zn=(40-14040 40+i4092 40+j40-j40 and the total impedance presented to the voltage source is z=2p+40-140=40+y40+40-140=809 Then I, the current leaving the voltage source, is j0 =3+j0A and by a current division (3+j0)=j(3+j0)=0+j3A 40-140+140 In Fig. 3. 12(b), the current source delivers current to the 40-Q2 resistor and to an impedance consisting of za=-140+Zp= -140+40+y40=40 Then, two current divisions give Ic 40 40+40[40-140+140/0-j6)=2(0-16)=3+10A The current I in the circuit of Fig. 3. 12(a) is I=I+I=0+j3+(3+和0)=3+j3A The Network Theorems of Thevenin and norton If interest is to be focused on the voltages and across the currents through a small portion of a network such as network B in Fig. 3. 13(a), it is convenient to replace network A, which is complicated and of little interest by a simple equivalent. The simple equivalent may contain a single, equivalent, voltage source in series with an equivalent impedance in series as displayed in Fig 3. 13(b). In this case, the equivalent is called a Thevenin equivalent. Alternatively, the simple equivalent may consist of an equivalent current source in parallel with equivalent impedance. This equivalent, shown in Fig. 3. 13(c), is called a Norton equivalent. Observe that as long as Zr(subscript T for Thevenin) is equal to ZN(subscript N for Norton), the two equivalents may be obtained from one another by a simple source transformation. Conditions of Application The Thevenin and Norton network equivalents are only valid at the terminals of network A in Fig 3. 13(a)and they do not extend to its interior. In addition, there are certain restrictions on networks A and B Network A may contain only linear elements but may contain both independent and dependent sources. Network B, on the other hand, is not restricted to linear elements; it may contain nonlinear or time-varying elements and may c 2000 by CRC Press LLC
© 2000 by CRC Press LLC and the total impedance presented to the voltage source is Z = ZP + 40 – j40 = 40 + j40 + 40 – j40 = 80 W Then ^ I1, the current leaving the voltage source, is and by a current division In Fig. 3.12(b), the current source delivers current to the 40-W resistor and to an impedance consisting of the capacitor and Zp . Call this impedance Za so that Za = –j40 + ZP = –j40 + 40 + j40 = 40 W Then, two current divisions give ^ IC The current ^ I in the circuit of Fig. 3.12(a) is ^ I = ^ IV + ^ IC = 0 + j3 + (3 + j0) = 3 + j3 A The Network Theorems of Thévenin and Norton If interest is to be focused on the voltages and across the currents through a small portion of a network such as network B in Fig. 3.13(a), it is convenient to replace network A, which is complicated and of little interest, by a simple equivalent. The simple equivalent may contain a single, equivalent, voltage source in series with an equivalent impedance in series as displayed in Fig. 3.13(b). In this case, the equivalent is called a Thévenin equivalent. Alternatively, the simple equivalent may consist of an equivalent current source in parallel with an equivalent impedance. This equivalent, shown in Fig. 3.13(c), is called a Norton equivalent. Observe that as long as ZT (subscript T for Thévenin) is equal to ZN (subscript N for Norton), the two equivalents may be obtained from one another by a simple source transformation. Conditions of Application The Thévenin and Norton network equivalents are only valid at the terminals of network A in Fig. 3.13(a) and they do not extend to its interior. In addition, there are certain restrictions on networks A and B. Network A may contain only linear elements but may contain both independent and dependent sources. Network B, on the other hand, is not restricted to linear elements; it may contain nonlinear or time-varying elements and may Z j j j j j P = - + - = + (40 40)( 40) 40 40 40 40 40 W ˆ I j j 1 240 0 80 = 3 0 + = + A ˆ I ( ) ( ) j j j j j j j V = - + È Î Í Í ˘ ˚ ˙ ˙ + = + = + 40 40 40 40 3 0 3 0 0 3A ˆ I ( ) ( ) j j j j j j j C = + È Î Í Í ˘ ˚ ˙ ˙ - + È Î Í Í ˘ ˚ ˙ ˙ - = - = + 40 40 40 40 40 40 40 0 6 2 0 6 3 0 A
FIGURE 3. 13 (a) Two one-port networks;(b)the Thevenin equivalent for network a; and (c) the Norton equivalent for also contain both independent and dependent sources. Together, there can be no controlled source coupling or magnetic coupling between networks A and B The Thevenin Theorem The statement of the Thevenin theorem is based on Fig 3.13(b): Insofar as a load which has no magnetic or controlled source coupling to a one-port is concerned, a network containing linear elements and both independent and controlled sources may be replaced by an ideal voltage source of strength, Vr, and an equivalent impedance Zr, in series with the source. The value of Vr is the open-circuit voltage, Voo appearing across the terminals of the network and Zr is the driving point imped ce at the terminals of the network, obtained with all independent sources set equal to zero. The Norton Theorem The Norton theorem involves a current source equivalent. The statement of the Norton theorem is based on ig.3.13(c): Insofar as a load which has no magnetic or controlled source coupling to a one-port is concerned, the network containing linear elements and both independent and controlled sources may be replaced by an ideal current source of strength, IN, and an equivalent impedance, ZN, in parallel with the source. The value of IN is the hort-circuit current, Isc, which results when the terminals of the network are shorted and Zx is the driving point impedance at the terminals when all independent sources are set equal to zero The Equivalent Impedance, Z=Z Three methods are available for the determination of Zr. All of them are applicable at the analyst's discretion. When controlled sources are present, however, the first method cannot be used The first method involves the direct calculation of Ze=Z ZN by looking into the terminals of the network after all independent sources have been nulled. Independent sources are nulled in a network by replacing all independent voltage sources with a short circuit and all independent current sources with an open circuit. