CIRCUITS, NODES, AND BRANCHES

Transcription

1 Background: CIRCUITS, NODES, AND BRANCHES We know the propertes of a few deal crcut elements (voltage source, current source, resstor) Ideal wres hook these elements up wth no loss of voltage or loss of power (wres are knd of lke resstors n the lmt R approaches 0) We saw ntutvely that conservaton of charge leads to a conservaton of current when wres ntersect, and conservaton of energy means we can add voltages as we move around n a crcut. Now we wll formalze the latter observatons Today: We begn wth elementary crcut analyss - methods for rgorous approach to determnng crcut response Crcut theorems for analyss of branch currents and voltages Krchhoff s current law and voltage law

2 BRANCHES AND NODES Crcut wth several branches connected at a node: branch (crcut element) KIRCHOFF s CURRENT LAW KCL : (see Text 1.2 and 1.3) (Sum of currents enterng node) (Sum of currents leavng node) = 0 q = charge stored at node s zero. If charge s stored, for example n a capactor, then the capactor s a branch and the charge s stored there NOT at the node.

3 WHAT IF THE NET CURRENT WERE NOT ZERO? Suppose mbalance n currents s 1µA = 1 µc/s (net current enterng node) Assumng that q = 0 at t = 0, the charge ncrease s 10 6 C each second or /1.6 = charge carrers each second But by defnton, the capactance of a node to ground s ZERO because we show any capactance as an explct crcut element (branch). Thus, the voltage would be nfnte (Q = CV). Somethng has to gve! In the lmt of zero capactance the accumulaton of charge would result n nfnte electrc felds there would be a spark as the ar around the node broke down. Charge s transported around the crcut branches (even stored n some branches), but t doesn t ple up at the nodes!

4 SIGN CONVENTIONS FOR SUMMING CURRENTS Krchhoff s Current Law (KCL) Sum of currents enterng node = sum of currents leavng node Use reference drectons to determne enterng and leavng currents no concern about actual polartes KCL yelds one equaton per node Alternatve statements of KCL 1 Algebrac sum of currents enterng node = 0 where algebrac sum means currents leavng are ncluded wth a mnus sgn 2 Algebrac sum of currents leavng node = 0 where currents enterng are ncluded wth a mnus sgn

5 KIRCHHOFF S CURRENT LAW EXAMPLE 24 µa -4 µa 10 µa Currents enterng the node: 24 µa Currents leavng the node: 4 µa 10 µa } 24 = 10 ( 4) = 18 µa Three statements of KCL IN n n ALL = out ALL = 0 = 0 out OUT 24 = ( 4) 10 = = 0 0 = 18 µ A = 18 µ A = 18 µ A EQUIVALENT

6 GENERALIZATION OF KCL Sum of currents enterng and leavng a closed surface s zero Physcs 7B Could be a bg chunk of crcut n here, e.g., could be a Black Box Note that crcut branches could be nsde the surface. The surface can enclose more than one node

7 KIRCHHOFF S CURRENT LAW USING SURFACES Example 5 µa surface 5 µ A 2 µ A = enterng leavng Another example 50 ma 2 µa =? =7µA? must be 50 ma

8 Example of the use of KCL At node X: R1 X Current nto X from the left: (V 1 - v X )/R1 V 1 - R2 Current out of X to the rght: v X /R2 KCL: (V 1 - v X )/R1 = v X /R2 Gven V 1, Ths equaton can be solved for v X Of course we just get the same result as we obtaned from our seres resstor v X = V 1 R2 /(R1 R2) formulaton. (Fnd the current and multply by R2)

9 BRANCH AND NODE VOLTAGES The voltage across a crcut element s defned as the dfference between the node voltages at ts termnals b v 1 a c v 2 d v 2 = vd va v e 0 (snce t s the reference) e select as ref. ground Specfyng node voltages: Use one node as the mplct reference (the common node attach specal symbol to label t) Now sngle subscrpts can label voltages: e.g., v b means v b v e, v a means v a v e, etc.

10 KIRCHHOFF S VOLTAGE LAW (KVL) The algebrac sum of the voltage drops around any closed loop s zero. Why? We must return to the same potental (conservaton of energy). Voltage drop defned as the branch voltage f the sgn s encountered frst; t s (-) the branch voltage f the sgn s encountered frst mportant bookkeepng Path V 1 - drop Path - V 2 rse or step up (negatve drop) Closed loop: Path begnnng and endng on the same node

11 Examples of Three closed paths: 1, 2, 3 Note that: v 2 = v a - v b v 3 = v c - v b KVL EXAMPLE v a v a v 2 v b v 3 v c 3 1 ref. node v b - 2 v c Path 1: va v2 vb = 0 Path 2: vb v3 vc = 0 Path 3: va v2 v3 vc = 0 v a v b YEP!

12 ALTERNATIVE STATEMENTS OF KIRCHHOFF S VOLTAGE LAW 1 For any node sequence A, B, C, D,, M around a closed path, the voltage drop from A to M s gven by v = v v v L AM AB BC CD v LM 2 For all pars of nodes and j, the voltage drop from to j s v j = v where the node voltages are measured wth respect to the common node. v j

13 FORMAL CIRCUIT ANALYSIS Systematc approaches to wrtng down KCL and KVL: Text 2.3 Nodal Analyss: Node voltages are the unknowns Text secton 2.3 Mesh Analyss: Branch currents are the unknowns Use one or the other for crcut analyss We wll do only nodal analyss (because voltages make more convenent varables than currents) So gnore Text 2.4