Mesh and Node Equations: Circuits Containing Dependent Sources
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1 Mesh nd Node Equtons: Crcuts Contnng Dependent Sources Introducton The crcuts n ths set of problems re smll crcuts tht contn sngle dependent source. These crcuts cn be nlyzed usng mesh equton or usng node equtons. When dong so, t s useful to express the controllng current or voltge of the dependent source s functon of the mesh currents or node voltges. Node equtons re dscussed n Sectons 4.3, 4.4 nd 4.5 of Introducton to Electrc Crcuts by R.C. Dorf nd J.A Svobod. Secton 4.5 consders crcuts tht contn dependent sources. Mesh equtons re dscussed n Sectons 4.6 nd 4.7. Secton 4.7 consders crcuts tht contn dependent sources. Worked Exmples Exmple : Consder the crcut shown n Fgure. Fnd the vlue of the gn, A, of the CCCS. Fgure The crcut consdered n Exmple. Soluton: Fgure 2 shows the crcut from Fgure fter replcng the mmeter by n equvlent short crcut nd lbelng the current mesured by the mmeter. Ths crcut cn be nlyzed usng mesh equtons. Fgure 3 shows the crcut fter numberng the meshes. Let nd 2 denote the mesh currents n meshes nd 2, respectvely.
2 Fgure 2 The crcut from Fgure fter replcng the mmeter by short crcut. Fgure 3 The crcut from Fgure 2 fter lbelng the meshes. Ths crcut contns dependent source. When nlyzng such crcut usng mesh equtons, t s mportnt to express the controllng current or voltge of the dependent source n terms of the mesh currents. In ths crcut, the controllng current of the dependent source s the current. The controllng current of the dependent source s relted to mesh current by = The mesh current 2 s the current mesured by the mmeter so Apply KVL to the supermesh to get Substtutng for the mesh currents gves ( ) ( ) 2 = 2 A = = 0 = A The current n the dependent source s relted to the mesh currents by 2
3 Substtutng the vlues of the currents gves A = 2 A ( ) = 2 A= 3 A/ A Exmple 2: Consder the crcut shown n Fgure 4. Fnd the vlue of the gn, A, of the CCVS. Fgure 4 The crcut consdered n Exmple 2. Soluton: Fgure 5 shows the crcut from Fgure 4 fter replcng the voltmeter by n equvlent open crcut nd lbelng the voltge mesured by the voltmeter. Ths crcut cn be nlyzed usng node equtons. Fgure 6 shows the crcut fter selectng reference node nd numberng the other nodes. Let v, v 2 nd v 3 denote the node voltges t nodes, 2 nd 3, respectvely. Fgure 5 The crcut from Fgure 4 fter replcng the voltmeter by n open crcut. 3
4 Fgure 6 The crcut from Fgure 5 fter lbelng the nodes. The voltge of the 2 V voltge source cn be expressed n terms of the node voltges s 2 = v 0 v = 2 V The voltmeter mesures the node voltge t node 3 so v 3 = 3.27 V Ths crcut contns dependent source. When nlyzng such crcut usng node equtons, t s mportnt to express the controllng current or voltge of the dependent source n terms of the node voltges. In ths crcut, the controllng current of the dependent source s the current. The controllng current of the dependent source s equl to the current drected from rght to left n the 8 Ω resstor. Hence s relted to node voltges by v2 v = = =.09 A 8 8 The dependent source voltge s relted to the node voltges by A = 0 v = v 2 2 The vlue of the current n n open crcut s lwys zero so pplyng KCL t node 3 gves v v = 0 Substtutng for the node voltges gves A 3.27 = 0 A(.09) 3.27 = 0 A= 3 V/A 7 4
5 Exmple 3: Consder the crcut shown n Fgure 7. Fnd the vlue of the gn, A, of the VCCS. Fgure 7 The crcut consdered n Exmple 3. Soluton: Fgure 8 shows the crcut from Fgure 7 fter replcng the mmeter by n equvlent short crcut nd lbelng the current mesured by the mmeter. Also, the 6 Ω nd 6 Ω seres resstors hve been replced by the equvlent 22 Ω resstor. Ths crcut cn be nlyzed usng mesh equtons. Fgure 9 shows the crcut fter numberng the meshes. Let nd 2 denote the mesh currents n meshes nd 2, respectvely. Fgure 8 The crcut from Fgure 7 fter replcng the mmeter by n short crcut. Fgure 9 The crcut from Fgure 8 fter lbelng the meshes. 5
6 The mesh current s equl to the current n the ndependent current source so = 0.2 A The mesh current 2 s the current mesured by the mmeter so 2 =. A Ths crcut contns dependent source. When nlyzng such crcut usng mesh equtons, t s mportnt to express the controllng current or voltge of the dependent source n terms of the mesh currents. In ths crcut, the controllng voltge of the dependent source s the voltge v. Applyng KVL to the supermesh gves ( ) v = 0 v = = 4.4 V The current n the dependent source s relted to the mesh currents by Av = 2 Substtutng the vlues of the currents nd voltge gves A ( ) ( ) 4.4 = 0.2. A= 0.3 A/V Exmple 4: Consder the crcut shown n Fgure 0. Fnd the vlue of the gn, A, of the VCVS. Fgure 0 The crcut consdered n Exmple 4. 6
7 Soluton: Fgure shows the crcut from Fgure 0 fter replcng the voltmeter by n equvlent open crcut nd lbelng the voltge mesured by the voltmeter. Also, the 3 Ω nd 8 Ω seres resstors hve been replced by the equvlent Ω resstor. Ths crcut cn be nlyzed usng node equtons. Fgure 2 shows the crcut fter selectng reference node nd numberng the other nodes. Let v nd v 2 denote the node voltges t nodes nd 2, respectvely. Fgure The crcut from Fgure 0 fter replcng the voltmeter by n open crcut. Fgure 2 The crcut from Fgure fter lbelng the nodes. Ths crcut contns dependent source. When nlyzng such crcut usng node equtons, t s mportnt to express the controllng current or voltge of the dependent source n terms of the node voltges. In ths crcut, the controllng voltge of the dependent source s the voltge v. Ths voltge s relted to node voltges by v = v 0 = v The dependent source voltge s relted to the node voltges by Av 0 v2 v 2 = = () The voltmeter mesures the node voltge t node 2 so v 2 = 47.4 V 7
8 Apply KCL t node to get v v 2 5+ = 0 Substtutng the vlue of v 2 gves v = v = = V Substtutng voltge vlues nto Equton gves ( ) A 7.86 = 47.4 A= 6 V/V 8
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