1. EGR 2201 Unit 2 Basic Laws Read Alexander & Sadiku, Chapter 2.
Homework #2 and Lab #2 due next week.
Quiz next week.
Handouts: Quiz 1, Unit 2 practice sheet.

2. Ohm’s Law
Ohm’s law says that the voltage v across a resistor is equal to the current i through the resistor times the resistor’s resistance R. In symbols:
v = i  R
The voltage’s polarity and current’s direction must obey the passive sign convention, as shown in the diagram at right. Otherwise, you need a negative sign in this equation: v = i  R.
Applies to resistors only, which is why we’ll focus on resistive circuits for the next few weeks.

3. Ohm’s Law Rearranged
The equation v = i  R is useful if we know i and R, and we’re trying to find v.
Often we’ll need to rearrange the equation to one of the following forms:
i = v  R or
R = v  i
Do practice question 1.

4. Ohm’s Law Game
Given values for two of the three quantities in Ohm’s law, you must be able to find the third quantity.
To practice, play my Ohm’s Law game.

5. Short Circuit
An element with R=0 (or with an extremely small resistance) is called a short circuit.
Since a short circuit’s resistance is zero, Ohm’s law tells us that the voltage across a short circuit must also be zero:
v = i  R = i  0 = 0
But we can’t use Ohm’s law to compute a short circuit’s current:
i = v  R = 0  0 = ???
Demo by measuring resistance of a wire.

6. Open Circuit
An element with R= (or with an extremely large resistance) is called an open circuit.
Since an open circuit’s resistance is infinite, Ohm’s law tells us that the current through an open circuit must be zero:
i = v  R = v   = 0
But we can’t use Ohm’s law to compute an open circuit’s voltage:
v = i  R = 0   = ???
-Demo by measuring resistance of air gap.
-Then do practice questions 2 and 3.

7. Power Dissipated by a Resistor
When current flows through a resistor, electric energy is converted to heat, at a rate given by the power law:
p = v  i
Once the energy has been given off as heat, we can’t easily reverse this process and convert the heat back to electric energy. We therefore say that resistors dissipate energy.
In contrast, we’ll see later that capacitors and inductors store energy, which can easily be recovered.

8. Other Power Formulas for Resistors
By combining the power law (p = v  i) with Ohm’s law (v = i  R or i = v  R), we can easily derive two other useful formulas for the power dissipated by a resistor:
p = i 2  R
p = v 2  R
Of course, each of these equations can in turn be rearranged, resulting in a number of useful equations that are summarized in the “power wheel”….
Do practice question 4.

9. The “Power Wheel”
This is a useful aid for people who aren’t comfortable with basic algebra, but you shouldn’t need it.
The important point is that if you know any two of these four quantities (P, V, I, and R), you can compute the other two, as long as you remember p=vi and v=iR.
Do practice question 5.

10. Power Calculation Games
Given values for two of the following four quantities—voltage, current, resistance, power—you must be able to find the other two quantities.
To practice, play these games:
Ohm’s Law
Power Law
Power-Current-Resistance
Power-Voltage-Resistance

11. Conductance
It’s sometimes useful to work with the reciprocal of resistance, which we call conductance.
The symbol for conductance is G:
G = 1  R
Its unit of measure is the siemens (S).
Example: If a resistor’s resistance is 20 , its conductance is 50 mS.

12. Review: Some Quantities and Their Units
Quantity
Symbol
SI Unit
Symbol for the Unit
Current
I or i
ampere A
Voltage
V or v
volt V
Resistance R
ohm 
Charge
Q or q
coulomb C
Time t
second s
Energy
W or w
joule J
Power
P or p
watt W
Conductance G
siemens S

13. Ohm’s Law and the Power Formulas Using Conductance
As the book discusses, Ohm’s law and our power formulas can be rewritten using conductance G instead of resistance R.
Example: Instead of writing
v = i  R
we can write
v = i  G
But I advise you to ignore this, and always use R instead of G.

14. Circuit Topology: Branches
When describing a circuit’s layout (or “topology”), it’s often useful to identify the circuit’s branches, nodes, and loops.
A branch represents a single circuit element such as a voltage source or a resistor.
Example: This circuit (from Figure 2.10) has five branches.

15. Dots or No Dots?
In schematic diagrams, our textbook sometimes draws dots at the points where two or more branches meet, as in this diagram.
But most of the time the book omits these dots, as in this diagram.
There’s no difference in meaning.

16. Circuit Topology: Nodes
A node is the point of connection between two or more branches.
Example: This circuit has three nodes, labeled a, b, and c.

17. Circuit Topology: Loops
A loop is any closed path in a circuit.
Example: This circuit has six loops.
Don’t worry about the book’s distinction between loops and independent loops.
Do practice question 6.

18. Elements in Series
Two elements are connected in series if they are connected to each other at exactly one node and there are no other elements connected to that node.
Example: In this circuit, the voltage source and the 5- resistor are connected in series.

