Electrical Circuits and Engineering EconomicsElectrical Circuits and

Engineering Economics

Electrical CircuitsElectrical Circuits

Interconnection of electrical components for the purpose of either generating and distributing electrical power; converting electrical power to some other useful form such as light, heat, or mechanical torque; or processing information contained in an electrical form (electrical signals)

Interconnection of electrical components for the purpose of either generating and distributing electrical power; converting electrical power to some other useful form such as light, heat, or mechanical torque; or processing information contained in an electrical form (electrical signals)

ClassificationClassification

Direct Current circuits DC Currents and voltages do not vary with time

Alternating Current circuits AC Currents and voltages vary sinusoidally with

time Steady state - when current/voltage time is

purely constant Transient circuit - When a switch is thrown

that turns a source on or off

Direct Current circuits DC Currents and voltages do not vary with time

Alternating Current circuits AC Currents and voltages vary sinusoidally with

time Steady state - when current/voltage time is

purely constant Transient circuit - When a switch is thrown

that turns a source on or off

Quantities Used in Electrical Circuits

Quantities Used in Electrical Circuits

QuantitySymb

olUnit

Defining Equation

Definition

Charge Q coulomb Q=∫Idt

Current I ampere I=dQ / dtTime rate of flow of

charge past a point in circuit

Voltage V volt V=dW / dQ

Energy per unit charge either gained or lost

through a circuit element

Energy W joule W= ∫VdQ =

∫Pdt

Power P watt P = dW / dt

= IVPower is the time rate

of energy flow

Circuit ComponentsCircuit Components

Resistors - Absorb energy and have a resistance value R measured in ohms I=V/R OR V=IR AMPERES=VOLTS/OHMS

Inductors - Store energy and have an inductance value L measured in henries V=L(dl/dt) VOLT=(AMPERES•HENRIES)/SECONDS

Capacitors - Store energy and have a capacitance value C measured in farads I=C(dV/dt) VOLT=(AMPERES•HENRIES)/SECONDS

Resistors - Absorb energy and have a resistance value R measured in ohms I=V/R OR V=IR AMPERES=VOLTS/OHMS

Inductors - Store energy and have an inductance value L measured in henries V=L(dl/dt) VOLT=(AMPERES•HENRIES)/SECONDS

Capacitors - Store energy and have a capacitance value C measured in farads I=C(dV/dt) VOLT=(AMPERES•HENRIES)/SECONDS

Sources of Electrical Energy

Sources of Electrical Energy

Independent of current and/or voltage values elsewhere in the circuit, or they can be dependent upon them

Page 443 (Figure 18.1) of the text shows both ideal and linear models of current and voltage sources

Independent of current and/or voltage values elsewhere in the circuit, or they can be dependent upon them

Page 443 (Figure 18.1) of the text shows both ideal and linear models of current and voltage sources

Kirchhoff’s Laws (Conservation of Energy)

Kirchhoff’s Laws (Conservation of Energy)

Kirchhoff’s Voltage Law (KVL) Sum of voltage

rises or drops around any closed path in an electrical circuit must be zero

∑VDROPS = 0

∑VRISES = 0 (around closed path)

Kirchhoff’s Voltage Law (KVL) Sum of voltage

rises or drops around any closed path in an electrical circuit must be zero

∑VDROPS = 0

∑VRISES = 0 (around closed path)

Kirchhoff’s Current Law (KCL) Flow of charges

either into (positive) or out of (negative) any node in a circuit must add zero

∑IIN = 0

∑IOUT = 0 (at node)

Kirchhoff’s Current Law (KCL) Flow of charges

either into (positive) or out of (negative) any node in a circuit must add zero

∑IIN = 0

∑IOUT = 0 (at node)

Ohm’s LawOhm’s Law

Statement of relationship between voltage across an electrical component and the current through the component

DC Circuits - resistors V = IR OR I = V/R

AC Circuits Resistors, capacitors, and inductors stated

in terms of component impedance Z V = IZ OR I = V/Z

Statement of relationship between voltage across an electrical component and the current through the component

DC Circuits - resistors V = IR OR I = V/R

AC Circuits Resistors, capacitors, and inductors stated

in terms of component impedance Z V = IZ OR I = V/Z

Reference Voltage Polarity and Current Direction

Reference Voltage Polarity and Current Direction

Arrow placed next to circuit component to show current direction

Polarity marks can be defined Current always flows from positive (+) to negative (-)

marks

Positive current value Current flows in reference direction Loss of energy or reduction in voltage from + to -

Negative current value Current flows opposite reference direction Gain of energy when moving through circuit from + to -

Arrow placed next to circuit component to show current direction

Polarity marks can be defined Current always flows from positive (+) to negative (-)

marks

Positive current value Current flows in reference direction Loss of energy or reduction in voltage from + to -

