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Transcript of Electrical Circuits and Engineering Economics. Electrical Circuits F Interconnection of electrical...
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