1. 1 ECE 3144 Lecture 12 Dr. Rose Q. Hu Electrical and Computer Engineering Department Mississippi State University

2. 2 Review for Chapter 1 Charge (q), current (i), and their relationship Voltage (v), work/energy (w), power (p), and their relationships. Steady state voltage V (voltage is constant), steady state current I (current is constant) and steady state power P (power is constant).

3. 3 Review for Chapter 1 Passive sign convention –The positive reference of voltage v(t) is at the same terminal the the current variable i(t) is entering. –How to determine absorbing or supplying energy? Active elements and passive elements –Active elements Independent sources  Independent voltage sources and independent current sources Dependent sources  Voltage controlled voltage source  Current controlled voltage source  Voltage controlled current source  Current controlled current source –Passive element: resistor

4. 4 Review for Chapter 2 Resistor, Resistance, Resistivity: R =  *L/A R=1/G Ohms’law v(t) = i(t)*R: only linear resistors satisfy Ohm’s law. Power p(t) = v(t)*i(t)= v(t) 2 /R = i(t) 2 R Kirchoff’s current law (KCL): The algebraic sum of the currents leaving (entering) a node is zero. Kirchoff’s voltage law: The algebraic sum of the voltages around any loop path is zero. Single loop circuit: resistors in series => voltage divider Voltage sources in series Single node circuit: resistors in parallel =>current divider Circuits containing a single source and a series-parallel interconnection of resistors How to perform series-parallel combinations between two terminals. Wye-delta or Delta-to-Wye transformations Circuits containing dependent sources: not covered in the exam 1.,,,,

5. 5 Review for Chapter 2 Current sources in parallel Circuits containing a single source and a series-parallel interconnection of resistors How to perform series-parallel combinations between two terminals. Wye-delta or Delta-to-Wye transformations Circuits containing dependent sources: not covered in the exam 1.

6. 6 Reminder from Lecture 11 Nodal analysis case 1: with independent current sources and resistors only GV = I. G matrix is symmetric. In general, KCL is applied to node j with node voltage v j. The coefficient of v j, which is the element g jj of G matrix, is the sum of all the conductances connected to node j. The coefficient of any other node voltage, say i (i  j), is the negative of the sum of the conductances connected directly between node j and node i. The right hand side of the equation is equal to the sum of the currents entering the node j via independent current resources. Nodal analysis case 2: with dependent current sources GV = I However G matrix may not be symmetric anymore Treat the dependent sources as independent sources first. Derive the KCL equations at each node. Replace the dependent sources by the control formula given. Solve the equations for nodal voltages. Nodal analysis case 3: with independent voltage sources Scenario 1:The independent voltage source is connected to the reference node.  Any time an independent voltage source is connected between an reference node and a nonreference node, the voltage for the nonreference node is known. Scenario 2: will be introduced in this lecture

7. 7 Reminder: nodal analysis with independent voltage sources scenario 1 -independent voltage sources are connected to the reference node Any time an independent voltage source is connected between a reference node and a nonreference node, the voltage for the nonreference node is known. So the presence of voltages sources in this case will make the voltage equations simpler. Say the given network has N nodes. If k independent voltages sources are connected between a reference node and a nonreference node, then the number of linearly independent equations needed to solve the node voltages is reduced from (N-1) to (N-1-k). For the circuit given, immediately we know V 1 = 12 V and V 2 = -6V. The only unknown voltage variable is V 3. Applying KCL at node 2 + - + - 9k  12k  6k  V1V1 V2V2 V3V3 12 V 6 V V 2 =1.5 V

8. 8 Scenario 2: independent voltage source are connected between the nonreference nodes If we follow the nodal analysis brute force manner, we will have a problem: the current on the branch where independent voltage source is can be not calculated, which means KCL equations at node 1 and node 2 can not be established. A method called supernode technique is introduced.  First circle the voltage source and the two connecting nonreference nodes to form a supernode.  Write the equations that defines the voltage relationships between the two nonreference nodes as a result of the presence of the voltage source. V 1 – V 2 = 6  Write the KCL equation for the super node. + - 6k  12k  6 mA 6 V 4 mA V1V1 V2V2 (1) (2)  Then we have two equations (1)& (2) and two unknowns => V 1 = 10 V, V 2 = 4 V

9. 9 Example 1: circuits with independent voltage sources connected between nonreference nodes 1 2 0 Problem 3.28: Using nodal analysis to find V o for the given network (1) (2) (3)  At super node:  At node 0: Put equations (1), (2) and (3) in matrix form: 

10. 10 Nodal analysis case 4: circuits containing dependent voltages sources Circuits containing dependent voltages sources are treated the same way as the independent voltage sources except that the voltage controlling equations for the dependent voltage sources should be used instead of the given independent voltage source values. –Scenario 1: if a dependent voltage source is connected between a reference node and a nonreference node, the controlling equation for voltage at the nonreference node is immediately known. –Scenario 2: a dependent voltage source is connected between a reference node and a nonreference node First circle the dependent voltage source and the two connecting nonreference nodes to form the supernode. Write the equation that defines the voltage relationship between the nonreference nodes as a result of the presence of the dependent voltage source. Write the KCL equation for the supernode and the rest normal nodes. Write the controlling equation for the dependent voltage source

11. 11 Case 4 example : nodal analysis with dependent voltage sources connected between the reference node and a nonreference node Problem 3.37: Find V 0 in the circuit given 0 1 The dependent voltage source is connected to the reference node At node 1:(1)  At node 0:  (2) Write the controlling equation for the dependent voltage source  (3) Put equations (1),(2) and (3) in matrix form 

12. 12 Case 4 example: nodal analysis with dependent voltage sources connected between two nonreference nodes Problem 3.35: Find V o in the circuit network given. 012 The dependent voltage source connected to two nonreference nodes 0 and 2. Thus node 0 and node 2 form a supernode.  (1) A voltage source is connected between the reference node and node 1. Apply KCL to the supernode: The controlling function for the voltage source is: (2)  (3)

13. 13 Example 3: cont’d Put the three equations (1), (2) and (3) into matrix format: 

14. 14 Homework for lecture 12 Problem 3.31, 3.32, 3.34, 3.36, 3.39 Due Feb 11