Basic Electrical Engineering Lecture # 04 Simple Resistive Circuits Course Instructor: Engr. Sana...

Basic Electrical Engineering

Lecture # 04Simple Resistive Circuits

Course Instructor:Engr. Sana Ziafat

Agenda

•Series / parallel (how voltage sources/ current sources in series/parallel may be combined).

•Resistance in series•Resistance in parallel•Voltage divider•Current divider

Voltages Sources and Current SourcesSeries and Parallel Combinations

•Two Types of Current:•DC—Direct Current—produced by solar cells and chemical cells (batteries)

•Current only flows in one direction.

•AC—Alternating Current

•Current flows back and forth (alternates)

•Found in homes•Generators produce AC current

Voltage Sources in SeriesDC voltage sources in series can be

combined and replaced with a single source.AC voltage sources in series can be combined

and replaced with a single source only if the angular frequency of operation are identical.

DC and AC voltage sources can be added together when calculating a total voltage.

AC voltage sources operating at different frequencies can be added together.The current flowing through one voltage

source must be equal to the current flowing through the other voltage source.

Example 1

Example 2

Or

Example 3

Or

Or

Example 4

Or

Or

DC and AC sources

•A 5V dc voltage source in series a 2V sin(10t) ac voltage source has a total combined voltage of 5V+2Vsin(10t). ▫However, we do not have a

symbol for a single voltage source that can replace the symbols for the dc and ac sources.

AC sources with Different •A 2V ac voltage source

operating at 10 rad/s in series with a 2V ac voltage source operating at 10.5 rad/s has a total voltage of 2Vsin(10t)+2Vsin(10t). ▫Again, there is not a symbol for

a single ac voltage source that can replace the symbols for the two ac sources operating at different frequencies.

Voltage Sources in Parallel

•Since the voltage sources share common nodes, the only time two or more voltage sources are allowed in parallel is when they have exactly the same voltage, polarity, and frequency of operation (if ac sources).▫The multiple voltage sources can be

replaced by a single source with the same voltage, polarity, and frequency of operation (if ac sources).

Example 5Allowed

Not Allowed

Example 6Allowed

Not Allowed

Current Sources in ParallelDC current sources in parallel can be

combined and replaced with a single source.AC current sources in parallel can be

combined and replaced with a single source only if the angular frequency of operation are identical.

DC and AC current sources in parallel can be added together when calculating a total current.

AC current sources operating at different frequencies can be added together.The voltage drop across one current source

must be equal to the voltage dropped across the other current sources in parallel.

Example 7

Example 8

Example 9

Or

DC and AC Current Sources

•A 5A dc current source in parallel a 2A sin(10t) ac current source means that they are contributing a total current of 5A+2Asin(10t) at that node. ▫However, we do not have a symbol for a

single current source that can replace the symbols for the dc and ac sources.

AC Sources with Different •A 2A ac current source operating at 10

rad/s in parallel with a 2V ac current operating at 10.5 rad/s means that they are contributing a total current of 2Asin(10t)+2Asin(10t) at that node. ▫Again, there is not a symbol for a single ac

current source that can replace the symbols for the two ac sources operating at different frequencies.

Current Sources in Series•Since components in series must have the

same current flowing through each component, the only time two or more currents sources are allowed in series is when they have exactly the same magnitude current, the current is flowing in the same direction, and frequency of operation (if ac sources).▫The multiple current sources in series can be

replaced by a single source with the same magnitude, direction of current flow, and frequency of operation (if ac sources).

Example 10

Allowed Not Allowed

SummaryVoltage sources in series can be added.Current sources in parallel can be added.Only in the case where the magnitude, polarity,

and frequency of operation are identical can multiple voltage sources be in parallel.They can be replaced with a single voltage source

of the same magnitude, polarity, and frequency of operation.

Only in the case where the magnitude, direction of current flow, and frequency of operation are identical can multiple currents sources be in series.They can be replaced with a single current source

of the same magnitude, direction of current flow, and frequency of operation.

Batteries in Series and Parallel

•In series—The voltage is increased.

