Networks I for M.E. ECE 09.201 - 2

1. Networks I for M.E.ECE 09.201 - 2 James K. Beard, Ph.D.

2. Passive Sign Convention • Always mark one terminal with a + sign • Voltage is positive measured from the other terminal • Current is positive going into the terminal marked with the + sign Networks I for M.E.

3. Source Sign Convention • Voltage sources marked with + and – signs inside a circle or diamond • Current sources marked with an arrow inside a square, circle or diamond • Positive • Current out of a voltage source • Voltage out of a current source Networks I for M.E.

4. General Methodology • Write the loop equations • Pick small or simple loops • Make all of them one direction – clockwise or counterclockwise • Make sure that every trace is covered once • Use node voltage equations • Leverage supernodes – treat as single nodes • Write the node equations -- Currents are positive into each node • Write the Ohm’s Law equations • Rearrange the equations to allow vector-matrix notation • Check your work • Use computer to solve the matrix equation • TI-89 for smaller problems • Matlab or Mathcad for matrices larger than 4 X 4 or so • Note that free linear algebra packages are available for most HLLs Networks I for M.E.

5. Approach Used Here for 4.4-7 • Use voltage node notation • Leverage supernodes • Write Ohm’s Law equations first • Knowns are • Resistances • Source voltages and currents • Controlled source multipliers • Unknowns are • Node voltages • Currents through voltage sources • Voltages across current sources Networks I for M.E.

6. Technique for Loop Equations • Add voltage drops around the loop • The sum of voltage drops around a closed loop must be zero • Voltage drop is voltage at the present node minus voltage on the other terminal of the resistor or source • Drop is positive through a resistor when the loop goes into the “+” terminal and out the “-” terminal and its current is positive • Look at the equation with and without Ohm’s Law • Supernodes will identify themselves • Node voltages instead of currents often gives simpler equations Networks I for M.E.

7. Direction of loop Voltage Drop Networks I for M.E.

8. Technique for Node Equations • Current into a node through each resistor is the voltage on the other side of each resistor divided by the resistance • Subtract the voltage at that node times the sum of the reciprocals of all the resistors connected to the node • Add currents through sources • Current sources connected to the node • Currents through voltage sources connected to the node • Use Ohm’s Law to pose the currents through resistors in terms of the node voltages Networks I for M.E.

9. Matrix Notation Is • A way of organizing several linear equations. • Nothing is changed from the original equations. • TI-89, Matlab, and Mathcad can solve the problem for you from there. Networks I for M.E.

10. Problem 4.4-7 Waaaay too messy for a quiz! Networks I for M.E.

11. Ohm’s Law Use to make node voltages the unknowns Networks I for M.E.

12. Loop Equations Networks I for M.E.

13. Node Equations Networks I for M.E.

14. Rearranging into Matrix Format Networks I for M.E.

15. Solution Networks I for M.E.

16. Problem 4.4-7 Solution Networks I for M.E.

17. Intuitive Approach • From Loop 2 plus Loop 3: • From Loop 2 • The rest is found from Ohm’s Law Good for many homework problems, but not for most real-world problems Networks I for M.E.

18. Homework Problem • Put problem 4.4-7 in Matlab and solve it with the matrix method. • Reproduce the results given here. • Save your code, print it out, and turn it in with the homework. • Use your program to solve problem 4.4-7 with the parameters to the right • Why can’t Kvv be 1.0? • What is special about the circuit that allows simple intuitive solutions? Networks I for M.E.