electric circuits ee 202 university of hail professor / mohamed a h eleiwa

Electric Circuits EE 202

University of Hail

Professor / Mohamed A H Eleiwa

Summary

Very large and very small numbers are represented with scientific and engineering notation.

Scientific and Engineering Notation

47,000,000 = 4.7 x 107 (Scientific Notation)

= 47 x 106 (Engineering Notation)

Summary

0.000 027 = 2.7 x 10-5 (Scientific Notation)

= 27 x 10-6 (Engineering Notation)

0.605 = 6.05 x 10-1 (Scientific Notation)

= 605 x 10-3 (Engineering Notation)

Scientific and Engineering Notation

Summary

Most scientific calculators can be placed in a mode that will automatically convert any decimal number entered into scientific notation or engineering notation.

Metric Conversions

Numbers in scientific notation can be entered in a scientific calculator using the EE key.

Summary

SI Fundamental Units

Length Mass

Time

Electric current

Temperature

Luminous intensity

Amount of substance

Quantity Unit Symbol

Meter m

Kilogram kg

Second s

Ampere A

Kelvin K

Candela cd

Mole mol

Summary

Some Important Electrical Units

Except for current, all electrical and magnetic units are derived from the fundamental units. Current is a fundamental unit.

CurrentCharge

Voltage

Resistance

Ampere A

Coulomb C

Volt V

Ohm WWatt W

Quantity Unit Symbol

Power

These derived units are based on fundamental units from the meter-kilogram-second system, hence are called mks units.

Summary

Engineering Metric Prefixes

peta

tera

giga

mega

kilo

1015

1012

109

106

103

P

T

G

M

k

Can you name the prefixes and their meaning?

Summary

Engineering Metric Prefixes

10-3

10-6

10-9

10-12

10-15

milli

micro

nano

pico

femto

m

m

n

p

f

Can you name the prefixes and their meaning?

Summary

When converting from a larger unit to a smaller unit, move the decimal point to the right. Remember, a smaller unit means the number must be larger.

Metric Conversions

0.47 MW = 470 kW

Larger number

Smaller unit

Summary

When converting from a smaller unit to a larger unit, move the decimal point to the left. Remember, a larger unit means the number must be smaller.

Metric Conversions

10,000 pF = 0.01 mF

Smaller number

Larger unit

Summary

When adding or subtracting numbers with a metric prefix, convert them to the same prefix first.

Metric Arithmetic

10,000 W + 22 kW =

10,000 W + 22,000 W = 32,000 W

Alternatively,

10 kW + 22 kW = 32 kW

Summary

When adding or subtracting numbers with a metric prefix, convert them to the same prefix first.

Metric Arithmetic

200 mA + 1.0 mA =

200 mA + 1,000 mA = 12,000 mA

Alternatively,

0.200 mA + 1.0 mA = 1.2 mA

Summary

WV

Q

One volt is the potential difference (voltage) between two points when one joule of energy is used to move one coulomb of charge from one point to the other.

Voltage

The defining equation for voltage is

Summary

Current (I) is the amount of charge (Q) that flows past a point in a unit of time (t). The defining equation is:

QI

t

One ampere is a number of electrons having a total charge of 1 C moving through a given cross section in 1 s.

0.4 AWhat is the current if 2 C passes a point in 5 s?

Current

Summary

Resistance is the opposition to current.

One ohm (1 W) is the resistance if one ampere (1 A) is in a material when one volt (1 V) is applied.

Conductance is the reciprocal of resistance.

1G

R

Components designed to have a specific amount of resistance are called resistors. Color bands

Resistance material(carbon composition)

Insulation coating

Leads

Resistance

Summary

Color

Black

Brown

Red

Orange

Yellow

Green

Blue

Violet

Gray

White

Gold

Silver

No band

0

1

2

3

4

5

6

7

8

9

±5%

±10%

Digit

±20%

100

101

102

103

104

105

106

107

108

109

10-1

10-2

Multiplier

1% (five band)

5% (four band)

Tolerance

2% (five band)

10% (four band)

Resistance value, first three bands:

First band – 1st digit

Second band – 2nd digit

*Third band – Multiplier (number of zeros following second digit)

Fourth band - tolerance

* For resistance values less than 10 W , the third band is either gold or silver. Gold is for a multiplier of 0.1 and silver is for a multiplier of 0.01.

