Electrical Circuits and Engineering Economics

Electrical Circuits F 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 F Direct Current circuits F DC F Currents and voltages do not vary with time F Alternating Current circuits F AC F Currents and voltages vary sinusoidally with time F Steady state - when current/voltage time is purely constant F Transient circuit - When a switch is thrown that turns a source on or off F Direct Current circuits F DC F Currents and voltages do not vary with time F Alternating Current circuits F AC F Currents and voltages vary sinusoidally with time F Steady state - when current/voltage time is purely constant F Transient circuit - When a switch is thrown that turns a source on or off

Quantities Used in Electrical Circuits QuantitySymbolUnit Defining Equation Definition ChargeQcoulombQ= ∫Idt CurrentIampereI=dQ / dt Time rate of flow of charge past a point in circuit VoltageVvolt V=dW / dQ Energy per unit charge either gained or lost through a circuit element EnergyWjoule W= ∫VdQ = ∫Pdt PowerPwatt P = dW / dt = IV Power is the time rate of energy flow

Circuit Components F Resistors - Absorb energy and have a resistance value R measured in ohms F I=V/R OR V=IR F AMPERES=VOLTS/OHMS F Inductors - Store energy and have an inductance value L measured in henries F V=L(dl/dt) F VOLT=(AMPERESHENRIES)/SECONDS F Capacitors - Store energy and have a capacitance value C measured in farads F I=C(dV/dt) F VOLT=(AMPERESHENRIES)/SECONDS F Resistors - Absorb energy and have a resistance value R measured in ohms F I=V/R OR V=IR F AMPERES=VOLTS/OHMS F Inductors - Store energy and have an inductance value L measured in henries F V=L(dl/dt) F VOLT=(AMPERESHENRIES)/SECONDS F Capacitors - Store energy and have a capacitance value C measured in farads F I=C(dV/dt) F VOLT=(AMPERESHENRIES)/SECONDS

Sources of Electrical Energy F Independent of current and/or voltage values elsewhere in the circuit, or they can be dependent upon them F Page 443 (Figure 18.1) of the text shows both ideal and linear models of current and voltage sources F Independent of current and/or voltage values elsewhere in the circuit, or they can be dependent upon them F 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) F Kirchhoff’s Voltage Law (KVL) F Sum of voltage rises or drops around any closed path in an electrical circuit must be zero F ∑V DROPS = 0 F ∑V RISES = 0 (around closed path) F Kirchhoff’s Voltage Law (KVL) F Sum of voltage rises or drops around any closed path in an electrical circuit must be zero F ∑V DROPS = 0 F ∑V RISES = 0 (around closed path) F Kirchhoff’s Current Law (KCL) F Flow of charges either into (positive) or out of (negative) any node in a circuit must add zero F ∑I IN = 0 F ∑I OUT = 0 (at node)

Ohm’s Law F Statement of relationship between voltage across an electrical component and the current through the component F DC Circuits - resistors F V = IR OR I = V/R F AC Circuits F Resistors, capacitors, and inductors stated in terms of component impedance Z F V = IZ OR I = V/Z F Statement of relationship between voltage across an electrical component and the current through the component F DC Circuits - resistors F V = IR OR I = V/R F AC Circuits F Resistors, capacitors, and inductors stated in terms of component impedance Z F V = IZ OR I = V/Z

Reference Voltage Polarity and Current Direction F Arrow placed next to circuit component to show current direction F Polarity marks can be defined F Current always flows from positive (+) to negative (-) marks F Positive current value F Current flows in reference direction F Loss of energy or reduction in voltage from + to - F Negative current value F Current flows opposite reference direction F Gain of energy when moving through circuit from + to - F Arrow placed next to circuit component to show current direction F Polarity marks can be defined F Current always flows from positive (+) to negative (-) marks F Positive current value F Current flows in reference direction F Loss of energy or reduction in voltage from + to - F Negative current value F Current flows opposite reference direction F Gain of energy when moving through circuit from + to -

Reference Voltage Polarity and Current Direction F Voltage Drops F Experienced when moving through the circuit from the plus (+) polarity to the minus (-) polarity mark F Voltage Drops F Experienced when moving through the circuit from the plus (+) polarity to the minus (-) polarity mark F Voltage Rises F Experienced when moving through the circuit from the minus (-) polarity to the plus (+) polarity mark