Negative current value Current flows opposite reference direction Gain of energy when moving through circuit from + to -

Reference Voltage Polarity and Current Direction

Reference Voltage Polarity and Current Direction

Voltage Drops Experienced when

moving through the circuit from the plus (+) polarity to the minus (-) polarity mark

Voltage Drops Experienced when

moving through the circuit from the plus (+) polarity to the minus (-) polarity mark

Voltage Rises Experienced when

moving through the circuit from the minus (-) polarity to the plus (+) polarity mark

Voltage Rises Experienced when

moving through the circuit from the minus (-) polarity to the plus (+) polarity mark

Circuit EquationsCircuit Equations

Current is assumed to have a positive value in reference direction and voltage is assumed to have a positive value as indicated by the polarity marks

For KVL circuit equation (Figure 18.2) Move around a closed circuit path in the circuit

and sum all the voltage rises and drops For ∑VRISES=0

VS - IR1 - IR2 - IR3 = 0 For ∑VDROPS=0

-Vs + IR1 + IR2 + IR3 = 0

Current is assumed to have a positive value in reference direction and voltage is assumed to have a positive value as indicated by the polarity marks

For KVL circuit equation (Figure 18.2) Move around a closed circuit path in the circuit

and sum all the voltage rises and drops For ∑VRISES=0

VS - IR1 - IR2 - IR3 = 0 For ∑VDROPS=0

-Vs + IR1 + IR2 + IR3 = 0

Circuit Equations Using Branch Currents

Circuit Equations Using Branch Currents

Figure 18.3 Unknown current with a reference

direction is at each branch Write two KVL equations, one around

each mesh - VS + I1R1 + I3R2 + I1R3 = 0 - I3R2 + I2R4 +I2R5 + I2R6 = 0

Write one KCL equation at circuit node a I1 - I2 - I3 = 0

Figure 18.3 Unknown current with a reference

direction is at each branch Write two KVL equations, one around

each mesh - VS + I1R1 + I3R2 + I1R3 = 0 - I3R2 + I2R4 +I2R5 + I2R6 = 0

Write one KCL equation at circuit node a I1 - I2 - I3 = 0

Circuit Equations Using Branch Currents

Circuit Equations Using Branch Currents

Use three equations to solve for I1, I2, and I3 Current I1 is:

|VS 0 R2|

|0 R4 + R5 + R6 -R2||0 -1 -1|

I1=______________________________________________

|R1 + R3 0 R2|

| 0 R4 + R5 + R6 -R2| | 1 -1 -1|

Use three equations to solve for I1, I2, and I3 Current I1 is:

|VS 0 R2|

|0 R4 + R5 + R6 -R2||0 -1 -1|

I1=______________________________________________

|R1 + R3 0 R2|

| 0 R4 + R5 + R6 -R2| | 1 -1 -1|

Circuit Equations UsingMesh Currents

Circuit Equations UsingMesh Currents

Simplification in writing the circuit equations occurs using mesh currents

I3 = I1 - I2Using Figure 18.3

Current through R1 and R3 is I1Current through R4, R5, and R6 is I2Current through R2 is I1 - I2

Simplification in writing the circuit equations occurs using mesh currents

I3 = I1 - I2Using Figure 18.3

Current through R1 and R3 is I1Current through R4, R5, and R6 is I2Current through R2 is I1 - I2

Circuit Equations UsingMesh Currents

Circuit Equations UsingMesh Currents

Write two KVL equations - VS + I1(R1 + R2 + R3) - I2R2 = 0 -I1R2 + I2(R2 + R4 + R5 + R6) = 0

Two equations can be solved for I1 and I2 Current I1 is equivalent to that of before

|VS -R2 |

|0 R2 + R4 + R5 +R6|I=____________________________________________________

|R1 + R2 + R3 -R2 |

| -R2 R2 + R4 + R5 + R6|

Write two KVL equations - VS + I1(R1 + R2 + R3) - I2R2 = 0 -I1R2 + I2(R2 + R4 + R5 + R6) = 0

Two equations can be solved for I1 and I2 Current I1 is equivalent to that of before

|VS -R2 |

|0 R2 + R4 + R5 +R6|I=____________________________________________________

|R1 + R2 + R3 -R2 |

| -R2 R2 + R4 + R5 + R6|

Circuit SimplificationCircuit Simplification

Possible to simplify a circuit by combining components of same kind that are grouped together in the circuit

Formulas for combining R’s, L’s and C’s to singles are found using Kirchhoff’s laws

Figure 18.5 with two inductors

Possible to simplify a circuit by combining components of same kind that are grouped together in the circuit

Formulas for combining R’s, L’s and C’s to singles are found using Kirchhoff’s laws

Figure 18.5 with two inductors

Circuit Components in Series and Parallel

Circuit Components in Series and Parallel

Component

Series Parallel

R Req = R1 + R2 + … + RN

1/Req = (1/R1) + (1/R2) + … + (1/RN)

