i v v 6 i 2 i 3 v + (1) (2) (3) (4) (5) Substituting (4) and (5) into (3) (6) = 2 (7) (5) and (6) (8) (4) and (6) ˆ

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Download "i v v 6 i 2 i 3 v + (1) (2) (3) (4) (5) Substituting (4) and (5) into (3) (6) = 2 (7) (5) and (6) (8) (4) and (6) ˆ"

Willis Lee
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1 5V 6 v 6 î v v Ω î Ω v v 8Ω V î v 5 6Ω 5 Mesh : 6ˆ ˆ = Mesh : ˆ 8ˆ = Mesh : ˆ ˆ ˆ 8 0 = 5 Solvng ˆ ˆ ˆ from () = Solvng ˆ ˆ ˆ from () = 7 7 Substtutng () and (5) nto () (5) and (6) 9 ˆ = A 8 ˆ = A 0 () and (6) ˆ = A 0 () () () () (5) (6) (7) (8)

2 5V 6 v 6 î v v Ω î Ω v v 8Ω V î v 5 6Ω 5 = ˆ ˆ = ˆ = ˆ ˆ = ˆ ˆ ˆ 5 = = ˆ 6,,,,,, ( ˆ ˆ) v = v = ˆ v = v = 8 ( ˆ ˆ ) v = 6ˆ v = Loop equaton around mesh : v v v = 0 ( ˆ ˆ) ˆ = 0 6ˆ ˆ = () Loop equaton around mesh : v v v = 0 ( ) 6ˆ 8( ˆ ˆ ) = 0 5 ˆ 8ˆ = Loop equaton around mesh : v v v = 0 5 ( ˆ ˆ) 8( ˆ ˆ ) = 0 6 ˆ 8ˆ 0ˆ = 5 () ()

3 6 5 We can redraw ths dgraph so that there are no ntersectng branches. 5 6 Hence the above dgraph s planar.

4 Wrtng NodeAdmttance Matrx Y n By Inspecton Node Votage Equaton: Y Y Y, n e s Y Y Y, n e s = (8) Y n, Yn, Yn, n en s n Y e n s where = ( a lgebrac sum of all current sources leavng node j ) s j Dagonal Elements of Yn Y = sum of admttances Y j of all resstors R j connected to node m, m =,,, n, where n s the total number of nodes. s called the nodeadmttance matrx. s the nodetodatum voltage vector. s called the node current source vector. mm OffDagonal Elements of Yn Y Y n e s jk = ( sum of admttances Y j of all resstors R j connected across node j and node k ) Symmetry Property: Y s a symmetrc matrx,.e., n jk kj Y = Y

5 Proof : Snce Y b n () s a dagonal matrx, Y b = Y T b Y T n = ( AY A T ) T b T b = AY A = AY A b T T = Y n

6 Wrtng MeshImpedance Matrx Z m By Inspecton Let m be the total number of meshes of a planar dgraph G, ncludng the exteror mesh formed by traversng the outer boundary branches (.e., those branches havng only onecrculatng current ˆ passng through them). Hence, MeshCurrent Equaton : Z Z Z ˆ v, m s Z ˆ Z Z, m v s = Z ˆ m, Zm, Zm, m v m sm Z m î vs where v = ( clockwse algebrac sum of all voltage sour ces around mesh j) s j Dagonal Elements of Zm Z Z m s called the meshmpedance matrx. ˆ s called the meshcurrent vector. v s m = number of nteror meshes (wndows) = sum of mpedances Z R of all resstors located along mesh " k", k=,,, m, where m s the total number of (nteror and exteror) meshes. kk j j s called the meshvoltage source vector. OffDagonal Elements of Zm j () () Z ( = sum of mpedances Z R of all resstors along both mesh j and mesh k ) jk j j Symmetry Property: Z s a symmetrc matrx,.e., m jk kj Z = Z () ()

7 Extended Mesh Current Method Crcut N v _ e e _ v = ˆ A Ω v î v v Ω î KVL around mesh : ( ) v 6V ( ) v = ˆ ˆ Step. When the crcut contans " β" current sources s, s,,, use s β ther assocated voltages vs, v s,, v when applyng KVL. s β ˆ ˆ = 0 () KVL around mesh : ( ˆ ) ˆ ˆ = 6 () Step. For each current source s, add an equaton. j s j s = j s j ˆ = () Step. Solve the ( m) β equatons for ˆ, ˆ, ˆ m, vs, v,. s v sβ Substtutng () nto (), we obtan : ( ) ˆ ˆ = 6 ˆ = 0 () Substtutng () and () nto (), we obtan : 0 v = 0 v = 6V (5) ( )

