Electric Circuits and Power

Electric Circuits and PowerChapter 201Series and Parallel Circuits20.12Series CircuitsHave only ONE LOOP or circuit for the current to travel through.

3Resistors in SeriesWhen two or more resistors are connected end-to-end, they are said to be in series

The current is the same in all resistors because any charge that flows through one resistor flows through the other

The sum of the voltages across the resistors is equal to the total voltages across the combination Kirchoffs Voltage Law, the Conservation of Voltage4Resistors in SeriesPotentials addV = IR1 + IR2 = I (R1+R2)Consequence of Conservation of EnergyThe equivalent resistance has the effect on the circuit as the original combination of resistors

5Equivalent Resistance SeriesRe = R1 + R2 + R3 +

The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistorsBatteries and even wires contribute small amounts of resistance but we can ignore this for now6Equivalent Resistance SeriesFour resistors are replaced with their equivalent resistance

An Example7Resistors in ParallelThe voltage across each resistor is the same because each is connected directly across the battery terminalsThe current, I, that enters a point must be equal to the total current leaving that pointI = I1 + I2The currents are generally not the sameConsequence of Kirchoffs Second Law, the Conservation of Charge8Equivalent Resistance Parallel Equivalent resistance replaces the two original resistancesHousehold circuits are wired so the electrical devices are connected in parallelCircuit breakers may be used in series with other circuit elements for safety purposes

An Example9Equivalent Resistance ParallelEquivalent Resistance

The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistanceThe equivalent is always less than the smallest resistor in the group

10Example 1

In the depicted circuit, the voltage supplied by the battery is 12V, and the resistors have values of R1 = 10, R2 = 5 and R3 = 15. What is the current flowing through each branch?11Example 1The current through each resistor can be found with Ohms LawI1 = 1.2A;I2 = 2.4A;I3 = 0.8A

To check this, find the current through the whole circuit by finding the total resistance, then using Ohms Law again.Rtot = 2.7Itot = 4.4A

This matches the sum of the individual currents12Example 2

If, V = 24V; R1 = 2; R2 = 3.3; R3 = 7 and R4 = 12.2Find the current through the circuit and the current through each resistor.

13Example 2Find the total resistance:Rtot = o.973Use this to find the current through the circuitI = 24.7AThe current through each resistor is just the voltage divided by each individual resistance:I1 = 12A;I2 = 7.3A;I3 = 3.4AI4 = 2A14Series CircuitA series of sources separated by resistors is equivalent to a single source having the net voltage and a single resistor having the combined resistance.R1ABR2R3R4V1V2V3R = R1 + R2 + R3 + R4V = V1 - V2 + V3The same current passes through every resistor in a given branch, regardless of the presence of sources in that branch, and the resistors are in series even though they are not directly connected to one another..15Parallel CircuitThe current in each branch of a parallel circuit depends inversely on the total resistance: the larger the resistance, the less current flows through the branchIf we know I but not VR1R2V

16Analysis Of Circuits20.217Review of Circuit RulesFor SERIES Circuits:There is ONE path for the currentCurrent is CONSTANTVoltage DROPS across each resistanceResistors add simplyAdditional resistances DECREASE current18Review of Circuit RulesFor PARALLEL Circuits:There are MULTIPLE paths for the currentCurrent may NOT be CONSTANTVoltage is CONSTANT to each branchResistors add RECIPROCALLYAdditional resistance INCREASES current19Review of Circuit RulesSometimes you will have BOTH series AND parallel resistors in the SAME circuit!! You then need to SIMPLIFY the circuit in your analysis.

20Example 3Consider this circuit. (a) If possible, simplify it and determine an equivalent resistance between C and G. (b) What current is provided by the source? (c) What is the voltage across points G and E?