L Leq = L1 + L2 + … + LN

1/Leq = (1/L1) + (1/L2) + … + (1/LN)

C1/Ceq = (1/C1) + (1/C2) + …

+ (1/CN)Ceq = C1 + C2 + … + CN

DC CircuitsDC Circuits

Only crucial components are resistors

InductorAppears as zero resistance connection

Short circuit

CapacitorAppears as infinite resistance

Open circuit

Only crucial components are resistors

InductorAppears as zero resistance connection

Short circuit

CapacitorAppears as infinite resistance

Open circuit

DC Circuit ComponentsDC Circuit Components

Component

Impedance Current PowerEnergy Stored

Resistor R I = V/RP = I2R =

V2/RNone

InductorZero

(short circuit)

Unconstrained

None dissipated

WL = (1/2)LI2

CapacitorInfinite(open circuit)

ZeroNone

dissipatedWC = (1/2)CV2

Engineering EconomicsEngineering Economics

Best design requires the engineer to anticipate the good and bad outcomes

Outcomes evaluated in dollars Good is defined as

positive monetary value

Best design requires the engineer to anticipate the good and bad outcomes

Outcomes evaluated in dollars Good is defined as

positive monetary value

Value and InterestValue and Interest

Value is not synonymous with amount

Value of an amount depends on when the amount is received and spent

InterestDifference between anticipated amount

and its current valueFrequently expressed as a time rate

Value is not synonymous with amount

Value of an amount depends on when the amount is received and spent

InterestDifference between anticipated amount

and its current valueFrequently expressed as a time rate

Interest ExampleInterest Example

What amount must be paid in two years to settle a current debt of $1,000 if the interest rate is 6%? Value after 1 year =

1000 + 1000 * 0.06 1000(1 + 0.06) $1060

Value after 2 years = 1060 + 1060 * 0.06 1000(1 + 0.06)2

$1124

$1124 must be paid in two years to settle the debt

What amount must be paid in two years to settle a current debt of $1,000 if the interest rate is 6%? Value after 1 year =

1000 + 1000 * 0.06 1000(1 + 0.06) $1060

Value after 2 years = 1060 + 1060 * 0.06 1000(1 + 0.06)2

$1124

$1124 must be paid in two years to settle the debt

Cash Flow DiagramsCash Flow Diagrams

An aid to analysis and communicationHorizontal

Time axisVertical

Dollar amounts

Draw a cash flow diagram for every engineering economy problem that involves amounts at different times

An aid to analysis and communicationHorizontal

Time axisVertical

Dollar amounts

Draw a cash flow diagram for every engineering economy problem that involves amounts at different times

Cash Flow PatternsFigure 18.7

Cash Flow PatternsFigure 18.7

P-pattern Single amount P occurring at the beginning of n years

P frequently represents “present” amounts

F-pattern Single amount F occurring at the end of n years

F frequently represents “future” amounts

A-pattern Equal amounts A occurring at the ends of each n years

A frequently used to represent “annual” amounts

G-pattern End-of-year amounts increasing by an equal annual

gradient G

P-pattern Single amount P occurring at the beginning of n years

P frequently represents “present” amounts

F-pattern Single amount F occurring at the end of n years

F frequently represents “future” amounts

A-pattern Equal amounts A occurring at the ends of each n years

A frequently used to represent “annual” amounts

G-pattern End-of-year amounts increasing by an equal annual

gradient G

Equivalence of Cash Flow Patterns

Equivalence of Cash Flow Patterns

Two cash flow patterns said to be equivalent if they have the same value

Most computational effort directed at finding cash flow pattern equivalent to a combination of other patterns

i = interestn = number of periods

Two cash flow patterns said to be equivalent if they have the same value

Most computational effort directed at finding cash flow pattern equivalent to a combination of other patterns

i = interestn = number of periods

Formulas for Interest Factors

Formulas for Interest Factors

Symbol To Find Given Formula

(F/P)in P F (1+i)n

(P/F)in P F

1_________ (1+i)n

(A/P)in A Pi(1+i)n

______________

(1+i)n - 1

Formulas for Interest Factors

Formulas for Interest Factors

Symbol To Find Given Formula

(P/A)in P A

(1+i)n - 1__________

i(1+i)

(A/F)in A F

i_________ (1+i)n

- 1

(F/A)in F A(1+i)n - 1______________

i

Formulas for Interest Factors

Formulas for Interest Factors

SymbolTo

FindGiven Formula

(A/G)in A G (1/i) - (n/(1+i)n - 1)

(F/G)in F G

(1/i) * [(((1+i)n - 1) / (i))-

1]

(P/G)in P G (1/i) * [(((1+i)n - 1) / (i(1+i)n)) - (n / (1+i)n)]