Resistive Network Analysis. The Node Voltage Method  1


 Silvia Bryant
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1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff s Lw  we wll look t seerl of these. Generl pproch: Defne ll relent rles n systemtc wy. Identfy the known nd unknown rles. Construct set of equtons reltng these rles. Sole the equtons, usng the smllest set of equtons needed to sole for ll the unknown rles. he Node Voltge Method  hs s the most generl method for nlysng crcuts. Bss: defne oltge t ech node s n ndependent rle. One node s selected s reference node (often, ut not necessrly, ground). Ech of the other node oltges s referenced to ths node. Ohm s Lw s ppled etween ny two djcent nodes to determne the oltge flowng n ech rnch. Ech rnch current s expressed n terms of one or more node oltges, so currents do not pper n the equtons. rnch current : Krchoff s Current Lw s ppled t ech of the n nodes. hs ges n ndependent lner equtons for the n ndependent node oltges (n th reference node).
2 he Node Voltge Method  Methodology: () Select reference node (usully ground). All other node oltges re referenced to ths node. () Defne the remnng n node oltges s the ndependent rles. () Apply Krchoff s Current Lw t ech of the n nodes, expressng ech current n terms of the djcent node oltges. (4) Sole the lner system of n equtons n n unknowns. Exmple of pplyng KCL t one node (): d 0 c d 0 c he Node Voltge Method  Exmple: Drecton of current flow chosen ssumes s s poste current. eference node c. wo other nodes: nd. Sole for these. Applcton of KCL t node : S 0 Applcton of KCL t node : 0 Applcton of KCL t node : 0 (not ndependent!) S node node s s s node c c 0
3 4 Exmple contnued: Defne currents s f(,): Susttute these n the two nodl equtons: Sole for, n terms of s,,, : c c 0 0 S S c s s 0 0 he Node Voltge Method  4 he Node Voltge Method  5 he node oltge method s esy to pply when current sources re present ecuse they re drectly ccounted for y KCL. Howeer, the node oltge method cn lso e ppled when oltge sources re present, nd ths cse s ctully smpler. Exmple: One node oltge s known: s. Cn sole for nd c n terms of s, nd the four resstnces. c s s 4 0 c : 0 : 4 c S c c S
4 he Mesh Current Method  hs s n effcent nd systemtc method for nlysng crcuts ecuse meshes re esly dentfed n crcut. Bss: defne current n ech mesh s n ndependent rle. he current flowng through resstor n specfed drecton defnes the polrty of the oltge cross the resstor. Current, defned s flowng from left to rght, determnes the polrty of the oltge cross. o od confuson, defne poste mesh currents s clockwse. he sum of the oltges round closed crcut must equl zero y Krchoff s Voltge Lw. Apply KVL to ech mesh to otn set of n equtons, one for ech mesh. Brnch currents nd oltges cn e dered from mesh currents. he Mesh Current Method  Methodology: () Defne ech mesh current consstently. Generlly defne mesh currents clockwse, for conenence. () Apply KVL round ech mesh, expressng ech oltge n terms of one or more mesh currents. () Sole the resultng lner system of equtons, wth mesh currents s the ndependent rles. Once the drecton of current flow s selected, KVL requres: 0 Note: n rtrry drecton cn e ssumed for ny current n crcut s long s sgns re ppled consstently. If the nswer for the current s negte then the chosen reference drecton s just opposte to the drecton of ctul current flow. mesh 5
5 he Mesh Current Method  Exmple: womesh crcut wth two unknowns: nd. Consder ech mesh ndependently. Mesh Voltges nd round the mesh he een ssgned ccordng the clockwse drecton of mesh current. Note: mesh current s flowng through ( rnch current for ) ut s not the rnch current for. hs s . So (  ). Apply KVL for mesh : ( ) s 0 s 4 s 4 Consder mesh he Mesh Current Method  4 Exmple contnued: Mesh Voltges,, nd 4 round the mesh he een ssgned ccordng the clockwse drecton of mesh current. Note: mesh current s the rnch current for nd 4, ut not for. hs s . So (  ). hs s the opposte to mesh ecuse the mesh currents flow through n opposng drectons. Apply KVL for mesh : Comne the equtons for the two meshes to get: ( ) ( Sole for mesh currents nd. Dere other currents nd oltges. ) 4 S ( ) 4 0 s 0 4 Consder mesh 4 6