Resistance color-code

Summary

What is the resistance and tolerance of each of the four-band resistors?

5.1 k W ± 5%

820 k W ± 5%

47 W ± 10%

1.0 W ± 5%

Summary

• Two or three digits, and one of the letters R, K, or M are used to identify a resistance value.

• The letter is used to indicate the multiplier, and its position is used to indicate decimal point position.

Alphanumeric Labeling

Summary

Variable resistors include the potentiometer and rheostat. The center terminal of a variable resistor is connected to the wiper.

13

2

Resistiveelement

Wiper

Shaft

Variable resistors

R

Variable resistor (potentiometer)

R

Variable resistor (rheostat)

To connect a potentiometer as a rheostat, one of the outside terminals is connected to the wiper.

Summary

The electric circuit

Circuits are described pictorially with schematics. For example, the flashlight can be represented by

Battery (2 cells)

Switch

Lamp

Summary

The DMM (Digital Multimeter) is an important multipurpose instrument which can measure voltage, current, and resistance. Many include other measurement options.

The DMM

V

Hz

10 A

40 mA

OFF

mV

A

V

H

H

V H

COM

VW

Summary

An analog multimeter is also called a VOM (volt-ohm-milliammeter). Analog meters measure voltage, current, and resistance. The user must choose the range and read the proper scale.

Analog meters

Photo courtesy of Triplett Corporation

Summary

Voltage is

Review of V, I, and R

the amount of energy per charge available to

move electrons from one point to another in a circuit and is measured in volts.

Current is the rate of charge flow and is measured in

amperes.

Resistance is the opposition to current and is measured

in ohms.

Summary

The most important fundamental law in electronics is Ohm’s law, which relates voltage, current, and resistance.

Georg Simon Ohm (1787-1854) formulated the equation that bears his name:

VI

R

What is the current in a circuit with a 12 V source if the resistance is 10 W? 1.2 A

Ohm’s law

Summary

Ohm’s law

If you need to solve for voltage, Ohm’s law is:

What is the voltage across a 680 W resistor if the current is 26.5 mA? 18 V

V IR

Summary

Ohm’s law

If you need to solve for resistance, Ohm’s law is:V

RI

115 V

V

1 s

1 s

40 m A

1 0 A

C O M

Ra ng eAuto ra ng eTo uc h /Ho ld

Fused

O FF V

V

Hz

m V

A

What is the (hot) resistance of the bulb? 132 W

In electrical work, the rate energy is dissipated can be determined from any of three forms of the power formula.

Summary

Energy and Power

2P I R P VI2V

PR

Together, the three forms are called Watt’s law.

Summary

What power is dissipated in a 27 W resistor is the current is 0.135 A?

2

2(0.135 A) 27

0.49 W

P I R

W

Given that you know the resistance and current, substitute the values into P =I 2R.

Energy and Power

Summary

All circuits have three common attributes. These are:

Summary

Series circuits

1. A source of voltage.

2. A load.

3. A complete path.

R1

VS R2

R3

+

SummarySummary

Series circuits

A series circuit is one that has only one current path.

VS VS VS

R1

R1

R1

R2 R2

R2

R3

R3

R3

SummarySummary

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

SummarySummary

Series circuits

The total resistance of resistors in series is

the sum of the individual resistors.

4.38 kW

For example, the resistors in a series circuit are 680 W, 1.5 kW, and 2.2 kW. What is the total resistance?

R

R

2

3

R 1

V S

6 8 0 W

2 .2 k W

1 .5 k W1 2 V

Series circuit

Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a series circuit. Continuing with the previous example, complete the parameters listed in the Table.