Circuit Equations F 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 F For KVL circuit equation (Figure 18.2) F Move around a closed circuit path in the circuit and sum all the voltage rises and drops  For ∑V RISES =0 F V S - IR 1 - IR 2 - IR 3 = 0  For ∑V DROPS =0 F -V s + IR 1 + IR 2 + IR 3 = 0 F 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 F For KVL circuit equation (Figure 18.2) F Move around a closed circuit path in the circuit and sum all the voltage rises and drops  For ∑V RISES =0 F V S - IR 1 - IR 2 - IR 3 = 0  For ∑V DROPS =0 F -V s + IR 1 + IR 2 + IR 3 = 0

Circuit Equations Using Branch Currents F Figure 18.3 F Unknown current with a reference direction is at each branch F Write two KVL equations, one around each mesh F - V S + I 1 R 1 + I 3 R 2 + I 1 R 3 = 0 F - I 3 R 2 + I 2 R 4 +I 2 R 5 + I 2 R 6 = 0 F Write one KCL equation at circuit node a F I 1 - I 2 - I 3 = 0 F Figure 18.3 F Unknown current with a reference direction is at each branch F Write two KVL equations, one around each mesh F - V S + I 1 R 1 + I 3 R 2 + I 1 R 3 = 0 F - I 3 R 2 + I 2 R 4 +I 2 R 5 + I 2 R 6 = 0 F Write one KCL equation at circuit node a F I 1 - I 2 - I 3 = 0

Circuit Equations Using Branch Currents F Use three equations to solve for I 1, I 2, and I 3 F Current I 1 is: |V S 0 R 2 | |0R 4 +R 5 + R 6 -R 2 | |0-1 -1| I 1 = ______________________________________________ |R 1 + R 3 0R 2 | | 0R 4 +R 5 + R 6 -R 2 | | | F Use three equations to solve for I 1, I 2, and I 3 F Current I 1 is: |V S 0 R 2 | |0R 4 +R 5 + R 6 -R 2 | |0-1 -1| I 1 = ______________________________________________ |R 1 + R 3 0R 2 | | 0R 4 +R 5 + R 6 -R 2 | | |

Circuit Equations Using Mesh Currents F Simplification in writing the circuit equations occurs using mesh currents F I 3 = I 1 - I 2 F Using Figure 18.3 F Current through R 1 and R 3 is I 1 F Current through R 4, R 5, and R 6 is I 2 F Current through R 2 is I 1 - I 2 F Simplification in writing the circuit equations occurs using mesh currents F I 3 = I 1 - I 2 F Using Figure 18.3 F Current through R 1 and R 3 is I 1 F Current through R 4, R 5, and R 6 is I 2 F Current through R 2 is I 1 - I 2

Circuit Equations Using Mesh Currents F Write two KVL equations F - V S + I 1 (R 1 + R 2 + R 3 ) - I 2 R 2 = 0 F -I 1 R 2 + I 2 (R 2 + R 4 + R 5 + R 6 ) = 0 F Two equations can be solved for I 1 and I 2 F Current I 1 is equivalent to that of before |V S -R 2 | |0 R 2 + R 4 + R 5 +R 6 | I= ____________________________________________________ |R 1 + R 2 + R 3 -R 2 | |-R 2 R 2 + R 4 + R 5 + R 6 | F Write two KVL equations F - V S + I 1 (R 1 + R 2 + R 3 ) - I 2 R 2 = 0 F -I 1 R 2 + I 2 (R 2 + R 4 + R 5 + R 6 ) = 0 F Two equations can be solved for I 1 and I 2 F Current I 1 is equivalent to that of before |V S -R 2 | |0 R 2 + R 4 + R 5 +R 6 | I= ____________________________________________________ |R 1 + R 2 + R 3 -R 2 | |-R 2 R 2 + R 4 + R 5 + R 6 |

Circuit Simplification F Possible to simplify a circuit by combining components of same kind that are grouped together in the circuit F Formulas for combining R’s, L’s and C’s to singles are found using Kirchhoff’s laws F Figure 18.5 with two inductors F Possible to simplify a circuit by combining components of same kind that are grouped together in the circuit F Formulas for combining R’s, L’s and C’s to singles are found using Kirchhoff’s laws F Figure 18.5 with two inductors