6 he Mesh Current Method  5 he mesh current method s ery effecte when ppled to crcuts tht contn only oltge sources. Howeer, t my lso e ppled to crcuts contnng oth oltge nd current sources  just e creful to dentfy the correct current n ech mesh. Exmple: Sole for unknown oltge x cross the current source. Presence of current source requres: S A KVL for mesh : 0 5 x 0 5 Ω Ω KVL for mesh : x 4 0 Sole to get: A s A x s 0 A 0 V 6 0 V x 4Ω Dependent or controlled sources Dependent Sources re sources whose current or oltge output s functon of some other oltge or current n crcut (unlke del sources whch re ndependent of ny other element n crcut) exmple: trnsstor mplfers s Source type oltgecontrolled oltge source (VCVS) currentcontrolled oltge source (CCVS) oltgecontrolled current source (VCCS) currentcontrolled current source (CCCS) eltonshp s A x s A x s A x I s A x s 7
7 Crcut Anlyss wth Dependent Sources  he node oltge nd mesh current methods cn lso e ppled to dependent sources, wth mnor modfcton. When dependent source s present n crcut, t cn e treted ntlly s n del source, wth the node or mesh equtons then wrtten down s preously descred. An ddtonl constrnt equton wll lso e needed, reltng the dependent source to one of the crcut oltges or currents. hs full set of equtons cn then soled. Note: once the constrnt equton hs een susttuted nto the ntl set of equtons, the numer of unknowns remns the sme. Crcut Anlyss wth Dependent Sources  Exmple: smplfed model of polr trnsstor mplfer wo oous nodes  pply node oltge nlyss. KCL t node : KCL t node : S β 0 S C Current cn e determned y consderng current dder: S S S Insert ths nto eqn to get two equtons tht cn e soled for nd. S s S node s node β c o 8
8 Prncple of Superposton  In lner crcut contnng N sources, ech rnch oltge nd current s the sum of N oltges nd currents, ech of whch my e computed y settng ll ut one source equl to zero nd solng the crcut contnng the sngle source. hs s conceptul d rther thn precse nlyss technque lke the mesh current nd node oltge methods. Useful n sulzng the ehour of crcut contnng multple sources. Apples to ny lner system. Whle t cn esly nd sometmes effectely e ppled to crcuts wth multple sources, other methods re often more effcent. Prncple of Superposton  Consder crcut wth two oltge sources connected n seres. B B B _ B B _ B (from zzon Fgure.6) Net current through sum of nddul source currents: B B B B B More formlly: B B hs crcut s equlent to the comnton of two crcuts, ech contnng sngle source. A short crcut s susttuted for the mssng source n ech sucrcut. A short crcut sees zero oltge cross tself, so ths s equlent to zerong the output of one of the oltge sources. B 9
9 Zerong Voltge nd Current Sources When pplyng the Prncple of Superposton oltge sources re zeroed y susttutng short crcuts current sources re zeroed y susttutng open crcuts (no current cn flow through n open crcut, so ths s equlent to zerong the output of the current source). In order to set oltge source equl to zero, we replce t wth short crcut. S _ S S A crcut he sme crcut wth S 0. In order to set current source equl to zero, we replce t wth n open crcut. S _ S S _ (from zzon Fgure.7) A crcut he sme crcut wth S 0 Equlent Crcuts Becuse Ohm s Lw nd Krchoff s Lws re lner, ny DC crcut cn e replced y smplfed equlent crcut. Applyng Ohm s Lw to comnton of resstors cn ge n equlent resstor. Applyng Krchoff s Lws to comnton of crcut elements cn ge n equlent crcut. It s useful to ew ech source or lod s twotermnl dece descred y n  cure. hs confgurton s clled oneport network. 0
10 Exmple of n Equlent Crcut S _ Source Lod (from zzon Fgure.9) EQ nd s EQ heenn s Equlent Crcut he heenn heorem As fr s lod s concerned, ny network composed of del oltge nd current sources, nd of lner resstors, my e represented y n equlent crcut consstng of n del oltge source,, n seres wth n equlent resstor,. Source Lod _ Lod (from zzon Fgure.)