I1= R1= 0.68 kW V1= P1=

I2= R2= 1.50 kW V2= P2=

I3= R3= 2.20 kW V3= P3=

IT= RT= 4.38 kW VS= 12 V PT= 2.74 mA

2.74 mA

2.74 mA

2.74 mA 1.86 V

4.11 V

6.03 V

5.1 mW

11.3 mW

16.5 mW

32.9 mW

SummarySummarySummaryR

R

2

3

R 1

V S

6 8 0 W

2 .2 k W

1 .5 k W1 2 V

SummarySummary

Voltage sources in series

Voltage sources in series add algebraically. For example, the total voltage of the sources shown is

+

+

+

9 V

9 V

9 V

27 V

9 VWhat is the total voltage if one battery is accidentally reversed?

+

+

+

9 V

9 V

9 V

SummarySummary

Kirchhoff’s voltage law

The sum of all the voltage drops around a single closed path in a circuit is equal to the total source voltage in that closed path.

KVL applies to all circuits, but you must apply it to only one closed path. In a series circuit, this is (of course) the entire circuit.

Kirchhoff’s voltage law (KVL) is generally stated as:

A mathematical shorthand way of writing KVL is1

0n

ii

V

Kirchhoff’s voltage law

Notice in the series example given earlier that the sum of the resistor voltages is equal to the source voltage.

I1= R1= 0.68 kW V1= P1=

I2= R2= 1.50 kW V2= P2=

I3= R3= 2.20 kW V3= P3=

IT= RT= 4.38 kW VS= 12 V PT= 2.74 mA

2.74 mA

2.74 mA

2.74 mA 1.86 V

4.11 V

6.03 V

5.1 mW

11.3 mW

16.5 mW

32.9 mW

R

R

2

3

R 1

V S

6 8 0 W

2 .2 k W

1 .5 k W1 2 V

SummarySummary

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 k . WApplying the voltage divider formula:

22 S

T

10 k20 V

25 k

RV V

R

W W

R1

VS R2+10 kW

15 kW

20 V

Voltage divider

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

8.0 V

SummarySummary

Resistors in parallel

Resistors that are connected to the same two points are said to be in parallel.

R1 R2 R3 R4

A

B

SummarySummary

Parallel circuits

A parallel circuit is identified by the fact that it hasmore than one current path (branch) connected to a common voltage source.

VS

+ R1 R2 R3 R4

SummarySummary

Parallel circuit rule for voltage

Because all components are connected across the same voltage source, the voltage across each is the same.

+ 5 .0 V+-

+ 5 .0 V+-

+ 5 .0 V+-

+ 5 .0 V+-

R 2 R 3R 1V S

6 8 0 W 2 .2 k W1 .5 k W+ 5 .0 V

For example, the source voltage is 5.0 V. What will a volt- meter read if it is placed across each of the resistors?

SummarySummary

Parallel circuit rule for resistance

The total resistance of resistors in parallel is

the reciprocal of the sum of the reciprocals of the individual resistors.

386 W

For example, the resistors in a parallel circuit are 680 W, 1.5 kW, and 2.2 kW. What is the total resistance?

VS

+ R1 R2 R3

680 W 1.5 kW 2.2 kW

SummarySummary

Special case for resistance of two parallel resistors

The resistance of two parallel resistors can be found by

either:

R1 R2

T

1 2

11 1

R

R R

or 1 2T

1 2

R RR

R R

18.2 kW

What is the total resistance if R1 = 27 kW and R2 = 56 kW?

SummarySummary

Parallel circuit

Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a parallel circuit.

Continuing with the previous example, complete the parameters listed in the Table.

