1. Power and Energy Consider the product of voltage and current. V  I
(J/C)  (C/s) = J/s (1 J/s = 1 W: power!)
Now multiply by time.
V  I  t
(J/C)  (C/s)  s = J (energy!)

2. Power and Energy P = V  I Recall that V = I R.
So P = V  I = I R  I = I2 R
Also, I = V/R.
So P = V  I = V  (V/R) = V2/R

3. Mathematical Conventions
If positive current leaves the positive voltage terminal, the element is delivering or furnishing power (active).
If positive current enters the positive voltage terminal, the element is absorbing or dissipating power (passive).

4. Independent Sources Independent Voltage Source
The voltage across the source is independent of the current drawn from the source.
Independent Current Source
The current delivered by the source is independent of the voltage across the source.
Strong, fresh battery
Lightning

5. Dependent Sources Voltage-Controlled Voltage Source Current-Controlled
Current Source
Current-Controlled
Current Source

6. Resistors in Series - Review

7. Resistors in Parallel - Review

8. Resistors Combination - Practice

9. RECALL: Types of connections
Resistors connected in series…
When JUST two elements connect at a SINGLE node, they are said to be in series and series-connected elements carry the SAME CURRENT (why? KCL)

10. Resistors connected in parallel…
RECALL: Types of connections
Resistors connected in parallel…
When two elements connect at a single node pair, they are said to be in parallel and they have the SAME VOLTAGE across their terminals ! (why? KVL)

11. Step by step process to find equivalent resistance

12. Resistors Combination - Practice

13. Voltage and Current Division
The current and voltage in a resistive circuit get divided between different resistors based on their resistance values.
How does that happen?
KVL and KCL are universal rules and should be satisfied in every circuit – including (obviously) resistive circuits.

14. Voltage Division From Ohm’s Law I = VT/Req
Req = R1 + R2 + R3
Voltage applied to series circuit applies a fraction of the voltage across each element

15. Voltage Division Section 2.3 Voltage Divider (Series Circuits)
Voltage Divider (Series Circuits)
Voltage Division Principle: the fraction of the total voltage across a single resistor in a series circuit is the ratio of the given resistor to the total series resistance
So if V1 = VT R1 / (R1 + R2 + R3) , then what is V2 and V3

16. VOLTAGE DIVISION V1 = VT R1/ (R1 + R2) …… …. V6 = VT R6/ (Req)
Voltage division is a simple method/procedure that allows one to determine the voltage across a resistor in a series combination, if the total voltage across all series-connected resistors is known. The relationships are shown below:
V1 = VT R1/ (R1 + R2) …… ….
V6 = VT R6/ (Req)

17. Current Division
Total current in a parallel circuit is divided among resistances.
What is i2?

18. Current Division Current Divider (Parallel Circuits)
Used to determine the current through one of several parallel resistors, if the total current entering the parallel combination is known

19. Once again… WHEN YOU HAVE RESISTORS CONNECTED IN SERIES
WHEN YOU HAVE RESISTORS CONNECTED IN PARALLEL
YOU CAN USE VOLTAGE DIVISION TO FIND INDIVIDUAL VOLTAGE VALUES ACROSS THE RESISTORS
YOU CAN USE CURRENT DIVISION TO FIND THE INDIVIDUAL CURRENT VALUES THROUGH THE RESISTORS

20. Let’s solve some examples…

21. And an example on current division…

22. How can we combine these techniques to solve circuits
How can we combine these techniques to solve circuits? Let’s try to find io and vo !

25. First thing we can do is to assign variables to each element – hopefully we’ll be able to solve for those !

26. First thing we can do is to assign variables to each element – hopefully we’ll be able to solve for those !

30. Node-Voltage Analysis
In addition to analyzing circuits by combining series and parallel resistors and applying the voltage and current – division principles, there is the NODE-VOLTAGE Analysis.
Recall a node is a point at which two or more circuit elements are joined together

31. Node-Voltage Analysis
CONVENTION: usually write expressions for I, current, leaving the node under consideration and set sum to zero
Step 1. Select or find reference node
Step 2. Label the node voltages - reference and other nodes

32. Node-Voltage Analysis
CONVENTION: usually write expressions for I, current, leaving the node under consideration and set sum to zero
Step 3. Observe node voltage relationship to element voltage (e.g. KVL says -v2 + vx + v3 = 0, then vx = v2 – v3)
Step 4. Write current equations at each of the nodes for ALL currents leaving the node

33. Node-Voltage Analysis
Step 5. Node-voltage equations can be written for each node in the form of current leaving one node and entering another node e.g. i4 through R4 leaving Node 2 and entering ground gives i4 = V2/R4

34. Node-Voltage Analysis
Node Voltage Equation for Node 3
KCL iR1 + iR5 + iR3 = 0
where
iR1 = (v3 – v1)/R1
iR5 = v3/R5
iR3 =( v3 – v2)/R3
Then the Node Voltage Equation for Node 3 is
(v3 – v1)/R v3/R (v3 – v2)/R3 = 0

35. Node-Voltage Analysis
What is the Node Voltage Equation for Node 1

36. Node-Voltage Analysis
What is the Node Voltage Equation for Node 2

37. Node-Voltage Analysis
RECALL: Node-voltage equations can be written for each node in the form of current leaving one node and entering another node
Another Example

38. Node-Voltage Analysis
Another Example

39. Section 2.4 Node Voltage Analysis
Another Example
CONVENTION: usually write expressions for I, current, leaving the node under consideration and set sum to zero

40. Mesh-Current Analysis
Then there is MESH- CURRENT Analysis.
Use KVL to write a vrises = vdrops equation for each mesh
mesh – a closed path that contains no other closed paths

41. Mesh-Current Analysis
1. Label the meshes. # of independent KVL equations for planar network are equal to # of open areas defined by the network layout (2 OPEN areas in this circuit, hence 2 mesh currents, i1, and i2)

42. Section 2.5 Mesh Current Analysis
2. Follow KVL around each mesh
Mesh 1
-Va + i1R1 + V3 = where v3 = R3(i1 - i2)
Thus
i1R1 + R3(i1 - i2) = Va
The same can be found for Mesh 2

43. Section 2.5 Mesh Current Analysis

44. Mesh-Current Analysis

45. Section 2.5 Mesh Current Analysis

46. Section 2.5 Mesh Current Analysis
Exercise. Find ia

47. Section 2.5 Mesh Current Analysis
Exercise. Find ib

48. Section 2.5 Mesh Current Analysis
Mesh with Controlled Sources
Combine MESH 1 and 2 – Supermesh
Voltage Controlled Current Source referenced as
i2 - i1 = vx/ where vx = 2i2