19. Current Through Series-Connected Elements
If two elements are connected in series, they must carry the same current.
Example: In this circuit, the current through the voltage source must equal the current through the 5- resistor.
But usually their voltages are different.
Example: In the circuit above, we wouldn’t expect the voltage across the 5- resistor to be 10 V.

20. Elements in Parallel
Two or more elements are connected in parallel if they are connected to the same two nodes.
Example: In this circuit, the 2- resistor, the 3- resistor, and the current source are connected in parallel.

21. Voltage Across Parallel-Connected Elements
If two elements are connected in parallel, they must have the same voltage across them.
Example: In this circuit, the voltage across the 2- resistor, the 3- resistor, and the current source must be the same.
But usually their currents are different.
Example: In the circuit above, we wouldn’t expect the current through the 3- resistor to be 2 A.

22. Some Connections are Neither Series Nor Parallel
Sometimes elements are connected to each other but are neither connected in series nor connected in parallel.
Example: In this circuit, the 5- resistor and the 2- resistor are connected, but they’re not connected in series or in parallel.
In such a case, we wouldn’t expect the two elements to have the same current or the same voltage.
This type of connection doesn’t have a special name.
Do practice question 7.

23. Another series circuit
Series Circuits
Two simple circuit layouts are series circuits and parallel circuits.
In a series circuit, each connection between elements is a series connection.
Therefore, current is the same for every element. (But usually voltage is different for every element.)
A series circuit is a “one-loop” circuit.
A series circuit:
Same circuit on the breadboard:
Another series circuit

24. Analyzing a Series Circuit
+ v1 
+ v2 
Analyzing a Series Circuit is i1 i2 i3
 v3 +
Find the total resistance by adding the series-connected resistors:
RT = R1 + R2 + R
Use Ohm’s law to find the current produced by the voltage source:
is = vs  RT
Recognize that this same current passes through each resistor:
is = i1 = i2 = i3 = ...
Use Ohm’s law to find the voltage across each resistor:
v1 = i1  R1 and v2 = i2  R2 and ...
Do practice question 8.

25. Another parallel circuit
Parallel Circuits
In a parallel circuit, every element is in parallel with every other element.
Therefore, voltage is the same for every element. (But usually current is different for every element.)
A parallel circuit:
Another parallel circuit
Same circuit on the breadboard:

26. Analyzing a Parallel Circuit+ v1  + v2  + v3 
Recognize that the voltage across each resistor is equal to the source voltage:
vs = v1 = v2 = v3 = ...
Use Ohm’s law to find the current through each resistor:
i1 = v1  R1 and i2 = v2  R2 and ...
Do practice question 9.

27. More Complicated Circuits
We’ve seen that series circuits and parallel circuits are easy to analyze, using little more than Ohm’s law.
But most circuits don’t fall into either of these categories, and are harder to analyze.
Some authors call these series-parallel circuits. Others call them complex circuits.
Do new practice questions reviewing terminology.
Examples that are neither series circuits nor parallel circuits

28. Kirchhoff’s Current Law (KCL)
Kirchhoff’s current law: The algebraic sum of currents entering a node is zero.
Example: In this figure, i1  i2 + i3 + i4  i5 = 0
Alternative Form of KCL: The sum of the currents entering a node is equal to the sum of the currents leaving the node.
Example: In the figure, i1 + i3 + i4 = i2 + i5
-For the first form, each current’s sign in the equation is + if the arrow points in, - if the arrow points out.
-Do practice questions 10 and 11.

29. KCL Applied to a Closed Boundary
You can also apply KCL to an entire portion of a circuit surrounded by an imaginary boundary. The sum of the currents entering the boundary is equal to the sum of the currents leaving the boundary.
Example: In this figure, i1 + i5 = i2 + i7 + i8

30. Kirchhoff’s Voltage Law (KVL)
Kirchhoff’s voltage law: Around any loop in a circuit, the algebraic sum of the voltages is zero.
Example: In this figure,
v1 + v2 + v3  v4 + v5 = 0
Alternative Form of KVL: Around any loop, the sum of the voltage drops is equal to the sum of the voltage rises.
Example: In the figure, v2 + v3 + v5 = v1 + v4
-For the first form, each voltage’s sign in the equation is the first terminal that you hit as you travel around the loop.
-For the alternative form, as you travel around the loop, if you move from the - side of an element to the + side, that’s a voltage rise. If you move from the + side to the - side, that’s a voltage drop.
-Do practice question 12.

31. Equivalent Resistance
In analyzing circuits we will often combine several resistors together to find their equivalent resistance.
Basic idea: What single resistor would present the same resistance to a source as the combination of resistors that the source is actually connected to?

32. Resistors in Series
The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances:
Req = R1 + R RN
We’ve already used this earlier in Step 1 of our analysis of series circuits.

