Schedule Date Day 24 Sept Wed 25 Sept

Schedule… Date Day 24 Sept Wed 25 Sept Thu 26 Sept Fri 27 Sept Sat 28 Sept Sun 29 Sept Mon Class No. 7 Title Network Analysis Chapters HW Due date Lab Due date Exam 3. 1 – 3. 3 LAB 2 Recitation 8 Network Analysis HW 3 3. 4 – 3. 5 NO LAB 30 Sept Tue 1 Oct Wed ECEN 301 9 Equivalent Circuits 3. 6 Discussion #7 – Node and Mesh Methods 1

Learning is good only when humble 2 Nephi 9: 28 -29 28 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 301 Discussion #7 – Node and Mesh Methods 2

Lecture 7 – Network Analysis Node Voltage and Mesh Current Methods ECEN 301 Discussion #7 – Node and Mesh Methods 3

Network Analysis u. Determining 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 301 Discussion #7 – Node and Mesh Methods 4

Network Analysis u. Network Analysis Methods: ÜNode voltage method ÜMesh current method ÙSuperposition ÙEquivalent circuits • Source transformation • Thévenin equivalent • Norton equivalent ECEN 301 Discussion #7 – Node and Mesh Methods 5

Node Voltage Method Network Analysis ECEN 301 Discussion #7 – Node and Mesh Methods 6

Node Voltage Method u. Identify all node and branch voltages a + v 1 – + v 3 – b R 1 vs + _ ia R 3 + v 2 – R 2 d ECEN 301 Node Voltages Branch Voltages c ic ib + v 4 – R 4 va = vs vs = va - vd = va vb = v 2 v 1 = va - vb vc = v 4 v 2 = vb - vd = vb vd = 0 v 3 = vb - vc v 4 = vc - vd = vc Discussion #7 – Node and Mesh Methods 7

Node Voltage Method u The most general method for electrical circuit analysis Ù Based on defining the voltage at each node Ù One 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) Ù Once node voltages are determined, Ohm’s law can determine branch currents • Branch currents are expressed in terms of one or more node voltages + R 3 – + R 1 – vb i va vd va vb i 3 + i 1 R R 2 i 2 – vc ECEN 301 Discussion #7 – Node and Mesh Methods 8

Node Voltage Method 1. 2. Label all currents and voltages (choose arbitrary orientations unless orientations are already given) Select a reference node (usually ground) Ù 3. Define the remaining n – 1 node voltages as independent or dependent variables Ù Ù 4. 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 Apply KCL at each node labeled as an independent variable Ù 5. This node usually has most elements tied to it Express currents in terms of node voltages Solve the linear system of n – 1 – m unknowns ECEN 301 Discussion #7 – Node and Mesh Methods 9

Node Voltage Method u. Example 1: find expressions for each of the node voltages and the currents + R 2 – + R 1 – is ECEN 301 + R 3 – Discussion #7 – Node and Mesh Methods 10

Node Voltage Method u. Example 1: find expressions for each of the node voltages and the currents 1. va + R – 2 Node a i 1 + R 1 – is Node b vb i 2 i 3 + R 3 – 2. 3. 4. vc ECEN 301 Node c Label currents and voltages (polarities “arbitrarily” chosen) Choose Node c (vc) as the reference node (vc = 0) Define remaining n – 1 (2) voltages • va is independent since it is not associated with a voltage source • vb is independent Apply KCL at nodes a and b (node c is not independent) to find expressions for i 1, i 2, i 3 Discussion #7 – Node and Mesh Methods 11

Node Voltage Method u. Example 1: find expressions for each of the node voltages and the currents 4. va + R – 2 Node a i 1 + R 1 – is vc ECEN 301 Apply KCL to find expressions for i 1, i 2, i 3 Node b vb i 2 i 3 + R 3 – Node c NB: whenever a node connects only 2 branches the same current flows in the 2 branches (EX: i 2 = i 3) Discussion #7 – Node and Mesh Methods 12

Node Voltage Method u. Example 1: find expressions for each of the node voltages and the currents 4. va + R – 2 Node a i 1 + R 1 – is vc ECEN 301 Express currents in terms of node voltages Node b vb i 2 i 3 + R 3 – Node c Discussion #7 – Node and Mesh Methods 13