•In parallel—No change in voltage; these batteries will last longer!

Circuit diagrams

• Minimum Three elements:-Source of electricity (battery)-Path or conductor on which electricity flows (wire)-Electrical resistor (lamp) which is any device that requires electricity to operate

• Pictorial way of showing circuitsThis is the Ammeter symbol

This is the Voltmeter symbol.

This is the resistor symbol.

This is the switch symbol.

This is the battery symbol.

Resistors in Series & Parallel

Series Circuit•Series circuit - has only one path through which the electricity can flow.

•When two circuit elements connect at single point

• In the above diagram, the electricity flows through both loads.• In series circuit current will remain same.

Series circuit rule for current:

Because there is only one path, the current everywhere is the same.

R 2

R 1

V S

+ _+ _

++ __

2.0 mA

For example, the reading on the first ammeter is 2.0 mA, What do the other meters read?

2.0 mA

2.0 mA2.0 mA

Equivalent Resistance:

We know the following for series resistors:

. . .

. . .R 1 R 2

R eq R N

Figure : Resistors in series.

Req = R1 + R2 + . . . + RN

1Can you prove HOW????

Equivalent Resistance•For the case of series circuit equivalent resistance is larger than largest resistance in a series connection.

Parallel Circuit•Parallel circuit -When two circuit elements connect at single Node pair.

• In parallel circuit voltage will remain same will remain same across their terminals.

Parallel Circuits

•A parallel circuit has multiple paths through which the electricity can flow.

•In a parallel circuit, the current though one path may be different than the current through the other path.


We know the following for parallel resistors:

. . .

. . .

R e q R 1 R 2 R N

Figure : Resistors in parallel.

Neq RRRR

1...

111

21


For the special case of two resistors in parallel:

R e q R 1 R 2

Figure: Two resistors in parallel.

21

21

RR

RRReq

Equivalent Resistance: Resistors in combination.

By combination we mean we have a mix of series and Parallel. This is illustrated below.

R 1

R 2

R 3

R 4 R 5R e q

Figure : Resistors In Series – Parallel Combination

To find the equivalent resistance we usually start atthe output of the circuit and work back to the input.


R 1

R 2

R 3

R xR e q54

54

RR

RRRx

R 1

R 2 R yR e q 3RRR xy

Figure : Resistance reduction.


R 1

R ZR e qY

YZ RR

RRR

2

2

R e q1

RRR Zeq

Figure : Resistance reduction, final steps.


It is easier to work the previous problem using numbers than towork out a general expression. This is illustrated below.

Example : Given the circuit below. Find Req.

R e q6 3

8

1 0

1 0

Figure : Circuit


Example : Continued . We start at the right hand side

of the circuit and work to the left.

1 0

1 0

8

2 R e q

1 0

5 R e q

Figure: Reduction steps

15eqRAns:

Series - Parallel Circuits

Current and Resistance in Series Circuits•For the series circuit the same current

flows through both loads. •The loads can be added together to

calculate the total load. •Rtot = R1 + R2, where Rtot is the total

resistance, R1 is the resistance of one load, and R2 is the resistance of the other.

•The total load (resistance) in a series circuit with “n” loads is the sum of the resistance of the “n” objects. Rtot = R1 + R2 + … + Rn.

Total Voltage in a Series Circuit

•Ohm’s Law can be used to calculate the total voltage in a series circuit by calculating the sum of the voltage parts.

•V = V1 + V2, where V is the total voltage (battery voltage), V1 is the voltage at the first load, and V2 is the voltage at the other load.

Current in a Parallel Circuit

•The total current in a parallel circuit is the sum of the two parts.

•I = I1 + I2, where I is the total current, I1 is the current through one load, and I2 is the current through the other load.

I1 I2

Parallel Circuits

Resistance in Parallel Circuits•Using Ohm’s Law you can derive a

formula for the equivalent resistance of two resistors in parallel.

