ECEN 301Discussion #7 – Node and Mesh Methods1 DateDayClass No. TitleChaptersHW Due date Lab Due date Exam 24 SeptWed7Network Analysis3.1 – SeptThu LAB 2 26 SeptFri Recitation HW 3 27 SeptSat 28 SeptSun 29 SeptMon8Network Analysis3.4 – 3.5 NO LAB 30 SeptTue 1 OctWed9Equivalent Circuits3.6 Schedule…

ECEN 301Discussion #7 – Node and Mesh Methods2 Learning is good only when humble 2 Nephi 9: O that cunning plan of the evil one! O the vainness, and the frailties, and the foolishness of men! When they are learned they think they are wise, and they hearken not unto the counsel of God, for they set it aside, supposing they know of themselves, wherefore, their wisdom is foolishness and it profiteth them not. And they shall perish. 29 But to be learned is good if they hearken unto the counsels of God.

ECEN 301Discussion #7 – Node and Mesh Methods3 Lecture 7 – Network Analysis Node Voltage and Mesh Current Methods

ECEN 301Discussion #7 – Node and Mesh Methods4 Network Analysis uDetermining the unknown branch currents and node voltages ÙImportant to clearly define all relevant variables ÙConstruct concise set of equations There are methods to follow in order to create these equations This is the subject of the next few lectures

ECEN 301Discussion #7 – Node and Mesh Methods5 Network Analysis uNetwork Analysis Methods: ÜNode voltage method ÜMesh current method ÙSuperposition ÙEquivalent circuits Source transformation Thévenin equivalent Norton equivalent

ECEN 301Discussion #7 – Node and Mesh Methods6 Node Voltage Method Network Analysis

ECEN 301Discussion #7 – Node and Mesh Methods7 Node Voltage Method uIdentify all node and branch voltages iaia +v2–+v2– vsvs + _ +v4–+v4– + v 1 – + v 3 – R2R2 R1R1 R3R3 R4R4 a b c d icic ibib Node VoltagesBranch Voltages v a = v s v s = v a - v d = v a v b = v 2 v 1 = v a - v b v c = v 4 v 2 = v b - v d = v b v d = 0v 3 = v b - v c v 4 = v c - v d = v c

ECEN 301Discussion #7 – Node and Mesh Methods8 Node Voltage Method uThe most general method for electrical circuit analysis ÙBased on defining the voltage at each node ÙOne node is selected as a reference node (often ground) All other voltages given relative to reference node n – 1 equations of n – 1 independent variables (node voltages) ÙOnce node voltages are determined, Ohm’s law can determine branch currents Branch currents are expressed in terms of one or more node voltages R i vava vbvb +R2–+R2– + R 1 – + R 3 – vava vdvd vbvb vcvc i1i1 i2i2 i3i3

ECEN 301Discussion #7 – Node and Mesh Methods9 Node Voltage Method 1.Label all currents and voltages (choose arbitrary orientations unless orientations are already given) 2.Select a reference node (usually ground) ÙThis node usually has most elements tied to it 3.Define the remaining n – 1 node voltages as independent or dependent variables  Each of the m voltage sources is associated with a dependent variable  If a node is not connected to a voltage source then its voltage is treated as an independent variable 4.Apply KCL at each node labeled as an independent variable  Express currents in terms of node voltages 5.Solve the linear system of n – 1 – m unknowns

ECEN 301Discussion #7 – Node and Mesh Methods10 Node Voltage Method uExample1: find expressions for each of the node voltages and the currents + R 2 – isis +R1–+R1– +R3–+R3–

ECEN 301Discussion #7 – Node and Mesh Methods11 Node Voltage Method uExample1: find expressions for each of the node voltages and the currents isis i1i1 i3i3 +R1–+R1– +R3–+R3– + R 2 – i2i2 Node a Node b Node c vava vbvb vcvc 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node c (v c ) as the reference node (v c = 0) 3.Define remaining n – 1 (2) voltages v a is independent since it is not associated with a voltage source v b is independent 4.Apply KCL at nodes a and b (node c is not independent) to find expressions for i 1, i 2, i 3