I1= R1= 0.68 kW V1= P1=

I2= R2= 1.50 kW V2= P2=

I3= R3= 2.20 kW V3= P3=

IT= RT= 386 W VS= 5.0 V PT=

5.0 V

5.0 V

5.0 V

13.0 mA

2.3 mA

3.3 mA

7.4 mA 36.8 mW

16.7 mW

11.4 mW

64.8 mW

VS

+ R1 R2 R3

680 W 1.5 kW 2.2 kW

SummarySummary

Kirchhoff’s current law

Kirchhoff’s current law (KCL) is generally stated as:

The sum of the currents entering a node is equal to the sum of the currents leaving the node.

Notice in the previous example that the current from the source is equal to the sum of the branch currents.

I1= R1= 0.68 kW V1= P1=

I2= R2= 1.50 kW V2= P2=

I3= R3= 2.20 kW V3= P3=

IT= RT= 386 W VS= 5.0 V PT=

5.0 V

5.0 V

5.0 V

13.0 mA

2.3 mA

3.3 mA

7.4 mA 36.8 mW

16.7 mW

11.4 mW

64.8 mW

SummarySummary

Current divider

When current enters a node (junction) it divides into currents with values that are inversely proportional to the resistance values.

Notice the subscripts. The resistor in the numerator is not the same as the one for which current is found.

21 T

1 2

RI I

R R

and 12 T

1 2

RI I

R R

The most widely used formula for the current divider is the two-resistor equation. For resistors R1 and R2,

SummarySummary

Current divider

21 T

1 2

4.7 k8.0 mA =

6.9 k

RI I

R R

W W

Assume that R1is a 2.2 kW resistor that is in parallel with R2, which is 4.7 kW. If the total current into the resistors is 8.0 mA, what is the current in each resistor?

5.45 mA

12 T

1 2

2.2 k8.0 mA =

6.9 k

RI I

R R

W W 2.55 mA

Notice that the larger resistor has the smaller current.

SummarySummary

Power in parallel circuits

Power in each resistor can be calculated with any of the standard power formulas. Most of the time, the voltage is

known, so the equation 2VP

R is most convenient.

As in the series case, the total power is the sum of the powers dissipated in each resistor.

1.04 W

What is the total power if 10 V is applied to the parallel combination of R1 = 270 W and R2 = 150 W?

SummarySummary

Assume there are 8 resistive wires that form a rear window defroster for an automobile.

(a) If the defroster dissipates 90 W when connected to a 12.6 V source, what power is dissipated by each resistive wire?

(b) What is the total resistance of the defroster?

(a) Each of the 8 wires will dissipate 1/8 of the total power or 90 W

8 wire11. W

s25

22 12.6 V

90 W1.76

VR

P W(b) The total resistance is

What is the resistance of each wire? 1.76 W x 8 = 14.1 W

SummarySummary

Most practical circuits have various combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components.

Combination circuits

An important analysis method is to form an equivalent circuit. An equivalent circuit is one that has characteristics that are electrically the same as another circuit but is generally simpler.

SummarySummary

Kirchhoff’s voltage law and Kirchhoff’s current law can be applied to any circuit, including combination circuits.

R5100

R3330

R 2470

R1270 W W

W

W

V S5 .0 V

R4

100

R6

100 Sta rt/Finish

W

WR5100

R3330

R 2470

R1270 W W

W

W

V S5 .0 V

R4

100

R6

100 Sta rt/Finish

W

W

So will this path!

For example, applying KVL, the path shown will have a sum of 0 V.

SummarySummary

I

+

+

26.5 mA

I

+18.5 mA

I

+8.0 mA

R5100 W

R3330 W

R2470 W

R1270 W

VS5.0 V

R4

100 W

R6

100 W

A- -

-

Kirchoff’s current law can also be applied to the same circuit. What are the readings for node A?

SummarySummary

Tabulating current, resistance, voltage and power is a useful way to summarize parameters. Solve for the unknown quantities in the circuit shown.