Node Voltage Method u. Example 1: find expressions for each of the node voltages and the currents 5. va + R – 2 Node a i 1 + R 1 – is vc ECEN 301 Solve the n – 1 – m equations Node b vb i 2 i 3 + R 3 – Node c Discussion #7 – Node and Mesh Methods 14

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ R 3 R 2 I 1 R 1 ECEN 301 R 4 I 2 Discussion #7 – Node and Mesh Methods 15

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ R 3 i 3 R 2 I 1 R 1 + R 2 – Node a R 4 I 2 + I 1 – + R 3 – + R 1 – i 1 Node b i 2 + R 4 – i 4 – I 2 + Node c ECEN 301 Discussion #7 – Node and Mesh Methods 16

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 1. 2. i 2 + I 1 – Node b + R 1 – i 1 + R 4 – i 4 – I 2 + 3. 4. vc ECEN 301 Label currents and voltages (polarities “arbitrarily” chosen) Choose Node c (vc) as the reference node (vc = 0) Define remaining n – 1 (2) voltages Ø va is independent Ø vb is independent Apply KCL at nodes a and b Node c Discussion #7 – Node and Mesh Methods 17

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 4. i 2 + I 1 – + R 1 – ECEN 301 Node b i 1 + R 4 – vc Node c i 4 Apply KCL at nodes a and b – I 2 + Discussion #7 – Node and Mesh Methods 18

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 4. i 2 + I 1 – + R 1 – ECEN 301 Node b i 1 + R 4 – vc Node c i 4 Express currents in terms of voltages – I 2 + Discussion #7 – Node and Mesh Methods 19

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 5. i 2 + I 1 – + R 1 – ECEN 301 Node b i 1 + R 4 – vc Node c i 4 Solve the n – 1 – m equations – I 2 + Discussion #7 – Node and Mesh Methods 20

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 5. i 2 + I 1 – + R 1 – ECEN 301 Node b i 1 + R 4 – vc Node c i 4 Solve the n – 1 – m equations – I 2 + Discussion #7 – Node and Mesh Methods 21

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – 5. i 2 + I 1 – + R 1 – ECEN 301 Node b i 1 + R 4 – vc Node c i 4 Solve the n – 1 – m equations – I 2 + Discussion #7 – Node and Mesh Methods 22

Node Voltage Method u Example 2: solve for all unknown currents and voltages Ù I 1 = 10 m. A, I 2 = 50 m. A, R 1 = 1 kΩ, R 2 = 2 kΩ, R 3 = 10 kΩ, R 4 = 2 kΩ Node a vb + R 3 – va i 3 + R 2 – i 2 + I 1 – + R 1 – ECEN 301 i 1 + R 4 – vc Node c i 4 Node b 5. Solve the n – 1 – m equations – I 2 + Discussion #7 – Node and Mesh Methods 23

Node Voltage Method u Example 5: find all node voltages Ù vs = 2 V, is = 3 A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω R 1 vs + – R 3 R 2 ECEN 301 R 4 is Discussion #7 – Node and Mesh Methods 34

Node Voltage Method u Example 5: find all node voltages Ù vs = 2 V, is = 3 A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω i 1 va + R 1 – vs + – i 3 vb + R 2 – 1. + R 3 – i 2 vd ECEN 301 vc + R 4 – 2. i 4 + is – 3. 4. Label currents and voltages (polarities “arbitrarily” chosen) Choose Node d (vd) as the reference node (vd = 0) Define remaining n – 1 (3) voltages Ø va is dependent (va = vs) Ø vb is independent Ø vc is independent Apply KCL at nodes b, and c Discussion #7 – Node and Mesh Methods 35

Node Voltage Method u Example 5: find all node voltages Ù vs = 2 V, is = 3 A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω i 1 va + R 1 – vs + – i 3 vb + R 2 – vc 4. Apply KCL at nodes b, and c + R 3 – i 2 + R 4 – i 4 + is – vd ECEN 301 Discussion #7 – Node and Mesh Methods 36

Node Voltage Method u Example 5: find all node voltages Ù vs = 2 V, is = 3 A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω i 1 va + R 1 – vs + – i 3 vb + R 2 – vc 4. Express currents in terms of voltages + R 3 – i 2 + R 4 – i 4 + is – vd ECEN 301 Discussion #7 – Node and Mesh Methods 37

