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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