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC also contain both independent and dependent sources. Together, there can be no controlled source coupling or magnetic coupling between networks A and B. The Thévenin Theorem The statement of the Thévenin theorem is based on Fig. 3.13(b): Insofar as a load which has no magnetic or controlled source coupling to a one-port is concerned, a network containing linear elements and both independent and controlled sources may be replaced by an ideal voltage source of strength, ^ VT , and an equivalent impedance ZT , in series with the source. The value of ^ VT is the open-circuit voltage, ^ VOC, appearing across the terminals of the network and ZT is the driving point impedance at the terminals of the network, obtained with all independent sources set equal to zero. The Norton Theorem The Norton theorem involves a current source equivalent. The statement of the Norton theorem is based on Fig. 3.13(c): Insofar as a load which has no magnetic or controlled source coupling to a one-port is concerned, the network containing linear elements and both independent and controlled sources may be replaced by an ideal current source of strength, ^ IN, and an equivalent impedance, ZN, in parallel with the source. The value of ^ IN is the short-circuit current, ^ ISC, which results when the terminals of the network are shorted and ZN is the driving point impedance at the terminals when all independent sources are set equal to zero. The Equivalent Impedance, ZT = ZN Three methods are available for the determination of ZT . All of them are applicable at the analyst’s discretion. When controlled sources are present, however, the first method cannot be used. The first method involves the direct calculation of Zeq = ZT = ZN by looking into the terminals of the network after all independent sources have been nulled. Independent sources are nulled in a network by replacing all independent voltage sources with a short circuit and all independent current sources with an open circuit. FIGURE 3.13 (a) Two one-port networks; (b) the Thévenin equivalent for network a; and (c) the Norton equivalent for network a
The second method, which may be used when controlled sources are present in the network, requires the computation of both the Thevenin equivalent voltage( the open-circuit voltage at the terminals of the network) and the Norton equivalent current(the current through the short-circuited terminals of the network). The The third method may also be used when controlled sources are present within the network. a tes may be placed across the terminals with a resulting current calculated or measured. Alternatively, a tes may be injected into the terminals with a resulting voltage determined. In either case, the equivalet can be obtained from the value of the ratio of the test voltage V to the resulting current I Example 3.2. The current through the capacitor with impedance -j35 Q2 in Fig 3. 14(a) may be found using Thevenin's theorem. The first step is to remove the -j35-Q2 capacitor and consider it as the load. When this is done, the network in Fig. 3. 14(b)result The Thevenin equivalent voltage is the voltage across the 40-Q2 resistor. The current through the 40-Q2 resist vas found in Example 3. 1 to be I=3+j3 Q2 Thus Vr=40(3+3)=120+j120 A The Thevenin equivalent impedance may be found by looking into the terminals of the network in 3. 14(c). Observe that both sources in Fig 3. 14(a) have been nulled and that, for ease of computation, npedances Za and Zh have been placed on Fig 3.14(c).Here, zn=(40-140)40)=40+140g2 0+140-140 b 0)(40) +j15g Both the Thevenin equivalent voltage and impedance are shown in Fig 3. 14(d), and when the load is attached, as in Fig 3. 14(d), the current can be computed as 120+20 20+15-35 0+j6A in Fig 3. 15. Here it is observed that a single current division gives ent source transformation and is shown The Norton equivalent circuit is obtained via a simple voltage-to-cur 20+n15 (672+0.96)=0+j6A +j15-j35 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC The second method, which may be used when controlled sources are present in the network, requires the computation of both the Thévenin equivalent voltage (the open-circuit voltage at the terminals of the network) and the Norton equivalent current (the current through the short-circuited terminals of the network). The equivalent impedance is the ratio of these two quantities The third method may also be used when controlled sources are present within the network. A test voltage may be placed across the terminals with a resulting current calculated or