I1= R1= 270 W V1= P1=

I2= R2= 330 W V2= P2=

I3= R3= 470 W V3= P3=

IT= RT= VS= 10 V PT=

4.18 V

4.18 V

5.82 V

21.6 mA

8.9 mA

12.7 mA

21.6 mA 126 mW

53.1 mW

37.2 mW

216 mW

R1

R3

470 W

270 W

R2

330 W

VS +10 V

464 W

Combination circuits

SummarySummary

Kirchhoff’s laws can be applied as a check on the answer.

I1= R1= 270 W V1= P1=

I2= R2= 330 W V2= P2=

I3= R3= 470 W V3= P3=

IT= RT= VS= 10 V PT=

4.18 V

4.18 V

5.82 V

21.6 mA

8.9 mA

12.7 mA

21.6 mA 126 mW

53.1 mW

37.2 mW

216 mW464 W

R1

R3

470 W

270 W

R2

330 W

VS +10 V

equal to the sum of the branch currents in R2 and R3.Notice that the current in R1 is

The sum of the voltages around the outside loop is zero.

SummarySummary

Thevenin’s theorem states that any two-terminal, resistive circuit can be replaced with a simple equivalent circuit when viewed from two output terminals. The equivalent circuit is:

Thevenin’s theorem

V T H

R T H

SummarySummary

V T H

R T H

VTH is defined as


RTH is defined as

the open circuit voltage between the two output terminals of a circuit.

the total resistance appearing between the two output terminals when all sources have been replaced by their internal resistances.

SummarySummary


R

R

1

R 2R 2 L

V SV S

1 2 V1 0 kW

6 8 kW2 7 kW

Output terminals

What is the Thevenin voltage for the circuit? 8.76 V

What is the Thevenin resistance for the circuit? 7.30 kW

Remember, the load resistor has no affect on the Thevenin parameters.

SummarySummary

Maximum power transfer

The maximum power is transferred from a source to a load when the load resistance is equal to the internal source resistance.

The maximum power transfer theorem assumes the source voltage and resistance are fixed.

RS

RL

VS +

SummarySummary

Maximum power transfer

What is the power delivered to the matching load?

The voltage to the load is 5.0 V. The power delivered is

RS

RL

VS + 50 W

50 W10 V

22

LL

5.0 V= 0.5 W

50

VP

R

W

SummarySummary

Superposition theoremThe superposition theorem is a way to determine currents and voltages in a linear circuit that has multiple sources by taking one source at a time and algebraically summing the results.

What does the ammeter read for I2? (See next slide for the method and the answer).

+-

-

+

-

+

R 1 R 3

R 2

I 2

V S 2V S 1

1 2 V

2 .7 kW 6 .8 kW

6 .8 kW

1 8 V

SummarySummary

6.10 kW

What does the ammeter read for I2?

1.97 mA 0.98 mA

8.73 kW 2.06 mA

+-

-

+

-

+

R 1 R 3

R 2

I 2

V S 2V S 1

1 2 V

2 .7 kW 6 .8 kW

6 .8 kW

1 8 V

0.58 mA

1.56 mA

Source 1: RT(S1)= I1= I2= Source 2: RT(S2)= I3= I2= Both sources I2=

Set up a table of pertinent information and solve for each quantity listed:

The total current is the algebraic sum.

+ -

-

+

R1 R3

R2

I2VS1

12 V

2.7 kW 6.8 kW

6.8 kW

+ -

-

+

R1 R3

R2

I2

VS2

2.7 kW 6.8 kW

6.8 kW

18 V+-

-

+

-

+

R 1 R 3

R 2

I 2

V S 2V S 1

1 2 V

2 .7 kW 6 .8 kW

6 .8 kW

1 8 V1.56 mA

Node Voltage Method (Nodal Analysis)

Summary

The Basic Capacitor

Capacitors are one of the fundamental passive components. In its most basic form, it is composed of two conductive plates separated by an insulating dielectric.

The ability to store charge is the definition of capacitance.

