z, ~ Fig. 6, Calculation of Z'—This is the impedance between, terminals A and B when load removed and so\ — { J-o, , AF source of e.m-f. 71, js short circuited (See Fig. 7) veofems :, © QZ., Thus Z=z,+72 a, | . 2 Z4Z wi) -], By using equations (4) and (5), equation (3) will be, —o, E' i, Le Fig.7, Lo Z'+Z, ©), The current for the Fig. 5(b) i.e., for equivalent circuit is, E', L=, Lo Z'+Z, ”), , Equations (6) and (7) are same., Therefore, the Thevenin’s theorem is proved., , §24, 4,Norton’s Theorem, This theorem gives an equivalent circuit where current sources are used rather than, , emé * os ., sources, Thus, the resultant equivalent cireutt contains a curren, , Voltage source,, , t source instead of, , , , , , » sources, , , , ing e”, , Fig. 8, inal network con taint, , Any two term, ith an equivalent cire, , e replaced W, * The value of l'is the sur, , wit Cont, < circuit current, , According to this theorem,, , (gen, , er, , Soret and impedances ca” b, Source I! in parallel with an impedance Z, , Scamec wan Cancer,