Crkt-1More CircuitsIn an electrical circuit, circuit elements such as resistors and batteries can be connected together in series or in parallel. Resistors in series are connected like links in a chain; resistors in parallel are side-by-side, like so:Series: tot 1 2 3R R R R= + + , Rtot > R1, R2, R3 Parallel: tot1 2 31R1 1 1R R R=+ + , Rtot < R1, R2, R3 Resistors in series act like a single large resistor.Resistors in parallel act like a single small resistor.Proof: Resistors in Series: Itot = I1 = I2 , Vtot = V1+V2 tot1 2 1 2tot 1 2tot tot 1 2VV V V VR R RI I I IDD +D D D= = = + = +Resistors in Parallel: Itot = I1 + I2 , Vtot = V1 = V2 tot1 2 1 2tot tot 1 2 tot 1 2II I I I 1 1 1V V V V R R R+= = + � = +D D D DLast update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado in series:R1R2R3=RtotR1R2R3Rtot=in parallel:Crkt-2Examples of parallel resistors: 1) Two 100 resistors in parallel:tot1 1 100R 501 122100 100100= = = = W� ���+����W W� �2) 10 in parallel with 0 wire:tot1 1R 01 10 10= = =�+Last update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado 100 50 =100 0 0 =R2Crkt-3Key points: The current is the same for resistors in series. Current is not "used up".Think of the water pipe analogy: two gravel plugs in series, same flow (same gal/min) through both plugs (assuming no leaks or bubble in the pipe) Adding another resistor in series always increases the total resistance. The voltage difference across each resistor is the same for resistors in parallel.Both resistors in parallel have the same big small small bigV V I R I RD = = =. Adding another R in parallel always decreases the total resistance. Like adding another pipe along side the original pipe allows more flow smaller total resistanceLast update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado R1R2I Igravel pluggravel plugRbigRsmallIsmallIbigV sameacross both R'ssmaller plug, more flowbigger plug, less flowpressure sameacross both plugshi Plo PCrkt-4 Kirchhoff's two rules for analyzing circuits (Kirchhoff is really spelled that way: 2 h's, 2 f 's ) Kirchhoff's Current Rule (also called the Junction Rule)The total current into any junction = total current out (junction = place where 3 or more wires meet)1 2 3I I I= +This is also called Conservation of Current. In steady-state, the charge is not building up anywhere, it is just flowing along at a steady rate. So the current into any portion of the circuit must equal the current coming out of that portion, otherwise charge would be building up in that part of the circuit.Kirchhoff's Voltage Rule (also called the Loop Rule)The sum of the voltage rises around any complete loop in a circuit = sum of the voltage drops around the same loop. Voltage rises and drops must sum to zero, since we must return to the same voltage after one complete loop.{{{1 2 1 2rise1 2fallfallVV I R I R I(R R ) IR R= + = + � =+Remember: voltage is a kind of "electrical pressure" or "electrical height" . If you go around a complete circuit and return to the same place, you are back at the same pressure (or height). So rises must equal drops.Last update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado R2 V I med Vlo VR1 + – hi VPumplo pressurehi P med PI2I3I1I1junction junctionCrkt-5 Ammeters and VoltmetersAn ammeter measures the current through itselfA voltmeter measures the voltage difference between its terminals.To measure the current through a resistor R, must place the ammeter in series with R.To measure the voltage across R, must place voltmeter in parallel with R.An ideal ammeter has zero internal resistance rinternal = 0, so current I is not affected.An ideal voltmeter has rinternal = , so no current flows through currents and voltages in rest of circuit are not affected.Last update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado R I AVIIAVAIi = 0Vrint=ideal ammeter: rint = 0ideal voltmeter: rint = Crkt-6Circuits with multiple loops and batteriesHave a circuit with known V'sand known R's. Seek the I's.Procedure:I. Guess direction of I through each R. Draw I arrows, label each (I1 thru R1, etc). (Directions ofcurrents not always obvious, so just guess. If you guess wrong, value of current I will come out with a negative value.)II. K's Current Law gives 1 or more equations:(eq'n 1) 1 2 3I I I+ = [3 unknowns ( I1 , I2 , I3 ) will need 3 eq'ns to solve]III. K's Voltage Law gives an equation for each complete loop in the circuit. 3 loops in this circuit. Only need 2 more equations,so only 2 of the 3 loop equations are needed.Loop 1: 1 1 1 3 3V I R I R= + (eq'n 2)Loop 2: 2 2 2 3 3V I R I R= + (eq'n 3)Loop 3: {{1 1 1 2 2 2rise drop rise dropV I R I R V 0+ - + - =1442443 1442443 (Moving CW around loop 3)Don't need the equation from loop 3, because already have 3 equations. Remember! In a resistor, if we move in the direction of the current, V drops, V is negative ;if we move in the direction opposite current, V rises, V is positive.In a battery, if move from (–) to (+) terminal, V rises ; if move from (+) to (–) terminal, V drops.We now have 3 equations in 3 unknowns ( I1 , I2 , I3 ) :Last update: 1/13/2019 Dubson Phys1120 Notes, University of Colorado 123hi Vlo Vhi Vlo VI3R1 = 10 I1I2V2 = 10 VR3 = 30 R2 = 10 V1 = 10 VCrkt-7(1) 1 2 3I I I+ =(2) 1 1 1 3 3V I R I R= +(3) 2 2 2 3 3V I R I R= +The physics part of this problem is over; now we have a messy algebra problem. How do we solve?Eqn (1) says we can substitute ( I1 + I2 ) for I3 . Eliminate I3 in equations (2), (3):1 1 1 1 2 3V I R (I I ) R= + +2 2 2 1 2 3V I R (I I ) R= + +Rearrange:A. 1 1 1 3 2 3V I (R R ) I R= + +B. 2 1 3 2 2 3V I R I (R R )= + +Now have 2 equations (A, B) in 2 unknowns (I1, I2). Now combine these to eliminate either I1 or I2. For instance, can solve eqn A for I1 : 1 2 311 3V I RIR R-=+ ( eqn C )Then plug this into eqn B: 1 2 32 3 2 2 31 3V I RV R I (R R )R R� �-���= + +�����+� �Can solve this for I2 (messy!). Then plug into (C) to get I1. Then plug back into (1) to get I3.Last update: 1/13/2019 Dubson Phys1120 Notes, University of ColoradoCrkt-8Household WiringWall socket = 3-prong plugThe short slot is the dangerous high-voltage one; shortslot is harder to stick your finger in.Standard electrical …
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