Dielectric

Conductors

Summary

Die le c tric

Pla te sLe a d s

Ele c tro ns

BA

-

-

--

+

+

+

+

-

-

+

+

+

+

-

Initially uncharged

+ -BA

-

-

-

-

-

-

-

+

+

+

--

-

-

- - - - - -

-

---

+

+

+

+

Charging

+ -BA

V S

+

+

+++++++++

-

-

---------

Fully charged

BA

VS

-+

-+

-+-+-+-+-+-+-+-+-+

Source removed

The charging process…

A capacitor with stored charge can act as a temporary battery.

The Basic Capacitor

Capacitance is the ratio of charge to voltage

QC

V

Rearranging, the amount of charge on a capacitor is determined by the size of the capacitor (C) and the voltage (V).

Q CV

If a 22 mF capacitor is connected to a 10 V source, the charge is 220 mC

Capacitance

A capacitor stores energy in the form of an electric field that is established by the opposite charges on the two plates. The energy of a charged capacitor is given by the equation

Capacitance

2

2

1CVW

where

W = the energy in joulesC = the capacitance in faradsV = the voltage in volts

The capacitance of a capacitor depends on three physical characteristics.

Summary

128.85 10 F/m r AC

d

-

C is directly proportional to

and the plate area.

the relative dielectric constant

C is inversely proportional to

the distance between the plates

Capacitance

128.85 10 F/m r AC

d

-

Summary

Find the capacitance of a 4.0 cm diameter sensor immersed in oil if the plates are separated by 0.25 mm.

The plate area is

The distance between the plates is

Capacitance

4.0 for oilr

3 2

123

4.0 1.26 10 m 8.85 10 F/m

0.25 10 mC

--

-

30.25 10 m-

178 pF

2 2 3 2π 0.02 m 1.26 10 mA r -

Summary

Series capacitors

When capacitors are connected in series, the total capacitance is smaller than the smallest one. The general equation for capacitors in series is

T

1 2 3 T

11 1 1 1

...C

C C C C

The total capacitance of two capacitors is

T

1 2

11 1

C

C C

…or you can use the product-over-sum rule

Summary

Series capacitors

If a 0.001 mF capacitor is connected in series with an 800 pF capacitor, the total capacitance is444 pF

0 .001 µ F 800 pF

C 1 C 2

Summary

Parallel capacitors

When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors. The general equation for capacitors in parallel is

T 1 2 3 ... nC C C C C

1800 pF

If a 0.001 mF capacitor is connected in parallel with an 800 pF capacitor, the total capacitance is

0 .001 µ F 800 pF

C 1 C 2

Summary

The Basic Inductor

The effect of inductance is greatly magnified by adding turns and winding them on a magnetic material. Large inductors and transformers are wound on a core to increase the inductance.

Magnetic core

One henry is the inductance of a coil when a current, changing at a rate of one ampere per second, induces one volt across the coil. Most coils are much smaller than 1 H.

Summary

Factors affecting inductance

Four factors affect the amount of inductance for a coil. The equation for the inductance of a coil is

2N AL

l

where L = inductance in henries N = number of turns of wire m = permeability in H/m (same as Wb/At-m) l = coil length on meters

Summary

What is the inductance of a 2 cm long, 150 turn coil wrapped on an low carbon steel core that is 0.5 cm diameter? The permeability of low carbon steel is 2.5 x10-4 H/m (Wb/At-m).

22 5 2π π 0.0025 m 7.85 10 mA r - 2N A

Ll

22 mH

2 4 5 2150 t 2.5 10 Wb/At-m 7.85 10 m

0.02 m

- -

Summary

Series inductorsWhen inductors are connected in series, the total inductance is the sum of the individual inductors. The general equation for inductors in series is

2.18 mH

T 1 2 3 ... nL L L L L

If a 1.5 mH inductor is connected in series with an 680 mH inductor, the total inductance is

L 1 L 2

1 .5 m H 680 H

Summary

Parallel inductorsWhen inductors are connected in parallel, the total inductance is smaller than the smallest one. The general equation for inductors in parallel is

The total inductance of two inductors is

…or you can use the product-over-sum rule.