Node Voltage Method u Example 5: find all node voltages Ù vs = 2 V, is = 3 A, R 1 = 1Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω i 1 va + R 1 – vs + – i 3 vb + R 2 – vc 5. Solve the n – 1 – m equations + R 3 – i 2 + R 4 – i 4 + is – vd ECEN 301 Discussion #7 – Node and Mesh Methods 38

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω R 3 Vs R 1 Is + – R 2 ECEN 301 iv R 4 Discussion #7 – Node and Mesh Methods 39

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω + R 3 – va + Is – + R 1 – vb i 1 + R 2 – 1. i 3 Vs vc + – i 2 iv 2. 3. i 4 + R 4 – vd 4. ECEN 301 Label currents and voltages (polarities “arbitrarily” chosen) Choose Node d (vd) as the reference node (vd = 0) Define remaining n – 1 (3) voltages Ø va is independent Ø vb is dependent (actually both vb and vc are dependent on each other so choose one to be dependent and one to be independent) (vb = vc + Vs) Ø vc is independent Apply KCL at nodes a, and c Discussion #7 – Node and Mesh Methods 40

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω + R 3 – va + Is – + R 1 – vb i 1 + R 2 – 4. i 3 Vs vc + – i 2 iv Apply KCL at nodes a, and c i 4 + R 4 – vd ECEN 301 Discussion #7 – Node and Mesh Methods 41

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω + R 3 – va + Is – + R 1 – vb i 1 + R 2 – 4. i 3 Vs vc + – i 2 iv Express currents in terms of voltages i 4 + R 4 – vd ECEN 301 Discussion #7 – Node and Mesh Methods 42

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω + R 3 – va + Is – + R 1 – vb i 1 + R 2 – 5. i 3 Vs vc + – i 2 iv Solve the n – 1 – m equations i 4 + R 4 – vd ECEN 301 Discussion #7 – Node and Mesh Methods 43

Node Voltage Method u Example 6: find the current iv Ù Vs = 3 V, Is = 2 A, R 1 = 2Ω, R 2 = 4Ω, R 3 = 2Ω, R 4 = 3Ω + R 3 – va + Is – + R 1 – vb i 1 + R 2 – 5. i 3 Vs vc + – i 2 iv Solve the n – 1 – m equations i 4 + R 4 – vd ECEN 301 Discussion #7 – Node and Mesh Methods 44

Mesh Current Method Network Analysis ECEN 301 Discussion #7 – Node and Mesh Methods 45

Mesh Current Method u. Write 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 i R i + v _ Positive voltage ECEN 301 NB: the direction of current defines the polarity of the voltage R + v _ Negative voltage Discussion #7 – Node and Mesh Methods 46

Mesh Current Method u. Write 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 vs + v 1 – + v 3 – R 1 R 3 + _ ECEN 301 ia + v 2 – R 2 ib Two meshes n=2 + v 4 R 4 – Discussion #7 – Node and Mesh Methods 47

Mesh Current Method u. Write 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 vs + v 1 – + v 3 – R 1 R 3 + _ ECEN 301 ia + v 2 – R 2 ib Two meshes + v 4 R 4 – n=2 i 1 (current through R 1) = ia but what about i 2? According to the current directions of ia and ib, and the polarity of v 2: i 2 (current through R 2) = ia – ib Discussion #7 – Node and Mesh Methods 48

Mesh Current Method u. Write 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 vs + v 1 – + v 3 – R 1 R 3 + _ ECEN 301 ia + v 2 – R 2 ib + v 4 R 4 – Discussion #7 – Node and Mesh Methods 49

Mesh Current Method u. Write 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 vs + v 1 – + v 3 – R 1 R 3 + _ ECEN 301 ia + v 2 – R 2 ib + v 4 R 4 – Discussion #7 – Node and Mesh Methods 50

Mesh Current Method 1. Label each mesh current consistently Ø 2. Label the voltage polarity of each circuit element Ø 3. 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 Apply KVL at each mesh associated with independent variables Ø 5. Strategically (based on current direction) choose polarity unless already given In a circuit with n meshes and m current sources n – m independent equations result Ø Ø 4. Current directions are chosen arbitrarily unless given Express voltages in terms of mesh currents Solve the linear system of n – m unknowns ECEN 301 Discussion #7 – Node and Mesh Methods 51

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω R 4 vs 1 +_ R 1 R 3 vs 2 +_ R 2 R 5 ECEN 301 Discussion #7 – Node and Mesh Methods 57