•I1 = V/R1•I2 = V/R2•I = I1 + I2 = V/R1 + V/R2 •= (VR2 + VR1)/R1R2 = V(R2+R1)/R1R2•Rtot = V/(V(R2+R1)/ R1R2 =

R1R2/(R1+R2)

Voltage Divider & Current Divider Circuits

SummarySummary

Voltage divider rule

The voltage drop across any given resistor in a series circuit is equal to the ratio of that resistor to the total resistance, multiplied by source voltage.

8 V

Assume R1 is twice the size of R2. What is the voltage across R1?

R 1

R 2R 2

1V SV S

1 2 V

SummarySummary

What is the voltage across R2?

The total resistance is 25 kApplying the voltage divider formula:

22 S

T

10 k20 V 8 V

25 k

RV V

R

R1

VS R2+10 k

15 k

20 V

Voltage divider

Notice that 40% of the source voltage is across R2, which represents 40% of the total resistance.

SummarySummary

Voltage divider

Voltage dividers can be set up for a variable output using a potentiometer. In the circuit shown, the output voltage is variable.

What is the largest output voltage available?

VS

VOUT

R2

R1

+

10 k

20 k

15 V

5.0 V

SummarySummary

Power in Series Circuits

Applying the voltage divider rule:

1

470 20 V 11.75 V

800 V

The power dissipated by each resistor is:

2

1

11.75 V0.29 W

470 P

Use the voltage divider rule to find V1 and V2. Then find the power in R1 and R2 and PT.

R1

VS R2+330

470

20 V

2

330 20 V 8.25 V

800 V

2

2

8.25 V0.21 W

330 P

PT = 0.5 W

}

Current Divider Rule

•Allows us to determine how the current flowing into a node is split between the various parallel resistors

Current Divider Rule•If current enters a parallel network with

a number of equal resistors, current will split equally between resistors

•In a parallel network, the smallest value resistor will have the largest current

•Most of the current will follow the path of least resistance▫Largest resistor will have the least current

Readings

•Chapter 3: 3.1, 3.2, 3.3, 3.4 (Electric Circuits)▫By James W. Nilson

Quiz

1. In a series circuit with more than one resistor, the current is

a. larger in larger resistors

b. smaller in larger resistors

c. always the same in all resistors

d. there is not enough information to say

Quiz

2. In a series circuit with more than one resistor, the voltage is

a. larger across larger resistors

b. smaller across larger resistors

c. always the same across all resistors


Quiz

3. If three equal resistors are in series, the total resistance is

a. one third the value of one resistor

b. the same as one resistor

c. three times the value of one resistor


Quiz

4. A series circuit cannot have

a. more than two resistors

b. more than one voltage source

c. more than one path

d. all of the above

Quiz

5. In a closed loop, the algebraic sum of all voltages (both sources and drops)

a. is 0

b. is equal to the smallest voltage in the loop

c. is equal to the largest voltage in the loop

d. depends on the source voltage

Quiz

6. The current in the 10 k resistor is

a. 0.5 mA

b. 2 mA

c. 2.4 mA

d. 10 mA

VS

R2

R1

+

2.0 k

10 k

24 V

Quiz

7. The output voltage from the voltage divider is

a. 2 V

b. 4 V

c. 12 V

d. 20 V

VS

VOUTR2

R1

+

2.0 k

10 k

24 V

Quiz

8. The smallest output voltage available from the voltage divider is

a. 0 V

b. 1.5 V

c. 5.0 V

d. 7.5 V

VS

VOUT

R2

R1

+

10 k

1 0 k

15 V

Quiz

9. The total power dissipated in a series circuit is equal to the

a. power in the largest resistor

b. power in the smallest resistor

c. average of the power in all resistors

d. sum of the power in all resistors

Quiz

10. The meaning of the voltage VAB is the voltage at

a. Point A with respect to ground

b. Point B with respect to ground

c. The average voltage between points A and B.

d. The voltage difference between points A and B.

Quiz

Answers:

1. c

2. a

3. c

4. c

5. a

6. b

7. b

8. a

9. d

10. d

Basic Electrical Engineering Lecture # 04 Simple Resistive Circuits Course Instructor: Engr. Sana...

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