33. Parallel Resistors 𝑅 𝑒𝑞 = 𝑅 1 𝑅 2 𝑅 1 + 𝑅 2
The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum:
𝑅 𝑒𝑞 = 𝑅 1 𝑅 2 𝑅 1 + 𝑅 2
Note that the equivalent resistance is always less than each of the original resistances.
-Do practice question 13.

34. More Than Two Resistors in Parallel
For more than two resistors in parallel, you cannot simply extend the product-over-sum rule like this:
𝑅 𝑒𝑞 = 𝑅 1 𝑅 2 𝑅 3 𝑅 1 + 𝑅 2 + 𝑅 3
Instead, either use the product-over-sum rule repeatedly (with two values at a time), or…
…use the so-called reciprocal formula:
𝑅 𝑒𝑞 = 𝑅 𝑅 𝑅 3 +…+ 1 𝑅 𝑁
-Do practice questions 14 and 15.

35. Parallel Resistors: Two Shortcut Rules for Special Cases
Special Case #1: For N parallel resistors, each having resistance R,
𝑅 𝑒𝑞 = 𝑅 𝑁
Special Case #2: When one resistance is much less than another one connected in parallel with it, the equivalent resistance is very nearly equal to the smaller one:
If R1 << R2, then Req  R1
-Do practice question 16.

36. Series-Parallel Combinations of Resistors
In many cases, you can find the equivalent resistance of combined resistors by repeatedly applying the previous rules for resistors in series and resistors in parallel.
Hint: Start farthest from the source or (in a case like the one above from Figure 2.34) farthest from the open terminals, and work your way back toward the source or open terminals.
-Do the one in the figure (=14.4 ohms).
-Then do practice questions 17 and 18.

37. Building Complex Circuits on the Breadboard
-Do practice questions 19, 20.

38. Voltage Division
For resistors in series, the total voltage across them is divided among the resistors in direct proportion to their resistances.
Example: In the circuit shown (Figure 2.29), if R1 is twice as big as R2, then v1 will be twice as big as v2.
See next slide for a formula that captures this…

39. The Voltage-Divider Rule
For N resistors in series, if the total voltage across the resistors is v, then the voltage across the nth resistor is given by:
𝑣 𝑛 = 𝑅 𝑛 𝑅 1 + 𝑅 2 +…+ 𝑅 𝑁 𝑣
-Do practice question 21.
Example: In the circuit shown,
𝑣 1 = 𝑅 1 𝑅 1 + 𝑅 2 𝑣 and 𝑣 2 = 𝑅 2 𝑅 1 + 𝑅 2 𝑣

40. The Voltage-Divider Rule in More Complex Circuits
The voltage-divider rule is easiest to apply in series circuits, but it also holds for series resistors in more complex circuits.
Example: In the circuit shown, suppose we know the value of the voltage v. Then we can say that
𝑣 1 = 𝑣 and 𝑣 2 = 𝑣
-Do practice question 23 (out of order).

41. Current Division
For resistors in parallel, the total current through them is shared by the resistors in inverse proportion to their resistances.
Example: In the circuit shown (Figure 2.31), if R1 is twice as big as R2, then i1 will be one-half as big as i2.
See next slide for a formula that captures this…

42. The Current-Divider Rule
For two resistors in parallel, if the total current through the resistors is i, then the current through each resistor is given by:
𝑖 1 = 𝑅 2 𝑅 1 + 𝑅 2 𝑖 and 𝑖 2 = 𝑅 1 𝑅 1 + 𝑅 2 𝑖
-Do practice question 22.

43. The Current-Divider Rule in More Complex Circuits
The current-divider rule is easiest to apply in parallel circuits, but it also holds for parallel resistors in more complex circuits.
Example: In the circuit shown, suppose we know the value of the current i. Then we can say that
𝑖 1 = 𝑖 and 𝑖 2 = 𝑖
-Do practice question 24.

44. Review of Short Circuits
Recall that an element with R=0 (or with an extremely small resistance) is called a short circuit.
Recall also that, since the resistance of a short circuit is zero, the voltage across it must always be zero.
Sometimes short circuits are introduced intentionally into a circuit, but often they result from a circuit failure.

45. New Observations about Short Circuits
The equivalent resistance of a short circuit in parallel with anything else is zero.
An element or portion of a circuit is short-circuited, or “shorted out,” when there is a short circuit in parallel with it.
No current flows in a short-circuited element; instead all current is diverted through the short circuit itself.
Example: In this circuit, no current will flow through R2 or R3.

46. Potentiometers & Rheostats
A potentiometer is an adjustable voltage divider that is widely used in a variety of circuit applications. It is a three-terminal device.
A rheostat is an adjustable resistance. It has only two terminals.

47. Useful Microsoft Word Features For Your Lab Reports
Bold
Italics
Superscript
Subscript

48. Useful Microsoft Word Features For Your Lab Reports (cont’d.)
Creating a Table

49. Useful Microsoft Word Features For Your Lab Reports (cont’d.)
Inserting Symbols
Then select the Symbol Font to find Greek letters and other symbols.