11 he Norton heorem Norton s Equlent Crcut As fr s lod s concerned, ny network composed of del oltge nd current sources, nd of lner resstors, my e represented y n equlent crcut consstng of n del current source, N, n prllel wth n equlent resstor, N. Source Lod N N Lod (from zzon Fgure.) Determnng heenn & Norton Equlent esstnce he frst step n computng heenn or Norton equlent crcut s to fnd the equlent resstnce presented y the crcut t ts termnls. hs s done y settng ll sources n the crcut equl to zero nd clcultng the effecte resstnce etween the termnls. Voltge nd current sources n the crcut re set to zero usng the sme pproch s wth the Prncple of Superposton oltge sources re replced y short crcuts current sources re replced y open crcuts
12 Exmple of heenn esstnce  Exmple: Wht s the equlent resstnce tht lod L sees etween termnls nd? emoe lod resstnce from the crcut nd replce oltge source s y short crcut. Exmple of heenn esstnce  Equlent resstnce seen y the lod: nd re n prllel (connected etween sme two nodes) Let e totl resstnce etween termnls nd. hen (from zzon Fgure.4)
13 Exmple of heenn esstnce  Alternte method of determnng the heenn resstnce: Hypothetcl A current source s connected etween nd. Voltge x cross  wll then equl (ecuse s A). he source current encounters s resstor n seres wth the prllel comnton of &. Wht s the totl resstnce the current S wll encounter n flowng round the crcut? x S S S (from zzon Fgure.5) Clcultng Equlent esstnce Methodology for clcultng equlent resstnce of oneport network (heenn or Norton): () emoe the lod. () Zero ll ndependent oltge nd current sources. () Compute the totl resstnce etween lod termnls, wth the lod remoed. hs resstnce s equlent to tht whch would e encountered y current source connected to the crcut n plce of the lod. Note tht ths procedure ges result tht s ndependent of the lod. hs s wht we wnt, ecuse once the equlent resstnce hs een clculted for source crcut, the equlent crcut s unchnged f dfferent lod s connected. 4
14 Determnng the heenn Voltge  he equlent heenn source oltge,, s equl to the open crcut oltge present t the lod termnls wth lod remoed..e., n order to clcute, t s suffcent to remoe the lod nd to compute the open crcut oltge t the oneport termnls he open crcut oltge, OC, nd the heenn oltge,, must e the sme f the heenn heorem s true (see elow). hs s ecuse n the crcut contnng nd, the oltge OC must equl ecuse no current flows through nd so the oltge cross s zero. From KVL: (0) OC OC Oneport network OC Determnng the heenn Voltge  Methodology: () emoe the lod, leng the lod termnls opencrcuted. () Defne the opencrcut oltge OC cross the open lod termnls. () Apply ny preferred method (e.g., nodl nlyss) to sole for OC. (4) he heenn oltge s OC. 5
15 A Crcut nd Its heenn Equlent L L S _ L S _ L (from zzon Fgure.49) A crcut Its héenn equlent hs s the crcut we consdered erler, long wth ts heenn equlent. he two crcuts re equlent n tht the current drwn y the lod, L, s the sme n oth: L S ( ) L L Determnng the Norton Current  he Norton equlent current, N, s equl to the short crcut current tht would flow f the lod were replced y short crcut. Consder the oneport network nd ts Norton equlent crcut: Current SC flowng through the short crcut replcng the lod s the sme s the Norton current N ecuse ll of the source current n ths crcut must flow through the short crcut. Oneport network SC N N SC (from zzon Fgure.57) 6
16 Determnng the Norton Current  Let s fnd current SC n ths crcut. S _ Short crcut SC (from zzon Fgure.58) replcng the lod Mesh Current Method: Let nd SC e the mesh currents n the crcut. wo mesh equtons (sole for SC ): Node Voltge Method: Nodl equton (sole for ): hus: N S ( ) ( S SC ) SC S 0 Determnng the Norton Current  Methodology: () eplce the lod wth short crcut. () Defne the short crcut current SC to e the Norton equlent current. () Apply ny preferred method (e.g., nodl nlyss) to sole for SC. (4) he Norton current s N SC. 7
17 Source rnsformtons  Source trnsformtons cn e useful for determnng equlent crcuts sometmes llow replcement of current sources wth oltge sources nd ce ers. he heenn nd Norton heorems stte tht ny oneport network cn e represented y oltge source n seres wth resstor, or y current source n prllel wth resstor, nd tht ether of these representtons s equlent to the orgnl crcut. Oneport heenn Norton network _ N equlent equlent (from zzon Fgure.6) Source rnsformtons  Implcton: ny heenn equlent crcut cn e replced y Norton equlent crcut, f we use the reltonshp: N S _ SC S SC he sucrcut on the left of the dshed lne cn e replced y ts Norton equlent, s shown. Current SC cn e esly found ecuse the three resstors re n prllel wth the current source  use smple current dder. SC flows through, so: SC N S (from zzon Fgure.64) S 8