ECEN 301Discussion #7 – Node and Mesh Methods12 Node Voltage Method uExample1: find expressions for each of the node voltages and the currents isis i1i1 i3i3 +R1–+R1– +R3–+R3– + R 2 – i2i2 Node a Node b Node c vava vbvb vcvc 4.Apply KCL to find expressions for i 1, i 2, i 3 NB: whenever a node connects only 2 branches the same current flows in the 2 branches (EX: i 2 = i 3 )

ECEN 301Discussion #7 – Node and Mesh Methods13 Node Voltage Method uExample1: find expressions for each of the node voltages and the currents isis i1i1 i3i3 +R1–+R1– +R3–+R3– + R 2 – i2i2 Node a Node b Node c vava vbvb vcvc 4.Express currents in terms of node voltages

ECEN 301Discussion #7 – Node and Mesh Methods14 Node Voltage Method uExample1: find expressions for each of the node voltages and the currents isis i1i1 i3i3 +R1–+R1– +R3–+R3– + R 2 – i2i2 Node a Node b Node c vava vbvb vcvc 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods15 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ I1I1 R1R1 R4R4 R 2 R 3 I2I2

ECEN 301Discussion #7 – Node and Mesh Methods16 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ I1I1 R1R1 R4R4 R 2 R 3 I2I2 +I1–+I1– –I2+–I2+ +R4–+R4– +R1–+R1– + R 2 – + R 3 – i3i3 i1i1 i2i2 i4i4 Node a Node b Node c

ECEN 301Discussion #7 – Node and Mesh Methods17 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node c (v c ) as the reference node (v c = 0) 3.Define remaining n – 1 (2) voltages  v a is independent  v b is independent 4.Apply KCL at nodes a and b +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods18 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c 4.Apply KCL at nodes a and b vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods19 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c 4.Express currents in terms of voltages vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods20 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c 5.Solve the n – 1 – m equations vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods21 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c 5.Solve the n – 1 – m equations vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods22 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node a Node b Node c 5.Solve the n – 1 – m equations vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods23 Node Voltage Method uExample2: solve for all unknown currents and voltages ÙI 1 = 10mA, I 2 = 50mA, R 1 = 1kΩ, R 2 = 2kΩ, R 3 = 10kΩ, R 4 = 2kΩ Node a 5.Solve the n – 1 – m equations +I1–+I1– +R1–+R1– +R4–+R4– + R 2 – + R 3 – –I2+–I2+ i3i3 i2i2 i1i1 i4i4 Node b Node c vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods24 Node Voltage Method uExample3: solve for all unknown voltages Ùi a = 1mA, i b = 2mA, R 1 = 1kΩ, R 2 = 500Ω, R 3 = 2.2kΩ, R 4 = 4.7kΩ iaia R1R1 R3R3 R 2 ibib R4R4

ECEN 301Discussion #7 – Node and Mesh Methods25 Node Voltage Method uExample3: solve for all unknown voltages Ùi a = 1mA, i b = 2mA, R 1 = 1kΩ, R 2 = 500Ω, R 3 = 2.2kΩ, R 4 = 4.7kΩ +ia–+ia– 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node c (v c ) as the reference node (v c = 0) 3.Define remaining n – 1 (2) voltages  v a is independent  v b is independent 4.Apply KCL at nodes a and b +R1–+R1– +R3–+R3– + R 2 – +ib–+ib– i2i2 i1i1 i3i3 +R4–+R4– i4i4 Node a Node b Node c vava vbvb vcvc