Given: Nine resistors, R = 1.0 kW each, and V = 12VFind: Re, I, and V between G and E 21Example 3Solution To solve this one we'll need the equivalent resistance of the circuitProcedure Redraw this circuit to make it look more manageableLift up the inside square, with the resistor and source attached, and place it outside E-F-G-H see diagram on rightBranches A-B-C and A-D-C are in parallel, as are E-F-G and E-H-G, and each has a resistance of 1.0 kW + 1.0 kW = 2.0 kW (resistors add in series)

22Example 3The equivalent circuit is shown to the rightThe resistance of each square E-F-G-H and A-B-C-D reduces to

and R = 1.0 kW. 23Example 3The three 1.0-kW resistors then are in series with the source, (a) The equivalent resistance is 3.0 kW. (b) Since V = IRe,

24Example 3

(c) A current of 4.0 mA leaves the battery and splits at CBecause the two branches C-D-A and C-B-A have the same resistance, the current divides into two equal streams of 2.0 mA eachThe voltage drop in going from C to A is given by VAC = IR = (2.0 mA)(1.0 kW + 1.0 kW) = 4.0 VIn going from A to E, there is another drop of VEA = IR = (4.0 mA)(1.0 kW) = 4.0 V. C is 12V above G, A is 8.0 V above G, and E is 4.0V above G. 25Example 4Consider the following circuit:A battery supplying 12 volts leads to a resistor (R1 = 1.3), then splits into three branches. The first branch has R2 = 4.5, the second branch has R3 = , and the third branch contains R4 = 5 AND R5 = 2.2 in series. Finally, the three branches reunite, and lead to R6 = 7 before reconnecting to the battery.Draw a diagram of this circuitFind the total resistanceFind the overall current in the circuit.26Problem-Solving Strategy, 1Combine all resistors in seriesThey carry the same currentThe potential differences across them are not the sameThe resistors add directly to give the equivalent resistance of the series combination: Re = R1 + R2 + 27Problem-Solving Strategy, 2Combine all resistors in parallelThe potential differences across them are the sameThe currents through them are not the sameThe equivalent resistance of a parallel combination is found through reciprocal addition:

28Problem-Solving Strategy, 3A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistorReplace any resistors in series or in parallel using steps 1 or 2. Sketch the new circuit after these changes have been madeContinue to replace any series or parallel combinations Continue until one equivalent resistance is found29Problem-Solving Strategy, 4If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuitsUse V = I R and the procedures in steps 1 and 230Equivalent Resistance Complex Circuit

31Capacitors in CircuitsA circuit is a collection of objects usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other formsA circuit diagram can be used to show the path of the real circuit32Capacitors in ParallelWhen capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively chargedThe flow of charges ceases when the voltage across the capacitors equals that of the batteryThe capacitors reach their maximum charge when the flow of charge ceases33Capacitors in ParallelThe total charge is equal to the sum of the charges on the capacitorsQtotal = Q1 + Q2The potential difference across the capacitors is the sameAnd each is equal to the voltage of the battery

34More About Capacitors in ParallelThe capacitors can be replaced with one capacitor with a capacitance of CeqThe equivalent capacitor must have exactly the same external effect on the circuit as the original capacitorsCapacitors in parallel all have the same voltage differences as does the equivalent capacitance

35Capacitors in ParallelThe equivalent capacitance of several capacitors in parallel is the sum of all the individual capacitors.

C = C1 + C2 +

The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors36Example 5The figure below shows two capacitors attached to a 12-V battery. Determine the equivalent capacitance and the charge it would carry. What is the charge on each of the capacitors in the figure? Given: C1 = 20 mF, C2 = 30 mF, and V = 12 VFind: C, Q, Q1, and Q2+++---20 mF30 mF12 V37Example 5Solution: Capacitors are in parallel and the potential across each capacitor is 12 V

Q = CV = (50 x 10-6 F)(12 V) = 6.0 x 10-4 C

38Capacitors in SeriesWhen a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the batteryAs this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive chargeAll of the right plates gain charges of Q and all the left plates have charges of +Q

39More About Capacitors in Series

An equivalent capacitor can be found that performs the same function as the series combinationThe potential differences add up to the battery voltage