measured. Alternatively, a test current may be injected into the terminals with a resulting voltage determined. In either case, the equivalent resistance can be obtained from the value of the ratio of the test voltage ^ Vo to the resulting current ^ Io Example 3.2. The current through the capacitor with impedance –j35 W in Fig. 3.14(a) may be found using Thévenin’s theorem. The first step is to remove the –j35-W capacitor and consider it as the load. When this is done, the network in Fig. 3.14(b) results. The Thévenin equivalent voltage is the voltage across the 40–W resistor. The current through the 40-W resistor was found in Example 3.1 to be I = 3 + j3 W. Thus, ^ VT = 40(3 + j3) = 120 + j120 V The Thévenin equivalent impedance may be found by looking into the terminals of the network in Fig. 3.14(c). Observe that both sources in Fig. 3.14(a) have been nulled and that, for ease of computation, impedances Za and Zb have been placed on Fig. 3.14(c). Here, and ZT = Zb + j15 = 20 + j15 W Both the Thévenin equivalent voltage and impedance are shown in Fig. 3.14(d), and when the load is attached, as in Fig. 3.14(d), the current can be computed as The Norton equivalent circuit is obtained via a simple voltage-to-current source transformation and is shown in Fig. 3.15. Here it is observed that a single current division gives Z Z Z V I V I T N eq T N OC SC = = = = ˆ ˆ ˆ ˆ Z V I T o o = ˆ ˆ Z j j j j j Z a b = - + - = + = + = ( )( ) ( )( ) 40 40 40 40 40 40 40 40 40 40 40 40 20 W W ˆ ˆ I V j j j j j T = + - = + - = + 20 15 35 120 120 20 20 0 6 A ˆ I ( . . ) j j j = j j + + - È Î Í Í ˘ ˚ ˙ ˙ + = + 20 15 20 15 35 6 72 0 96 0 6 A
409 -40240+0v j159 (c) 120+1 FIGURE 3. 14 (a)A network in the phasor domain;(b)the network with the load removed; (c)the network for the computation of the Thevenin equivalent impedance; and(d)the Thevenin equivalent 套20 -j359 j159 FIGURE 3. 15 The Norton equivalent of Fig. 3. 14(d)
© 2000 by CRC Press LLC FIGURE 3.14 (a) A network in the phasor domain; (b) the network with the load removed; (c) the network for the computation of the Thévenin equivalent impedance; and (d) the Thévenin equivalent. FIGURE 3.15 The Norton equivalent of Fig. 3.14(d)
ellegen's Theorem Tellegen's theorem state In an arbitrarily lumped network subject to KVL and KCL constraints, with reference directions of the branch currents and branch voltages associated with the KVl and KCl constraints, the product of all branch currents and branch voltages must equal zero Tellegen's theorem may be summarized by the equation VIJk where the lower case letters v and j represent instantaneous values of the branch voltages and branch currents, respectively, and where b is the total number of branches. A matrix representation employing the branch current and branch voltage vectors also exists. Because V and j are column vectors V·J==JV The prerequisite concerning the KVL and KCL constraints in the statement of Tellegen's theorem is of crucial Example 33. Figure 3. 16 displays an oriented graph of a particular network in which there are six branche labeled with numbers within parentheses and four nodes labeled by numbers within circles. Several know branch currents and branch voltages are indicated. Because the type of elements or their values is not germane to the construction of the graph, the other branch currents and branch voltages may be evaluated from repeated applications of KCL and KVL. KCL may be used first at the various nodes node3:i2=i-j4=4-2=2A J3 8-2=-10A node2:j=j3-i4=-10-2=-12A Then KVl gives 8-6=2V 10-6=-16V v1=v2+v6=8-16=-8V FIGURE 3.16 An oriented graph of a particular network with some known branch currents and branch voltages c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Tellegen’s Theorem Tellegen’s theorem states: In an arbitrarily lumped network subject to KVL and KCL constraints,with reference directions of the branch currents and branch voltages associated with the KVL and KCL constraints, the product of all branch currents and branch voltages must equal zero. Tellegen’s theorem may be summarized by the equation where the lower case letters v and j represent instantaneous values of the branch voltages and branch currents, respectively, and where b is the total number of branches.A matrix representation employing the branch current and branch voltage vectors also exists. Because V and J are column vectors V · J = VT J = J T V The prerequisite concerning the KVL and KCL constraints in the statement of Tellegen’s theorem is of crucial importance. Example 3.3. Figure 3.16 displays an oriented graph of a particular network in which there are six branches labeled with numbers within parentheses and four nodes labeled by numbers within circles. Several known branch currents and branch voltages are indicated. Because the type of elements or their values is not germane to the construction of the graph, the other branch currents and branch voltages may be evaluated from repeated applications of KCL and KVL. KCL may be used first at the various nodes. node 3: j2 = j6 – j4 = 4 – 2 = 2 A node 1: j3 = –j1 – j2 = –8 – 2 = –10 A node 2: j5 = j3 – j4 = –10 – 2 = –12 A Then KVL gives v3 = v2 – v4 = 8 – 6 = 2 V v6 = v5 – v4 = –10 – 6 = –16 V v1 = v2 + v6 = 8 – 16 = –8 V FIGURE 3.16 An oriented graph of a particular network with some known branch currents and branch voltages. v j k k k b = = Â 0 1