# Systematic Circuit Analysis (T&R Chap 3)

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1 Systematc Crcut Analyss TR Chap ) Nodeoltage analyss Usng the oltages of the each node relate to a ground node, wrte down a set of consstent lnear equatons for these oltages Sole ths set of equatons usng, say, Cramer s Rule Mesh current analyss Usng the loop currents n the crcut, wrte down a set of consstent lnear equatons for these arables. Sole. Ths ntroduces us to procedures for systematcally descrbng crcut arables and solng for them MAE4 Lnear Crcuts 4

2 Node oltages Nodal Analyss Pck one node as the ground node Label all other nodes and assgn oltages, B,, N and currents wth each branch,, M Recognze that the oltage across a branch B B s the dfference between the end node oltages A Thus B C wth the drecton as ndcated Wrte down the KCL relatons at each node Wrte down the branch relatons to express branch currents n terms of node oltages Accommodate current sources Obtan a set of lnear equatons for the node oltages 4 C C MAE4 Lnear Crcuts 4

3 Apply KCL Nodal Analyss Example p. Node A: Node B: Node C: 4 Wrte the element/branch eqns S G B ) G C ) G B 4 G 4 C S Substtute to get node oltage equatons Node A: G G ) G B G C S Node B: G G G ) B s Node C: G G G 4 ) C S Sole for, B, C then,,,, 4, G G G G B R R S R S 4 Reference node G G G G G G4 B C R 4 C S S S MAE4 Lnear Crcuts 4

4 G G G G G G G Systematc Nodal Analyss G G G4 B C S S S B R R S C S 4 Wrtng node equatons by nspecton Note that the matrx equaton looks just lke G for matrx G and ector and G s symmetrc and nonnegate defnte) R Reference node R 4 Dagonal,) elements: sum of all conductances connected to node Offdagonal,j) elements: conductance between nodes and j Rghthand sde: current sources enterng node There s no equaton for the ground node the column sums ge the conductance to ground MAE4 Lnear Crcuts 4

5 Nodal Analyss Example p.4 R/ Node A: Conductances G/ B G C.G Source currents enterng S Node B: S R B R R/ R C Conductances Node C: G/ A G/ C G ground G Source currents enterng Conductances G A G/ B G ground.g Source currents enterng MAE4 Lnear Crcuts.G.G G A S.G G.G B G.G.G C 44

6 Nodal Analyss some ponts to watch. The formulaton gen s based on KCL wth the sum of currents leang the node total G AtoB B ) G AtoC C ) G AtoGround leanga Ths yelds G AtoB G AtoGround ) G AtoB B G AtoC C enternga G AtoB G AtoGround ) G AtoB B G AtoC C enternga. If n doubt about the sgn of the current source, go back to ths basc KCL formulaton. Ths formulaton works for ndependent current source For dependent current sources ntroduced later) use your wts MAE4 Lnear Crcuts 4

7 Nodal Analyss Exercse p. KΩ B S KΩ S Ω O _ Sole ths usng standard lnear equaton solers Cramer s rule Gaussan elmnaton Matlab.. MAE4 Lnear Crcuts.. A B S S 46

8 Nodal Analyss wth Voltage Sources Current through oltage source s not computable from oltage across t. We need some trcks They actually help us smplfy thngs Method source transformaton R S Rest of the _ S crcut S RS R S Rest of the crcut B B Then use standard nodal analyss one less node MAE4 Lnear Crcuts 4

9 Nodal Analyss wth Voltage Sources Method groundng one node _ S B Rest of the crcut Ths remoes the B arable smpler analyss But can be done once per crcut MAE4 Lnear Crcuts 48

10 Nodal Analyss wth Voltage Sources Method Create a supernode Act as f A and B were one node KCL stll works for ths node Sum of currents enterng supernode box s Wrte KCL at all N other nodes N nodes less Ground node usng and B as usual Wrte one supernode KCL _ S B supernode Rest of the crcut Add the constrant B S These three methods allow us to deal wth all cases MAE4 Lnear Crcuts 49

11 Nodal Analyss Example 4 p. 6 _ S R _ R R S _ S R R S R R R _ Ths s method transform the oltage sources Applcable snce oltage sources appear n seres wth Rs Now use nodal analyss wth one node, A G G G ) A A G S G S G S G G G G S MAE4 Lnear Crcuts

12 S n _ Nodal Analyss Example p. R n R R/ B R R/ R C What s the crcut nput resstance ewed through S? S.G G B.G C G.G B. C Rewrte n terms of S, B, C Ths s method Sole. S 6. S B, C.. S B S C. S n R R /.R.R R n.8r. MAE4 Lnear Crcuts G B.G C.G S.G B.G C G S

13 Nodal Analyss Example 6 p. 8 S _ B supernode C Other consttuent relaton C S Two equatons n two unknowns MAE4 Lnear Crcuts R R R _ O S R 4 reference Method supernodes KCL for supernode: 4 Or, usng element equatons G G B ) G C B ) G 4 C Now use B S G G ) G G4 ) C G G ) S 4