ECEN 301Discussion #7 – Node and Mesh Methods26 Node Voltage Method uExample3: solve for all unknown voltages Ùi a = 1mA, i b = 2mA, R 1 = 1kΩ, R 2 = 500Ω, R 3 = 2.2kΩ, R 4 = 4.7kΩ +ia–+ia– +R1–+R1– +R3–+R3– + R 2 – +ib–+ib– i2i2 i1i1 i3i3 +R4–+R4– i4i4 Node a Node b Node c vava vbvb vcvc 4.Apply KCL at nodes a and b

ECEN 301Discussion #7 – Node and Mesh Methods27 Node Voltage Method uExample3: solve for all unknown voltages Ùi a = 1mA, i b = 2mA, R 1 = 1kΩ, R 2 = 500Ω, R 3 = 2.2kΩ, R 4 = 4.7kΩ +ia–+ia– +R1–+R1– +R3–+R3– + R 2 – +ib–+ib– i2i2 i1i1 i3i3 +R4–+R4– i4i4 Node a Node b Node c vava vbvb vcvc 4.Express currents in terms of voltages

ECEN 301Discussion #7 – Node and Mesh Methods28 Node Voltage Method uExample3: solve for all unknown voltages Ùi a = 1mA, i b = 2mA, R 1 = 1kΩ, R 2 = 500Ω, R 3 = 2.2kΩ, R 4 = 4.7kΩ +ia–+ia– +R1–+R1– +R3–+R3– + R 2 – +ib–+ib– i2i2 i1i1 i3i3 +R4–+R4– i4i4 Node a Node b Node c vava vbvb vcvc 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods29 Node Voltage Method uExample4: find v 2 ÙI 1 = 2A, I 2 = 3A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω I1I1 R4R4 +R2–+R2– R1R1 R3R3 I2I2

ECEN 301Discussion #7 – Node and Mesh Methods30 Node Voltage Method uExample4: find v 2 ÙI 1 = 2A, I 2 = 3A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +I1–+I1– +R4–+R4– +R2–+R2– + R 1 – + R 3 – I2I2 i3i3 i1i1 i2i2 i4i4 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node d (v d ) as the reference node (v d = 0) 3.Define remaining n – 1 (3) voltages  v a is independent  v b is independent  v c is independent 4.Apply KCL at nodes a, b, and c vava vbvb vcvc vdvd

ECEN 301Discussion #7 – Node and Mesh Methods31 Node Voltage Method uExample4: find v 2 ÙI 1 = 2A, I 2 = 3A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +I1–+I1– +R4–+R4– +R2–+R2– + R 1 – + R 3 – I2I2 i3i3 i1i1 i2i2 i4i4 vava vbvb vcvc vdvd 4.Apply KCL at nodes a, b, and c

ECEN 301Discussion #7 – Node and Mesh Methods32 Node Voltage Method +I1–+I1– +R4–+R4– +R2–+R2– + R 1 – + R 3 – I2I2 i3i3 i1i1 i2i2 i4i4 vava vbvb vcvc vdvd 4.Express currents in terms of voltages uExample4: find v2 ÙI 1 = 2A, I 2 = 3A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω

ECEN 301Discussion #7 – Node and Mesh Methods33 Node Voltage Method +I1–+I1– +R4–+R4– +R2–+R2– + R 1 – + R 3 – I2I2 i3i3 i1i1 i2i2 i4i4 vava vbvb vcvc vdvd uExample4: find v2 ÙI 1 = 2A, I 2 = 3A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods34 Node Voltage Method uExample5: find all node voltages Ùv s = 2V, i s = 3A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω R2R2 R4R4 R 3 isis R 1 +–+– vsvs

ECEN 301Discussion #7 – Node and Mesh Methods35 Node Voltage Method uExample5: find all node voltages Ùv s = 2V, i s = 3A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +R2–+R2– +R4–+R4– + R 3 – +is–+is– i3i3 i2i2 i4i4 vava vcvc vdvd + R 1 – +–+– i1i1 vsvs vbvb 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node d (v d ) as the reference node (v d = 0) 3.Define remaining n – 1 (3) voltages  v a is dependent (v a = v s )  v b is independent  v c is independent 4.Apply KCL at nodes b, and c