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω + R 4 – vs 1 +_ ib + R 1 – ic vs 2 +_ ia + R 2 – + R 3 – 1. 2. 3. 4. Mesh current directions given Voltage polarities chosen and labeled Identify n – m (3) mesh currents Ø ia is independent Ø ic is independent Apply KVL around meshes a, b, and c – R 5 + ECEN 301 Discussion #7 – Node and Mesh Methods 58

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω + R 4 – vs 1 +_ ib 4. + R 1 – ic vs 2 +_ ia + R 2 – Apply KVL at nodes a, b, and c + R 3 – – R 5 + ECEN 301 Discussion #7 – Node and Mesh Methods 59

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω + R 4 – vs 1 +_ ib 4. + R 1 – ic vs 2 +_ ia + R 2 – Express voltages in terms of currents + R 3 – – R 5 + ECEN 301 Discussion #7 – Node and Mesh Methods 60

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω + R 4 – vs 1 +_ ib 5. + R 1 – ic vs 2 +_ ia + R 2 – Solve the n – m equations + R 3 – – R 5 + ECEN 301 Discussion #7 – Node and Mesh Methods 61

Mesh Current Method u Example 8: find the voltages across the resistors Ù Vs 1 = Vs 2 = 110 V, R 1 = 15Ω, R 2 = 40Ω, R 3 = 16Ω, R 4 = R 5 = 1. 3Ω + R 4 – vs 1 +_ ib 5. + R 1 – ic vs 2 +_ ia + R 2 – Solve the n – m equations + R 3 – – R 5 + ECEN 301 Discussion #7 – Node and Mesh Methods 62

Mesh Current Method u. Example 9: find the mesh currents ÙVs = 6 V, Is = 0. 5 A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω R 4 R 1 Is ia ECEN 301 ic R 2 R 3 ib vs – + Discussion #7 – Node and Mesh Methods 63

Mesh Current Method u. Example 9: find the mesh currents ÙVs = 6 V, Is = 0. 5 A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R 4– + R 1 – Is ia ECEN 301 ic + R 2 – 1. 2. 3. +R 3 – ib vs – + 4. Mesh current directions given Voltage polarities chosen and labeled Identify n – m (3) mesh currents Ø ia is dependent (ia = is) Ø ia is independent Ø ic is independent Apply KVL around meshes b and c Discussion #7 – Node and Mesh Methods 64

Mesh Current Method u. Example 9: find the mesh currents ÙVs = 6 V, Is = 0. 5 A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R 4– + R 1 – Is ia ECEN 301 ic + R 2 – 4. Apply KVL at nodes b and c +R 3 – ib vs – + Discussion #7 – Node and Mesh Methods 65

Mesh Current Method u. Example 9: find the mesh currents ÙVs = 6 V, Is = 0. 5 A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R 4– + R 1 – Is ia ECEN 301 ic + R 2 – 4. Express voltages in terms of currents +R 3 – ib vs – + Discussion #7 – Node and Mesh Methods 66

Mesh Current Method u. Example 9: find the mesh currents ÙVs = 6 V, Is = 0. 5 A, R 1 = 3Ω, R 2 = 8Ω, R 3 = 6Ω, R 4 = 4Ω +R 4– + R 1 – Is ia ECEN 301 ic + R 2 – 5. Solve the n – m equations +R 3 – ib vs – + Discussion #7 – Node and Mesh Methods 67

Equation Solver Methods Calculator Matrix Cramer’s Rule Brute Force (Substitution) ECEN 301 Discussion #7 – Node and Mesh Methods 68

TI-89 Equation Solver 1. Press F 1 u Select option number 8 (Clear Home) 2. Press APPS u Select option number 1 (Flash. Apps) u Select Simultaneous Eqn Solver u Select New… 3. In the new box enter n (the number of equations and unknowns) ECEN 301 Discussion #7 – Node and Mesh Methods 69

TI-89 Equation Solver 4. Input the equation coefficients u For the set of 3 equations and 3 unknowns from example 8: Input the coefficients as follows: 1 2 3 ECEN 301 a 2 41. 3 0 -40 0 16. 3 -15 a 3 -40 -15 71 b 1 110 0 Discussion #7 – Node and Mesh Methods 70

TI-89 Equation Solver 5. Press F 5 (Solve) ECEN 301 Discussion #7 – Node and Mesh Methods 71