14 Summary of Nodal Analyss. Smplfy the cct by combnng elements n seres or parallel. Select as reference node the one wth most oltage sources connected. Label node oltages and supernode oltages do not label the nodes drectly connected to the reference 4. Use KCL to wrte node equatons. Express element currents n terms of node oltages and ICSs. Wrte expressons relatng node oltages and IVSs 6. Substtute from Step nto equatons from Step 4 Wrte the equatons n standard form. Sole usng Cramer, gaussan elmnaton or matlab MAE4 Lnear Crcuts

15 MAE4 Lnear Crcuts 4 Solng sets of lnear equatons Cramer s Rule Thomas Rosa Appendx B pp. A to A x x x ) ) ) ) ) ) ) ) 8 ) ) ) 8 ) 8 ) 8 8 ) ) ) ) ) ) 6 ) ) 8 ) ) ) ) x

16 Solng sets of lnear equatons contd) ) ) x ) ) x.68 Notes: Ths Cramer s not as much fun as Cosmo Kramer n Senfeld I do not know of any trcks for symmetrc matrces MAE4 Lnear Crcuts

17 MAE4 Lnear Crcuts 6 Solng Lnear Equatons gaussan elmnaton x x x Augment matrx Row operatons only row row row row / row row row / row row row ) row row row )

18 Solng Lnear Equatons matlab A[ ; ; 8] A 8 B[4;;6] B 4 6 na) na)*b ans A\B ans ans MAE4 Lnear Crcuts

19 Mesh Current Analyss Dual of Nodal Voltage Analyss wth KCL Mesh Current Analyss wth KVL Mesh loop enclosng no elements Restrcted to Planar Ccts no crossoers unless you are really cleer) Key Idea: If element K s contaned n both mesh and mesh j then ts current s k j where we hae taken the reference drectons as approprate Same old trcks you already know R R Mesh A: Mesh B: 4 R S 4 S A B R A R B R A B ) S 4 S R R ) A R B S R A R R ) B S R R R R R R A B S S MAE4 Lnear Crcuts 8

20 Mesh Analyss by nspecton Matrx of Resstances R Dagonal elements: sum of resstances around loop Offdagonal j elements: resstance shared by loops and j Vector of currents As defned by you on your mesh dagram R S Voltage source ector S Sum of oltage sources assstng the current n your mesh R S R R S A If ths s hard to fathom, go back to the basc KVL to sort these drectons out C B R4 R R R R R R 4 R R R R A B C S S S MAE4 Lnear Crcuts 9

21 Mesh Equatons wth Current Sources Duals of trcks for nodal analyss wth oltage sources. Source transformaton to equalent TR Example 8 p. 9 KΩ KΩ KΩ KΩ 4KΩ V KΩ 4KΩ ma V O A O KΩ 8V B KΩ KΩ 6 A B 8 A.69 ma B.69 ma O A B.49 ma MAE4 Lnear Crcuts 6

22 Mesh Analyss wth ICSs method Current source belongs to a sngle mesh KΩ KΩ V O KΩ 4KΩ A B C ma Same example KΩ A 6 A B B 4 C C ma Same equatons Same soluton MAE4 Lnear Crcuts 6

23 Mesh Analyss wth ICSs Method Supermeshes easer than supernodes Current source n more than one mesh and/or not n parallel wth a resstance. Create a supermesh by elmnatng the whole branch noled. Resole the nddual currents last Supermesh R B R Excluded branch S S R B A ) R B R4 C R C A) A S A R C R 4 O B C S MAE4 Lnear Crcuts 6

24 Summary of Mesh Analyss. Check f cct s planar or transformable to planar. Identfy meshes, mesh currents supermeshes. Smplfy the cct where possble by combnng elements n seres or parallel 4. Wrte KVL for each mesh. Include expressons for ICSs 6. Sole for the mesh currents MAE4 Lnear Crcuts 6

25 Lnearty Superposton Lnear cct modeled by lnear elements and ndependent sources Lnear functons Homogenety: Addtty: fkx)kfx) fxy)fx)fy) Superposton follows from lnearty/addtty Lnear cct response to multple sources s the sum of the responses to each source. Turn off all ndependent sources except one and compute cct arables. Repeat for each ndependent source n turn. Total alue of all cct arables s the sum of the alues from all the nddual sources MAE4 Lnear Crcuts 64

26 Turnng off sources Voltage source Superposton Turned off when for all a short crcut S _ Current source Turned off when for all an open crcut S We hae already used ths n Théenn and Norton equ MAE4 Lnear Crcuts 6

27 Where are we now? Fnshed resste ccts wth ICS and IVS Two analyss technques nodal oltage and mesh current Preference depends on smplcty of the case at hand The am has been to deelop generalzable technques for access to analytcal tools lke matlab Where to now? Acte ccts wth resste elements transstors, opamps Lfe starts to get nterestng desgn ntroduced Capactance and nductance dynamc ccts Frequency response sdoman analyss Flters MAE4 Lnear Crcuts 66

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