1. Lesson 4: Series DC Circuits and Kirchhoff’s Voltage Law (KVL)

2. Learning Objectives Identify elements that are connected in series.
State and apply KVL in analysis of a series circuit.
Determine the net effect of series-aiding and series-opposing voltage sources.
Compute the power dissipated by each element and the total power in a series circuit.
Compute voltage drops across resistors using the voltage divider formula.
Describe the basic function of a fuse and a switch.
Draw a schematic of a typical electrical circuit, and explain the purpose of each component and indicate the polarity and current direction.
Explain and compute how voltage divides between elements in a series circuit.
Apply concept of voltage potential between two points to the use of subscripts and the location of the reference voltage.
Analyze a series resistive circuit with the ground placed at various points.

3. Series Circuits
A series circuit is defined by two elements in a series:
Connected at a single point (node).
No other current-carrying connections at this node.
The same current flows through series connected circuits.

4. Series Circuits Normally;
Current will leave the positive terminal of a voltage source and move through the resistors.
Current will return to the negative terminal of the voltage source.
Current is the same everywhere in a series circuit. Is
Load

5. Series Circuits Current is similar to water flowing through a pipe.
Current leaving the element must be the same as the current entering the element.
Current = water flow rate
Pressure = potential difference = voltage
Same current passes through every element of a series circuit.

6. Series Resistors
Now that the series connection has been described, yo can now recognize that every fixed resistor has only two terminals to connect in a configuration
referred to as a two-terminal device.
Series connection of resistors.

7. Series Resistors Series connection of resistors.
Series connection of four
resistors of the same value
Series connection of resistors.

8. Series Resistors Example
In each circuits below, list the resistors in series with R2.
R1 and R3
None
Here is a look at the current flow, which we will talk about later R1

9. Two Resistors in Series
Two resistors in series can be replaced by an equivalent resistance Req. 9

10. N Resistors in Series
The equivalent resistance Req of any number of resistors in series is the sum of the individual resistances. 10

11. Resistors in Series
The equivalent resistance Req of any number of resistors in series is the sum of the individual resistances.

12. Kirchhoff’s Voltage Law (1)
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.
Mathematically, KVL implies:
Where ET => ET= E1+E2+E3+…+En 12

13. Kirchhoff’s Voltage Law (2)
Another way of stating KVL is: KVL is the summation of voltage rises is equal to the summation of voltage drops around a closed loop. 13

14. Kirchhoff’s Voltage Law (3)
A closed loop is any path that:
Originates at a point.
Travels around a circuit.
Returns to the original point without retracing any segments.

15. Kirchhoff’s Voltage Law (4)
Summation of voltage rises is equal to the summation of voltage drops around a closed loop.

16. Example Problem 1 Determine the unknown voltages in the network below:
First, determine the value of V3:
Next, determine the value of V1:
Because you know I, V1 and V2 we can also find R1 and R1:
Verify:

17. Example Problem 2
Use Kirchhoff’s Voltage Law to determine the magnitude and polarity of the unknown voltage ES in the circuit below:
Knowns: R1 = 12kΩ, Is = 2 mA, E2 = 62 V, R2 = 6k Ω, and R3 = 3 k Ω
First, determine the value of v1, v2 and v3:
Using KVL, find Es:

18. VOLTAGE DIVISION IN A SERIES CIRCUIT Voltage Divider Rule (VDR)
The voltage divider rule states that:
The voltage across a resistor (Vx) in a series circuit is equal to the value of that resistor (Rx) times the total applied voltage divided (E) by the total resistance (RT) of the series configuration.

19. The Voltage Divider Rule
Voltage applied to a series circuit:
Voltage drop across any series resistor is proportional to the magnitude of the resistor.
The voltage divider rule allows us to calculate the voltage across any series resistance in a single step, without first calculating the current.

20. The Voltage Divider Rule
Voltage applied to a series circuit:
Voltage drop across each resistor may be determined by the proportion of its resistance to the total resistance:

21. Voltage Divider Rule
If a single resistor is very large (~100x) compared to the other series resistors, the voltage across that resistor will be the source voltage.
Voltage across the small resistors will be essentially zero. V2 V2

22. Example Problem 3 For the circuit below (with Rtot =800Ω), determine:
Direction and magnitude of current.
Voltage drop across each resistor.
Value of the unknown resistance.
Verify:

23. Power in a Series Circuit
Power dissipated by each resistor is determined by the power formulas:
P = VI = V2/R = I2R
Since energy must be conserved, power delivered by voltage source is equal to total power dissipated by resistors.
PT = P1 + P2 + P3 + ∙∙∙ + Pn

24. Example Problem Using VDR to Solve For Power
For the circuit below, determine:
Power dissipated by each resistor and total power dissipated by the circuit.
Verify that the summation of the power dissipated by each resistor equals the total power delivered by the voltage source.
Another way to solve would have been to use the voltage divider rule if you wanted…

25. Interchanging Series Components
Order of series components
May be changed without affecting operation of circuit.
Sources may be interchanged, but their polarities can not be reversed.
After circuits have been redrawn, it may become easier to visualize circuit operation.