T

1 2 3 T

11 1 1 1

...L

L L L L

T

1 2

11 1

L

L L

Summary

Parallel inductors

If a 1.5 mH inductor is connected in parallel with an 680 mH inductor, the total inductance is 468 mH

L 1 L 2

1 .5 m H 680 H

OBJECTIVES• Become familiar with the characteristics of a sinusoidal waveform, including its general format, average value, and effective value.• Be able to determine the phase relationship between two sinusoidal waveforms of the same frequency.• Understand how to calculate the average and effective values of any waveform.• Become familiar with the use of instruments designed to measure ac quantities.

SINUSOIDAL ac VOLTAGE CHARACTERISTICS AND DEFINITIONS

Generation

• Sinusoidal ac voltages are available from a variety of sources. • The most common source is the typical home outlet, which provides an ac voltage that originates at a power plant. • Most power plants are fueled by water power, oil, gas, or nuclear fusion.

SINUSOIDAL ac VOLTAGE CHARACTERISTICS AND DEFINITIONS

Definitions

FIG. 13.3 Important parameters for a sinusoidal voltage.

AVERAGE POWER AND POWER FACTOR• Resistor• Inductor• Capacitor• Power Factor

AVERAGE POWER AND POWER FACTOR

FIG. 14.33 Purely resistive load with Fp = 1.

FIG. 14.34 Purely inductive load with Fp = 0.

COMPLEX NUMBERS• A complex number represents a point in a two-dimensional plane located with reference to two distinct axes. • This point can also determine a radius vector drawn from the origin to the point. • The horizontal axis is called the real axis, while the vertical axis is called the imaginary axis.

COMPLEX NUMBERS

FIG. 14.38 Defining the real and imaginary axes of a complex plane.


Chapter 1


POLAR FORM

• The format for the polar form is:

MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS

• Complex Conjugate• Reciprocal• Addition• Subtraction• Multiplication• Division

IMPEDANCE AND THE PHASOR DIAGRAMResistive Elements

FIG. 15.1 Resistive ac circuit.


Chapter 1



• In phasor form,


FIG. 15.2 Example 15.1.


Chapter 1


IMPEDANCE AND THE PHASOR DIAGRAMCapacitive Reactance

• We learned in Chapter 13 that for the pure capacitor in Fig. 15.13, the current leads the voltage by 90° and that the reactance of the capacitor XC is determined by 1/ψC.

• We have

SERIES CONFIGURATIONR-L-C

FIG. 15.36 Applying phasor notation to the circuit in Fig. 15.35.

VOLTAGE DIVIDER RULE


FREQUENCY RESPONSE FOR SERIES ac CIRCUITS

Series R-C ac Circuit

FIG. 15.47 Determining the frequency response of a series R-C circuit.

ILLUSTRATIVE EXAMPLES



FIG. 16.2 Network in Fig. 16.1 after assigning the block impedances.



FIG. 16.4 Network in Fig. 16.3 after assigning the block impedances.


Chapter 1


Δ-Y, Y-Δ CONVERSIONS

• The Δ-Y, Y-Δ (or p-T, T-p as defined in Section 8.12) conversions for ac circuits are not derived here since the development corresponds exactly with that for dc circuits.

FIG. 17.45 Δ-Y configuration.


FIG. 17.46 The T and π configurations.


FIG. 17.47 Converting the upper Δ of a bridge configuration to a Y.


Chapter 1



FIG. 17.48 The network in Fig. 17.47 following the substitution of the Y configuration.


FIG. 17.50 Converting a Δ configuration to a Y configuration.


FIG. 17.51 Substituting the Y configuration in Fig. 17.50 into the network in Fig. 17.49.


FIG. 17.52 Converting the Y configuration in Fig. 17.49 to a Δ.


FIG. 17.53 Substituting the Δ configuration in Fig. 17.54 into the network in Fig. 17.49.

electric circuits ee 202 university of hail professor / mohamed a h eleiwa

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