Capacitors in series all have the same charge, Q, as does their equivalent capacitance

40Capacitors in SeriesThe equivalent capacitance of several capacitors in series

The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

41Example 6The circuit shown in the figure consists of a 12-V battery and three capacitors. It is redrawn from Fig. 12.27a in the book. Determine both the voltage across and charge on each capacitor after the switch S is closed and electrostatic equilibrium is established. Find the equivalent capacitance of the network.Given: C1 = 2.0 mF, C2 = 2.0 mF, C3 = 5.0 mF, and V = 12 VFind: C, V1, V2, V3, Q1, Q2, and Q3+2.0 mF12 V2.0 mF5.0 mFC3C1C242Example 6Combining C1 and C2 which are in series+12 V1.0 mF5.0 mFC3C1 + C2

43Example 6Combining C3 and (C1 + C2) which are in parallel+12 V6.0 mFC

The equivalent capacitance of the network44Example 6There is 12 V across C3 soQ3 = C3V3 = (5.0 mF)(12 V) = 60 mC +2.0 mF12 V2.0 mF5.0 mFC3C1C2There is 12 V across the combination of the two 2.0 mF capacitors so there is a potential difference of 6.0 V across eachQ1 = Q2 = (2.0 mF)(6.0 V) = 12 mC 45Problem-Solving StrategyBe careful with the choice of unitsCombine capacitors following the formulasWhen two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the sameThe capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor46Problem-Solving Strategy, contCombining capacitorsWhen two or more capacitors are connected in parallel, the potential differences across them are the sameThe charge on each capacitor is proportional to its capacitanceThe capacitors add directly to give the equivalent capacitance47Problem-Solving Strategy, finalRepeat the process until there is only one single equivalent capacitorA complicated circuit can often be reduced to one equivalent capacitorReplace capacitors in series or parallel with their equivalentRedraw the circuit and continueTo find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions48Electric Power, AC and DC Electricity20.349Household CircuitsThe utility company distributes electric power to individual houses with a pair of wiresElectrical devices in the house are connected in parallel with those wiresThe potential difference between the wires is about 120V

50Household CircuitsA meter and a circuit breaker are connected in series with the wire entering the houseWires and circuit breakers are selected to meet the demands of the circuitIf the current exceeds the rating of the circuit breaker, the breaker acts as a switch and opens the circuitHousehold circuits actually use alternating current and voltage51Household UsageElectricity usage is measured in kilowatt-hours (kWh)Watts are units of PowerP = VI (for DC)Usually measured in Kilowatts or Horsepower1hp = 746W1kWh = 3.6 x 106J52Types of CurrentDirect Current (DC) charge flows uniformly in ONE directionAlternating Current (AC) charge flows in opposite directions alternating in a regular, periodic way with a given frequency.Peak vs. Average Voltage in AC the difference is the Average Voltage coming out of the wall is a percentage of the Peak Voltage supplied.**Alternating Current is both easier to generate AND to transmit long distances**53Power in ACPower in AC circuits is calculated in the same way as in DC circuits but using average voltage values instead of peak. Peak values for current (I) can be found using Peak Voltage when resistance (R) is known.

I freakin HATE Circuit analysis!!54Reactance in ACIn AC circuits, where the current is constantly (and ver rapidly) reversing, if you have anything other than simple resistance (Capacitors or Inductors), the response time or Reactance of those components will create a lag time in voltage response to the current shift.Inductive ReactanceCapacitive Reactance55Electrical SafetyElectric shock can result in fatal burnsElectric shock can cause the muscles of vital organs (such as the heart) to malfunctionThe degree of damage depends onthe magnitude of the currentthe length of time it actsthe part of the body through which it passes56Effects of Various Currents5 mA or lessCan cause a sensation of shockGenerally little or no damage10 mAHand muscles contractMay be unable to let go a of live wire100 mA If passes through the body for just a few seconds, can be fatal57Ground WireElectrical equipment manufacturers use electrical cords that have a third wire, called a case groundPrevents shocks

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