ECEN 301Discussion #7 – Node and Mesh Methods36 Node Voltage Method uExample5: find all node voltages Ùv s = 2V, i s = 3A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +R2–+R2– +R4–+R4– + R 3 – +is–+is– i3i3 i2i2 i4i4 vava vcvc vdvd + R 1 – +–+– i1i1 vsvs vbvb 4.Apply KCL at nodes b, and c

ECEN 301Discussion #7 – Node and Mesh Methods37 Node Voltage Method uExample5: find all node voltages Ùv s = 2V, i s = 3A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +R2–+R2– +R4–+R4– + R 3 – +is–+is– i3i3 i2i2 i4i4 vava vcvc vdvd + R 1 – +–+– i1i1 vsvs vbvb 4.Express currents in terms of voltages

ECEN 301Discussion #7 – Node and Mesh Methods38 Node Voltage Method uExample5: find all node voltages Ùv s = 2V, i s = 3A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +R2–+R2– +R4–+R4– + R 3 – +is–+is– i3i3 i2i2 i4i4 vava vcvc vdvd + R 1 – +–+– i1i1 vsvs vbvb 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods39 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω IsIs R4R4 R2R2 R 1 R 3 VsVs + – iviv

ECEN 301Discussion #7 – Node and Mesh Methods40 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +Is–+Is– +R4–+R4– +R2–+R2– + R 1 – + R 3 – VsVs i3i3 i1i1 i2i2 i4i4 + – iviv vava vbvb vcvc vdvd 1.Label currents and voltages (polarities “arbitrarily” chosen) 2.Choose Node d (v d ) as the reference node (v d = 0) 3.Define remaining n – 1 (3) voltages  v a is independent  v b is dependent (actually both v b and v c are dependent on each other so choose one to be dependent and one to be independent) (v b = v c + V s )  v c is independent 4.Apply KCL at nodes a, and c

ECEN 301Discussion #7 – Node and Mesh Methods41 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +Is–+Is– +R4–+R4– +R2–+R2– + R 1 – + R 3 – VsVs i3i3 i1i1 i2i2 i4i4 + – iviv vava vbvb vcvc vdvd 4.Apply KCL at nodes a, and c

ECEN 301Discussion #7 – Node and Mesh Methods42 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +Is–+Is– +R4–+R4– +R2–+R2– + R 1 – + R 3 – VsVs i3i3 i1i1 i2i2 i4i4 + – iviv vava vbvb vcvc vdvd 4.Express currents in terms of voltages

ECEN 301Discussion #7 – Node and Mesh Methods43 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +Is–+Is– +R4–+R4– +R2–+R2– + R 1 – + R 3 – VsVs i3i3 i1i1 i2i2 i4i4 + – iviv vava vbvb vcvc vdvd 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods44 Node Voltage Method uExample6: find the current i v ÙV s = 3V, I s = 2A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω +Is–+Is– +R4–+R4– +R2–+R2– + R 1 – + R 3 – VsVs i3i3 i1i1 i2i2 i4i4 + – iviv vava vbvb vcvc vdvd 5.Solve the n – 1 – m equations

ECEN 301Discussion #7 – Node and Mesh Methods45 Mesh Current Method Network Analysis

ECEN 301Discussion #7 – Node and Mesh Methods46 Mesh Current Method uWrite n equations of n unknowns in terms of mesh currents (where n is the number of meshes) ÙUse of KVL to solve unknown currents ÙImportant to be consistent with current direction +v_+v_ R i NB: the direction of current defines the polarity of the voltage Positive voltage +v_+v_ R i Negative voltage

