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

Transcription

1 Crcut Analyss I (ENG 405) Chapter Method of Analyss Nodal(KCL) and Mesh(KVL)

2 Nodal Analyss If nstead of focusng on the oltages of the crcut elements, one looks at the oltages at the nodes of the crcut, the number of smultaneous equatons to sole for can be reduced. Gen a crcut wth n nodes, wthout oltage sources, the nodal analyss s accomplshed a three steps:. Select a node as the reference node. Assgn oltages,, n to the remanng n- nodes, oltages are relate to the reference node.. Apply KCL to each of the n- non-reference nodes. Use Ohm s law to express the branch currents n terms of node oltages. Sole the resultng n- smultaneous equatons to obtan the unknown node oltages. The reference, or datum, node s commonly referred to as the ground snce ts oltage s by default zero.

3 Applyng Nodal Analyss Let s apply nodal analyss to ths crcut to see how t works. Ths crcut has a node that s desgned as ground. We wll use that as the reference node (node 0) The remanng two nodes are desgned and and assgned oltages and. Now apply KCL to each node: At node At node I I I

4 Apply Nodal Analyss II We can now use OHM s law to express the unknown currents,, and n terms of node oltages. In dong so, keep n mnd that current flows from hgh potental to low From ths we get: The last step s to sole the system of equatons 4 or 0 or or 0 G G G Substtutng back nto the node equatons I I I G G I G G I I or

5 Includng oltage sources Dependng on what nodes the source s connected to, the approach ares Between the reference node and a non-reference mode: Set the oltage at the nonreference node to the oltage of the source In the example crcut =0V Between two non-reference nodes The two nodes form a supernode. 5

6 Supernode A supernode s formed by enclosng a oltage source (dependant or ndependent) connected between two nonreference nodes and any elements connected n parallel wth t. Why? Nodal analyss requres applyng KCL The current through the oltage source cannot be known n adance (Ohm s law does not apply) By lumpng the nodes together, the current balance can stll be descrbed In the example crcut node and form a supernode The current balance would be: Or ths can be expressed as:

7 Analyss wth a supernode In order to apply KVL to the supernode n the example, the crcut s redrawn as shown. Gong around ths loop n the clockwse drecton ges: Note the followng propertes of a supernode:. The oltage source nsde the supernode prodes a constrant equaton needed to sole for the node oltages. A supernode has no oltage of ts own. A supernode requres the applcaton of both KCL and KVL 7

8 Mesh Analyss Another general procedure for analyzng crcuts s to use the mesh currents as the crcut arables. ecall: A loop s a closed path wth no node passed more than once A mesh s a loop that does not contan any other loop wthn t Mesh analyss uses KVL to fnd unknown currents Mesh analyss s lmted n one aspect: It can only apply to crcuts that can be rendered planar. A planar crcut can be drawn such that there are no crossng branches. 8

9 Planar s Nonpalanar The fgure on the left s a nonplanar crcut: The branch wth the Ω resstor preents the crcut from beng drawn wthout crossng branches The fgure on the rght s a planar crcut: It can be redrawn to aod crossng branches 9

10 Mesh Analyss Steps Mesh analyss follows these steps:.assgn mesh currents,, n to the n meshes.apply KVL to each of the n mesh currents..sole the resultng n smultaneous equatons to get the mesh currents 0

11 Mesh Analyss Example The aboe crcut has two paths that are meshes (abefa and bcdeb) The outer loop (abcdefa) s a loop, but not a mesh Frst, mesh currents and are assgned to the two meshes. Applyng KVL to the meshes: 0 0 V V V V

12 Mesh Analyss wth Current Sources The presence of a current source makes the mesh analyss smpler n that t reduces the number of equatons. If the current source s located on only one mesh, the current for that mesh s defned by the source. For example: Here, the current s equal to -A

13 Supermesh Smlar to the case of nodal analyss where a oltage source shared two non-reference nodes, current sources (dependent or ndependent) that are shared by more than one mesh need specal treatment The two meshes must be joned together, resultng n a supermesh. The supermesh s constructed by mergng the two meshes and excludng the shared source and any elements n seres wth t A supermesh s requred because mesh analyss uses KVL But the oltage across a current source cannot be known n adance. Intersectng supermeshes n a crcut must be combned to for a larger supermesh.

14 Creatng a Supermesh In ths example, a 6A current course s shared between mesh and. The supermesh s formed by mergng the two meshes. The current source and the Ω resstor n seres wth t are remoed. 4

15 Supermesh Example Usng the crcut from the last slde: Apply KVL to the supermesh or We next apply KCL to the node n the branch where the two meshes ntersect. 6 Solng these two equatons we get:.a.8a Note that the supermesh requred usng both KVL and KCL 5

16 Selectng an Approprate Approach In prncple both the nodal analyss and mesh analyss are useful for any gen crcut. What then determnes f one s gong to be more effcent for solng a crcut problem? There are two factors that dctate the best choce: The nature of the partcular network s the frst factor The second factor s the nformaton requred 6

17 Mesh analyss when If the network contans: Many seres connected elements Voltage sources Supermeshes A crcut wth fewer meshes than nodes If branch or mesh currents are what s beng soled for. Mesh analyss s the only sutable analyss for transstor crcuts It s not approprate for operatonal amplfers because there s no drect way to obtan the oltage across an op-amp. 7

18 Nodal analyss f If the network contans: Many parallel connected elements Current sources Supernodes Crcuts wth fewer nodes than meshes If node oltages are what are beng soled for Non-planar crcuts can only be soled usng nodal analyss Ths format s easer to sole by computer 8

19 Applcaton: DC transstor crcut Here we wll use the approaches learned n ths chapter to analyze a transstor crcut In general, there are two types of transstors commonly used: Feld Effect (FET) and Bpolar Juncton (BJT). Ths problem wll use a BJT. A BJT s a three termnal dece, where the nput current nto one termnal (the base) affects the current flowng out of a second termnal (the collector). The thrd termnal (the emtter) s the common termnal for both currents 9

20 KCL and KVL for a BJT The currents from each termnal can be related to each other as follows: I E I B I C The base and collector current can be related to each other by the parameter, whch can range from I Applyng KVL to the BJT ges: V C I B CE V EB V BC 0 0

21 DC model of a BJT A transstor has a few operatng modes dependng on the appled oltages/currents. In ths problem, we wll be nterested n the operaton n acte mode. Ths s the mode used for amplfyng sgnals. The fgure below shows the equalent DC model for a BJT n acte mode Note that nodal analyss can only be appled to the BJT after usng ths model

22 Settng up a BJT crcut Below are three approaches to solng a transstor crcut. Note when the equalent model s used and when t s not. Orgnal crcut Mesh analyss Nodal analyss PSpce analyss