18 Source rnsformtons  Sucrcuts menle to source trnsformton: Node or _ S S or S S _ Node heenn é sucrcuts Norton sucrcuts (from zzon Fgure.65) Fndng heenn & Norton Equlents Expermentlly heenn nd Norton equlent crcuts cn e eluted expermentlly usng smple technques. Bsc de: heenn oltge s n opencrcut oltge Norton current s shortcrcut current herefore possle to mke mesurements to determne these qunttes. Once nd N re known, the heenn resstnce of the crcut cn e found usng Need to mesure nd N. N 9
19 Fndng heenn & Norton Equlents Expermentlly Mesurement of opencrcut oltge nd shortcrcut current for n rtrry network connected to ny lod: Unknown network An unknown network connected to lod Unknown network Network connected for mesurement of shortcrcut current Unknown network OC Network connected for mesurement of opencrcut oltge SC A V rm Lod Note: Do not short crcut network y nsertng n mmeter n seres  ths could dmge the crcut or the mmeter! rm (from zzon Fgure.7) Fndng heenn & Norton Equlents Expermentlly hese mesurements requre cre ecuse the mesurng nstruments re nondel. In the presence of fnte meter resstnce r m, ths quntty must e tken nto ccount when determnng the opencrcut oltge nd the shortcrcut current. Qunttes OC nd SC he quotton mrks to ndcte tht the mesured lues re ffected y r m nd re not the true lues. he true lues cn e clculted usng (proe ths to yourself!): N r m "SC" " OC" rm where N del Norton current, del heenn oltge, nd true heenn resstnce. 0
20 Fndng heenn & Norton Equlents Expermentlly 4 r m N "SC" " OC" rm ecll For n del mmeter, r m should pproch zero (short crcut). For n del oltmeter, r m should pproch nfnty (open crcut). So these two equtons cn e used to fnd the true heenn nd Norton equlent sources from n mperfect mesurement of the opencrcut oltge nd the shortcrcut current, proded tht the nternl meter resstnce r m s known. In prctce, the nternl resstnce of oltmeters s hgh enough to e consdered nfnte relte to the equlent resstnce of most crcuts. Howeer, t s mpossle to uld n mmeter wth zero nternl resstnce: need to know r m to determne the short crcut current. Mxmum Power rnsfer  he heenn nd Norton models mply tht some of the power generted y source wll e dsspted y the nternl crcuts wthn the source. Gen ths unodle power loss, how much power cn e trnsferred to the lod from the source under the most del condtons? Or, wht s the lod resstnce tht wll sor mxmum power from the source? Mxmum Power rnsfer heorem
21 Mxmum Power rnsfer  Power trnsfer etween source nd lod: Prctcl source s represented y ts heenn equlent crcut Gen nd, wht lue of L wll llow for mxmum power trnsfer? (from zzon Fgure.7) Prctcl source L L _ L Power sored y the lod: Lod current: L Comne to get lod power: L P P L L L Lod L ( ) L L Source equlent Mxmum Power rnsfer  Dfferentte P L w.r.t. L to fnd fnd the lue of L tht mxmzes the lod power (ssumng constnt nd ). ( ) ( ) dpl L L L 0 4 dl ( L ) Hence: ( L ) L ( L ) 0 For whch the soluton s: L o trnsfer the mxmum power to lod, the equlent source nd lod resstnces must e mtched (.e., equl). hus, n order to trnsfer mxmum power to lod, gen fxed equlent source resstnce, the lod resstnce must mtch ths equlent source resstnce.
22 Voltge Source Lodng Effects Voltge source lodng: When prctcl oltge source s connected to lod, the current tht flows from the source to the lod wll cuse oltge drop cross the nternl source resstnce, nt. As result, the oltge seen y the lod wll e lower thn the opencrcut (heenn) oltge of the source. Lod oltge s then: L nt hus wnt smll nternl resstnce n prctcl oltge source. nt _ L Source Lod (from zzon Fgure.74) Current Source Lodng Effects Current source lodng: When prctcl current source s connected to lod, the nternl source resstnce wll drw some current, nt, wy from the lod. As result, the lod wll recee only prt of the shortcrcut (Norton) current lle from the source. Lod current s then: L N hus wnt lrge nternl resstnce n prctcl current source. nt N L (from zzon Fgure.74) Source Lod
+ + +   This circuit than can be reduced to a planar circuit
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