ECEN 301Discussion #7 – Node and Mesh Methods47 Mesh Current Method uWrite n equations of n unknowns in terms of mesh currents (where n is the number of meshes) ÙUse of KVL to solve unknown currents ÙImportant to be consistent with current direction iaia +v2–+v2– vsvs + _ +v4–+v4– + v 1 – + v 3 – R2R2 R1R1 R3R3 R4R4 ibib Two meshes n = 2

ECEN 301Discussion #7 – Node and Mesh Methods48 Mesh Current Method uWrite n equations of n unknowns in terms of mesh currents (where n is the number of meshes) ÙUse of KVL to solve unknown currents ÙImportant to be consistent with current direction iaia +v2–+v2– vsvs + _ +v4–+v4– + v 1 – + v 3 – R2R2 R1R1 R3R3 R4R4 ibib Two meshes n = 2 i 1 (current through R 1 ) = i a but what about i 2 ? According to the current directions of i a and i b, and the polarity of v 2 : i 2 (current through R 2 ) = i a – i b

ECEN 301Discussion #7 – Node and Mesh Methods49 Mesh Current Method uWrite n equations of n unknowns in terms of mesh currents (where n is the number of meshes) ÙUse of KVL to solve unknown currents ÙImportant to be consistent with current direction iaia +v2–+v2– vsvs + _ +v4–+v4– + v 1 – + v 3 – R2R2 R1R1 R3R3 R4R4 ibib

ECEN 301Discussion #7 – Node and Mesh Methods50 Mesh Current Method uWrite n equations of n unknowns in terms of mesh currents (where n is the number of meshes) ÙUse of KVL to solve unknown currents ÙImportant to be consistent with current direction iaia +v2–+v2– vsvs + _ +v4–+v4– + v 1 – + v 3 – R2R2 R1R1 R3R3 R4R4 ibib

ECEN 301Discussion #7 – Node and Mesh Methods51 Mesh Current Method 1.Label each mesh current consistently  Current directions are chosen arbitrarily unless given 2.Label the voltage polarity of each circuit element  Strategically (based on current direction) choose polarity unless already given 3.In a circuit with n meshes and m current sources n – m independent equations result  Each of the m current sources is associated with a dependent variable  If a mesh is not connected to a current source then its voltage is treated as an independent variable 4.Apply KVL at each mesh associated with independent variables  Express voltages in terms of mesh currents 5.Solve the linear system of n – m unknowns

ECEN 301Discussion #7 – Node and Mesh Methods52 Mesh Current Method uExample7: find the values of the mesh currents ÙV s1 = 12V, V s2 = 6V, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω R2R2 R 1 R 4 R 3 v s1 + _ v s2 –+–+ iaia ibib icic

ECEN 301Discussion #7 – Node and Mesh Methods53 Mesh Current Method uExample7: find the values of the mesh currents ÙV s1 = 12V, V s2 = 6V, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – v s1 + _ v s2 –+–+ iaia ibib icic 1.Mesh current directions given 2.Voltage polarities chosen and labeled 3.Identify n – m (3) mesh currents  i a is independent  i c is independent 4.Apply KVL around meshes a, b, and c

ECEN 301Discussion #7 – Node and Mesh Methods54 Mesh Current Method uExample7: find the values of the mesh currents ÙV s1 = 12V, V s2 = 6V, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – v s1 + _ v s2 –+–+ iaia ibib icic 4.Apply KVL at nodes a, b, and c

ECEN 301Discussion #7 – Node and Mesh Methods55 Mesh Current Method uExample7: find the values of the mesh currents ÙV s1 = 12V, V s2 = 6V, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – v s1 + _ v s2 –+–+ iaia ibib icic 4.Express voltages in terms of currents

ECEN 301Discussion #7 – Node and Mesh Methods56 Mesh Current Method uExample7: find the values of the mesh currents ÙV s1 = 12V, V s2 = 6V, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – v s1 + _ v s2 –+–+ iaia ibib icic 5.Solve the n – m equations

ECEN 301Discussion #7 – Node and Mesh Methods57 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω R2R2 R 4 v s2 + _ R1R1 R3R3 v s1 + _ R 5