8 Note: The unknown varables n the extended mesh current method consst of the usual m mesh currents, plus the unknown voltages assocated wth the current sources. Hence, f there are β current sources, the extended mesh current method would consst of (m)β ndependent lnear equatons nvolvng (m)β unknown varables { ˆ } ˆ ˆ m vs vs vs,,,,,,. β (m) mesh current varables β voltage varables

9 All branch voltages and currents can be trvally calculated from and v. î = î = 0 A, v = = 0V = ˆ ˆ = A, v = = 6V = î = 0 A, = î = A, v v = 6 V = 6 V Verfcaton of Soluton by Tellegen's Theorem : j= v= ( v) ( v) ( v) ( v) j j = (0)(0) (6)() (6)(0) ( 6)()? 0 =

10 Extended Node Voltage Method Crcut N v _ e e e _ = A Ω v v v Ω _ Step. When the crcut contans " α" voltages vs, v s,, v s α, use ther assocated currents s, s,, s α when applyng KCL. e ( e e) KCL at : = () ( e e) KCL at : = 0 () Step. For each voltage source v, add an equaton e e = v. _ 6V = e s j j j s e = 6 () Step. Solve the ( n) α equatons for e, e, en, s,,. s s α Substtutng () nto (), we obtan : e ( e 6) = e = 6V () Substtutng () nto (), we obtan : = 0 (5) e j

11 Note: The unknown varables n the extended node voltage method consst of the usual n nodetodatum voltages, plus the unknown currents assocated wth the voltage sources. Hence, f there are α voltage sources, the modfed node voltage method would consst of (n)α ndependent lnear equatons nvolvng (n)α unknown varables { e } e en s s s,,,,,,. α (n) nodetodatum voltage varables α current varables

12 Suffcent Condton for G to be Planar If G has less than 9 branches, t s planar. Proof. The basc nonplanar graphs have 9 and 0 branches, respectvely.

13 Mappng a planar dgraph on a sphere A dgraph G s planar f, and only f, t can be drawn on the surface of a sphere such that G can be parttoned nto contguous regons and colored (as n a map of countres) such that no two regons have overlappng colors.

14 Mappng a planar dgraph on a sphere 6 m m m 5 exteror mesh m m, m, and m are nteror clockwse meshes. We can always redraw a planar dgraph on the surface of a sphere wthout nteror branches and vceversa, as llustrated below. Although mesh m (formed by tracng along branches {,, 6} n the above planar dgraph n a clockwse drecton) appears to be counterclockwse on the sphere, t actually moves n a clockwse drecton when vewed from behnd. Although the counterclockwse loop m formed by exteror branches (.e, boundary branches wth only crculatng current) {6, 5, } does not look lke a mesh on the planar dgraph, t s n fact a clockwse mesh when the dgraph s mapped on the surface of a sphere.

15 6 m 5 m m Just as all countres on a globe can be mapped onto a flat plane of paper, any planar dgraph drawn on a sphere can always be redrawn as a planar dgraph on a plane.

16 Mappng a planar dgraph on a sphere 6 m m m 5 exteror mesh m 6 m 5 m m

17 Dualty Prncple There are many physcal varables, concepts, propertes, and theorems n electrcal crcut theory whch appear n pars, henceforth called dual pars, such that for each crcut theorem, property, concept, etc., there s a dual theorem, dual property, and dual concept, respectvely. Ths dualty prncple s extremely useful snce we only need to learn and memorze half of them! Varable, concept, property KVL KCL Voltage Current Seres Parallel Resstance (Ohms) Impedance Admttance Node (nondatum) Node voltage Datum node Dual varable, concept, property KCL KVL Current Voltage Parallel Seres Conductance (Semens) Admttance Impedance Mesh (nteror) Mesh current Exteror mesh

18 Example of Dualty e v s î R R v v s G G seres crcut parallel crcut KVL : ( R R) = vs voltage dvder: ˆ G G e KCL : ( ) = s current dvder: v R = R R v s G = G G s

19 Dualty Theorem For planar crcuts, the nodevoltage method and the meshcurrent method, as well as ther extended versons, are dual sets of equatons whch can be derved from each other va ther dual varables. Nodevoltage Equaton Meshcurrent equaton Ye= Z ˆ n s m = v s NodeAdmttance matrx Yn MeshImpedance matrx Zm

Circuit Analysis I (ENGR 2405) Chapter 3 Method of Analysis Nodal(KCL) and Mesh(KVL)

Circuit Analysis I (ENGR 2405) Chapter 3 Method of Analysis Nodal(KCL) and Mesh(KVL) Crcut Analyss I (ENG 405) Chapter Method of Analyss Nodal(KCL) and Mesh(KVL) Nodal Analyss If nstead of focusng on the oltages of the crcut elements, one looks at the oltages at the nodes of the crcut,