Download Electrical Circuits and Engineering Economics

Electrical Circuits and
Engineering Economics
Electrical 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)
Classification
 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
Quantity
Symbol
Unit
Defining
Equation
Charge
Q
coulomb
Q=∫Idt
Current
I
ampere
Voltage
V
volt
Energy
W
joule
Power
P
watt
Definition
I=dQ / dt
Time rate of flow of
charge past a point in
circuit
V=dW / dQ
Energy per unit charge
either gained or lost
through a circuit element
W= ∫VdQ =
∫Pdt
P = dW / dt
= IV
Power is the time rate of
energy flow
Circuit 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
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
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 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 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
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 -
Reference Voltage Polarity and
Current Direction
 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
Circuit 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
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
 I1 - I2 - I3 = 0
a
Circuit Equations Using Branch
Currents
 Use three equations to solve for I1, I2, and I3
 Current I1 is:
|VS
|0
|0
0
R4 + R5 + R 6
-1
R2|
-R2|
-1|
I1=______________________________________________
|R1 + R3
0
R2|
|
0
R4 + R5 + R 6
-R2|
|
1
-1
-1|
Circuit Equations Using
Mesh Currents
 Simplification in writing the circuit
equations occurs using mesh currents
 I3 = I1 - I 2
 Using Figure 18.3
Current through R1 and R3 is I1
Current through R4, R5, and R6 is I2
Current through R2 is I1 - I2
Circuit Equations Using
Mesh 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
|0
-R2
|
R2 + R4 + R5 +R6|
I=____________________________________________________
|R1 + R2 + R3
-R2
|
|
-R2
R2 + R4 + R5 + R6|
Circuit 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
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)
C
1/Ceq = (1/C1) + (1/C2) + … +
(1/CN)
Ceq = C1 + C2 + … + CN
DC Circuits
 Only crucial components are resistors
 Inductor
Appears as zero resistance connection
Short circuit
 Capacitor
Appears as infinite resistance
Open circuit
DC Circuit Components
Component
Impedance
Current
Power
Energy Stored
None
Resistor
R
I = V/R
P = I2R =
V2/R
Inductor
Zero
(short circuit)
Unconstrained
None
dissipated
WL = (1/2)LI2
Capacitor
Infinite
(open circuit)
Zero
None
dissipated
WC = (1/2)CV2
Engineering Economics
 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 Interest
 Value is not synonymous with amount
 Value of an amount depends on when
the amount is received and spent
 Interest
Difference between anticipated amount
and its current value
Frequently expressed as a time rate
Interest 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
Cash Flow Diagrams
 An aid to analysis and communication
Horizontal
Time axis
Vertical
Dollar amounts
 Draw a cash flow diagram for every
engineering economy problem that
involves amounts at different times
Cash Flow Patterns
Figure 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
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 = interest
 n = number of periods
Formulas for Interest Factors
Symbol
To Find
Given
Formula
(F/P)in
P
F
(1+i)n
(P/F)in
(A/P)in
P
A
F
P
1
_________
(1+i)n
i(1+i)n
______________
(1+i)n - 1
Formulas for Interest Factors
Symbol
To Find
Given
Formula
(1+i)n - 1
(P/A)in
P
A
(A/F)in
A
F
(F/A)in
F
A
__________
i(1+i)
i
_________
(1+i)n - 1
(1+i)n - 1
______________
i
Formulas for Interest Factors
Symbol
To Find
Given
Formula
(A/G)in
A
G
(1/i) - (n/(1+i)n - 1)
G
(1/i) *
[(((1+i)n - 1) / (i))-1]
G
(1/i) * [(((1+i)n - 1) /
(i(1+i)n)) - (n / (1+i)n)]
(F/G)in
(P/G)in
F
P