26. Interchanging Series Components
Be cautious that you recognize the polarity of the source calculated:

27. Voltage Sources in Series
In a circuit with more than one source in series sources can be replaced by a single source having a value that is the sum or difference of the individual sources.
Polarities must be taken into account.
ET= E1+E2+E3+…+En

28. Simplifying Voltage Sources
Polarities must be taken into account.

29. Voltage Sources in Series
Resultant source
Sum of the rises in one direction minus the sum of the voltages in the opposite direction.
ET = 5V
16Ω
ET= E1+E2+E3+…+En
ET = 2V - 6V - 1V = 5V

30. Example Problem 4
Redraw the circuit below, showing a single voltage source and single resistor. Solve for the current in the circuit.
ET= E1+E2+E3+…+En
ET = 12V + 3V - 6V = 9V
RT= R1+R2+R3+…+Rn
RT = 27kΩ + 33kΩ + 18kΩ = 78kΩ
= 9V
= 78kΩ

31. Notation
How circuits are drawn and how they are referenced is an important part of being able to analyze a circuit.
Some standard industry notation follows:

32. Grounds
Ground: Is a point of reference or a common point in a circuit for making measurements.
COMMON TYPES:
Chassis ground
Common point of circuit is often the metal chassis of the piece of equipment.
Often connected to Earth Ground.
If a fault occurs within a circuit, the current is redirected to the earth.
Earth ground
Physically connected to the earth by a metal pipe or rod.

33. NOTATION Voltage Sources and Grounds
Three ways to sketch the same series dc circuit.

34. NOTATION Voltage Sources
Replacing the special notation for a dc voltage source with the standard symbol.
Replacing the notation for a negative dc supply with the standard notation.

35. NOTATION Single-Subscript Notation
If point b of the notation Vab is specified as ground with potential (zero volts), then a single-subscript notation can be used that provides the voltage at a point with respect to ground.
Defining the use of single-subscript notation for voltage levels.

36. NOTATION Double-Subscript Notation
The fact that voltage is across a variable and exists between two points has resulted in a double-subscript notation that defines the first subscript as the higher potential.
Defining the sign for double-subscript notation.

37. NOTATION Double-Subscript Notation
The double-subscript notation Vab specifies point a as the higher potential than point b.
If this is not the case, a negative sign must be associated with the magnitude of Vab.
In other words, the voltage Vab is the voltage at point a with respect to point b.
A particularly useful relationship can now be established that has extensive applications in the analysis of electronic circuits.
For the above notational standards, the following relationship exists:

38. NOTATION General Comments
Another way to view the voltage
The impact of positive and negative voltages on the total voltage drop.

39. NOTATION Examples

40. Voltage Divider Rule Application Using Notation
Given the circuit below, find the output voltage Vab across the R2 resistor.
= 75 Ω
= 25 Ω
Incidentally; how would you find the current (I) for this circuit using the voltage divider rule?

41. Example Problem 5 Find the output voltage Vab across the R2 resistor.
= 50 Ω
= 20 Ω
= 30 Ω

42. Example Problem 6
Use the Voltage Divider rule to find ES.

43. Example Problem 7 Find Va, Vb, Vc, Vd .
Incidentally, we could also find v2:
ET= V1+V2+V3
V2 = ET - (V1+V3) = 100V-10V-30V = 60V

44. Other Basic Circuit Components
Beyond the resistor we have discussed extensively, there are some other circuit components that you may encounter:
Switch
Fuse
Circuit Breaker

45. Switches The most basic circuit components is a switch.
The switch below is known as a single-pole, single-throw (SPST) switch. 45

46. Fuses
A fuse is a device that prevents excessive current to protect against overloads or possible fires.
A fuse literally “blown” and can not be reset and must be replaced. 46

47. Circuit breakers
A circuit breaker (CB), like a fuse, will prevent excessive current from damaging circuits.
However, a CB uses an electro-mechanical mechanism that opens a switch.
A big difference between a fuse and CB is that a “popped” CB can be reset. 47

48. Consolidated Schematic
Circuit Breaker
Ammeter
Voltmeter
Lamp
Fuse
Battery

49. QUESTIONS?