ECEN 301Discussion #7 – Node and Mesh Methods58 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω 1.Mesh current directions given 2.Voltage polarities chosen and labeled 3.Identify n – m (3) mesh currents  i a is independent  i c is independent 4.Apply KVL around meshes a, b, and c +R2–+R2– + R 4 – v s2 + _ iaia ibib icic +R1–+R1– +R3–+R3– v s1 + _ – R 5 +

ECEN 301Discussion #7 – Node and Mesh Methods59 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω +R2–+R2– + R 4 – v s2 + _ iaia ibib icic +R1–+R1– +R3–+R3– v s1 + _ – R Apply KVL at nodes a, b, and c

ECEN 301Discussion #7 – Node and Mesh Methods60 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω +R2–+R2– + R 4 – v s2 + _ iaia ibib icic +R1–+R1– +R3–+R3– v s1 + _ – R Express voltages in terms of currents

ECEN 301Discussion #7 – Node and Mesh Methods61 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω +R2–+R2– + R 4 – v s2 + _ iaia ibib icic +R1–+R1– +R3–+R3– v s1 + _ – R Solve the n – m equations

ECEN 301Discussion #7 – Node and Mesh Methods62 Mesh Current Method uExample8: find the voltages across the resistors ÙV s1 = V s2 = 110V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1.3Ω +R2–+R2– + R 4 – v s2 + _ iaia ibib icic +R1–+R1– +R3–+R3– v s1 + _ – R Solve the n – m equations

ECEN 301Discussion #7 – Node and Mesh Methods63 Mesh Current Method uExample9: find the mesh currents ÙV s = 6V, I s = 0.5A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω R2R2 R 1 R 4 R 3 IsIs vsvs –+–+ iaia ibib icic

ECEN 301Discussion #7 – Node and Mesh Methods64 Mesh Current Method uExample9: find the mesh currents ÙV s = 6V, I s = 0.5A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – IsIs vsvs –+–+ iaia ibib icic 1.Mesh current directions given 2.Voltage polarities chosen and labeled 3.Identify n – m (3) mesh currents  i a is dependent (i a = i s )  i a is independent  i c is independent 4.Apply KVL around meshes b and c

ECEN 301Discussion #7 – Node and Mesh Methods65 Mesh Current Method uExample9: find the mesh currents ÙV s = 6V, I s = 0.5A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – IsIs vsvs –+–+ iaia ibib icic 4.Apply KVL at nodes b and c

ECEN 301Discussion #7 – Node and Mesh Methods66 Mesh Current Method uExample9: find the mesh currents ÙV s = 6V, I s = 0.5A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – IsIs vsvs –+–+ iaia ibib icic 4.Express voltages in terms of currents

ECEN 301Discussion #7 – Node and Mesh Methods67 Mesh Current Method uExample9: find the mesh currents ÙV s = 6V, I s = 0.5A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R2–+R2– + R 1 – +R 4 – +R 3 – IsIs vsvs –+–+ iaia ibib icic 5.Solve the n – m equations

ECEN 301Discussion #7 – Node and Mesh Methods68 Equation Solver Methods Calculator Matrix Cramer’s Rule Brute Force (Substitution)

ECEN 301Discussion #7 – Node and Mesh Methods69 TI-89 Equation Solver 1.Press F1  Select option number 8 ( Clear Home ) 2.Press APPS  Select option number 1 ( FlashApps )  Select Simultaneous Eqn Solver  Select New… 3.In the new box enter n (the number of equations and unknowns)

ECEN 301Discussion #7 – Node and Mesh Methods70 TI-89 Equation Solver 4.Input the equation coefficients uFor the set of 3 equations and 3 unknowns from example 8: Input the coefficients as follows: a1a2a3b1

ECEN 301Discussion #7 – Node and Mesh Methods71 TI-89 Equation Solver 5.Press F5 ( Solve )