Circuit Components in Series and Parallel ComponentSeriesParallel RR eq = R 1 + R 2 + … + R N 1/R eq = (1/R 1 ) + (1/R 2 ) + … + (1/R N ) LL eq = L 1 + L 2 + … + L N 1/L eq = (1/L 1 ) + (1/L 2 ) + … + (1/L N ) C 1/C eq = (1/C 1 ) + (1/C 2 ) + … + (1/C N ) C eq = C 1 + C 2 + … + C N

DC Circuits F Only crucial components are resistors F Inductor F Appears as zero resistance connection F Short circuit F Capacitor F Appears as infinite resistance F Open circuit F Only crucial components are resistors F Inductor F Appears as zero resistance connection F Short circuit F Capacitor F Appears as infinite resistance F Open circuit

DC Circuit Components ComponentImpedanceCurrentPowerEnergy Stored ResistorRI = V/R P = I 2 R = V 2 /R None Inductor Zero (short circuit) Unconstrained None dissipated W L = ( 1 / 2 )LI 2 Capacitor Infinite (open circuit) Zero None dissipated W C = ( 1 / 2 )CV 2

Engineering Economics F Best design requires the engineer to anticipate the good and bad outcomes F Outcomes evaluated in dollars F Good is defined as positive monetary value F Best design requires the engineer to anticipate the good and bad outcomes F Outcomes evaluated in dollars F Good is defined as positive monetary value

Value and Interest F Value is not synonymous with amount F Value of an amount depends on when the amount is received and spent F Interest F Difference between anticipated amount and its current value F Frequently expressed as a time rate F Value is not synonymous with amount F Value of an amount depends on when the amount is received and spent F Interest F Difference between anticipated amount and its current value F Frequently expressed as a time rate

Interest Example F What amount must be paid in two years to settle a current debt of $1,000 if the interest rate is 6%? F Value after 1 year = F * 0.06 F 1000( ) F $1060 F Value after 2 years = F * 0.06 F 1000( ) 2 F $1124 F $1124 must be paid in two years to settle the debt F What amount must be paid in two years to settle a current debt of $1,000 if the interest rate is 6%? F Value after 1 year = F * 0.06 F 1000( ) F $1060 F Value after 2 years = F * 0.06 F 1000( ) 2 F $1124 F $1124 must be paid in two years to settle the debt

Cash Flow Diagrams F An aid to analysis and communication F Horizontal F Time axis F Vertical F Dollar amounts F Draw a cash flow diagram for every engineering economy problem that involves amounts at different times F An aid to analysis and communication F Horizontal F Time axis F Vertical F Dollar amounts F Draw a cash flow diagram for every engineering economy problem that involves amounts at different times

Cash Flow Patterns Figure 18.7 F P-pattern F Single amount P occurring at the beginning of n years F P frequently represents “present” amounts F F-pattern F Single amount F occurring at the end of n years F F frequently represents “future” amounts F A-pattern F Equal amounts A occurring at the ends of each n years F A frequently used to represent “annual” amounts F G-pattern F End-of-year amounts increasing by an equal annual gradient G F P-pattern F Single amount P occurring at the beginning of n years F P frequently represents “present” amounts F F-pattern F Single amount F occurring at the end of n years F F frequently represents “future” amounts F A-pattern F Equal amounts A occurring at the ends of each n years F A frequently used to represent “annual” amounts F G-pattern F End-of-year amounts increasing by an equal annual gradient G

Equivalence of Cash Flow Patterns F Two cash flow patterns said to be equivalent if they have the same value F Most computational effort directed at finding cash flow pattern equivalent to a combination of other patterns F i = interest F n = number of periods F Two cash flow patterns said to be equivalent if they have the same value F Most computational effort directed at finding cash flow pattern equivalent to a combination of other patterns F i = interest F n = number of periods

Formulas for Interest Factors SymbolTo FindGivenFormula (F/P) i n PF(1+i) n (P/F) i n PF 1 _________ (1+i) n (A/P)inAP i(1+i) n ______________ (1+i) n - 1

Formulas for Interest Factors SymbolTo FindGivenFormula (P/A) i n PA (1+i) n - 1 __________ i(1+i) (A/F) i n AF i _________ (1+i) n - 1 (F/A)inFA (1+i) n - 1 ______________ i

Formulas for Interest Factors SymbolTo FindGivenFormula (A/G) i n AG(1/i) - (n/(1+i) n - 1) (F/G) i n FG (1/i) * [(((1+i) n - 1) / (i))-1] (P/G)inPG (1/i) * [(((1+i) n - 1) / (i(1+i) n )) - (n / (1+i) n )]