Circuit Analysis Basics, Cont.Elements in ParallelElements in Parallel--ExamplesResistors in ParallelEquivalent Resistance of Resistors in ParallelSlide 6Current division between just two paralleled resistorsCurrent Division—Other CasesIssues with Series and Parallel CombinationSlide 10Slide 11Approximating Resistor CombinationIdeal Voltage SourceRealistic Voltage SourceRealistic Current SourceTaking MeasurementsMeasuring CurrentMeasuring VoltageMeasuring ResistanceExampleSlide 21EE 42 Lecture 3 1Spring 2005Circuit Analysis Basics, Cont.Resistors in ParallelCurrent DivisionRealistic Models of SourcesMaking MeasurementsTips and Practice ProblemsEE 42 Lecture 3 2Spring 2005Elements in ParallelKVL tells us that any set of elements that are directly connected by wire at both ends carry the same voltage.We say these elements are in parallel.KVL clockwise, start at top:Vb – Va = 0Va = VbEE 42 Lecture 3 3Spring 2005Elements in Parallel--ExamplesWhich of these resistors are in parallel?R1R2R3R4R5R6R7R8NoneR4 and R5R7 and R8EE 42 Lecture 3 4Spring 2005Resistors in ParallelResistors in parallel have the same voltages across them. All of the resistors below have voltage VR .The current flowing through each resistor could definitely be different. Even though they have the same voltage, the resistances could be different.R1 R2 R3 + VR _i1i2i3i1 = VR / R1 i2 = VR / R2 i3 = VR / R3EE 42 Lecture 3 5Spring 2005Equivalent Resistance of Resistors in ParallelIf we view the three resistors as one unit, with a current iTOTAL going in, and a voltage VR, this unit has the following I-V relationship:iTOTAL = i1 + i2 + i3 = VR(1/R1 + 1/R2 + 1/R3) in other words,VR = (1/R1 + 1/R2 + 1/R3)-1 iTOTAlSo to the outside world, the parallel resistors look like one:R1 R2 R3 + VR _i1i2i3iTOTALREQ + VR _iTOTAL1/REQ = 1/R1 + 1/R2 +1/R3EE 42 Lecture 3 6Spring 2005Case of Just Two Resistors in ParallelWe’ll often find circuits where jut two resistors are connected in parallel, and it is convenient to replace them with their equivalent resistance. From our earlier general formula for any number in parallel, we see that for just R1 and R2 in parallel we obtain REQ = (1/R1 + 1/R2) -1 REQ = R1 R2 /(R1 + R2) Equivalent resistance of two resistors in parallelEE 42 Lecture 3 7Spring 2005Current division between just two paralleled resistorsIf we know the total current flowing into two parallel resistors, we can easily find out how the current will divide between the two resistors:The expression derived on the previous page applies to the circuit below. The current through resistor R1, for example, is just the total applied voltage itotal x REQ divided by R1, or i1 = itotal x [R1 R2/(R1 + R2)] (1/ R1), Thus the fraction of the currently flowing through R1 is i1 / itotal = R2 /(R1 + R2). Likewise, the fraction of the total current that flows through R2 is I2/ itotal = R1 / (R1 + R2)Note that this differs slightlyfrom the voltage divisionformula for series resistors.R1 R2i1i2itotalEE 42 Lecture 3 8Spring 2005Current Division—Other CasesIf more than two resistors are in parallel, one can:Find the voltage across the resistors, VR, by combining the resistors in parallel and computing VR = iTOTAL REQ.Then, use Ohm’s law to find i1 = VR / R1, etc.Or, leave the resistor of interest alone, and combine other resistors in parallel. Use the equation for two resistors.R1 R2 R3 + VR _i1i2i3REQ + VR _iTOTALiTOTALEE 42 Lecture 3 9Spring 2005Issues with Series and Parallel CombinationResistors in series and resistors in parallel, when considered as a group, have the same I-V relationship as a single resistor.If the group of resistors is part of a larger circuit, the rest of the circuit cannot tell whether there are separate resistors in series (or parallel) or just one equivalent resistor. All voltages and currents outside the group are the same whether resistors are separate or combined.Thus, when you want to find currents and voltages outside the group of resistors, it is good to use the simpler equivalent resistor.Once you simplify the resistors down to one, you (temporarily) lose the current or voltage information for the individual resistors involved.EE 42 Lecture 3 10Spring 2005Issues with Series and Parallel CombinationFor resistors in series:The individual resistors have the same current as the single equivalent resistor.The voltage across the single equivalent resistor is the sum of the voltages across the individual resistors.Individual voltages and currents can be recovered using Ohm’s law or voltage division.iR1R2R3v-+i+ v -REQEE 42 Lecture 3 11Spring 2005Issues with Series and Parallel CombinationFor resistors in parallel:The individual resistors have the same voltage as the single equivalent resistor.The current through the equivalent resistor is the sum of the currents through the individual resistors.Individual voltages and currents can be recovered using Ohm’s law or current division.R1 R2 R3 + VR _i1i2i3iTOTALREQ + VR _iTOTALEE 42 Lecture 3 12Spring 2005Approximating Resistor CombinationSuppose we have two resistances, RSM and RLG, where RLG is much larger than RSM. Then:RSMRLG≈RLGRSM RLG≈RSMEE 42 Lecture 3 13Spring 2005Ideal Voltage SourceThe ideal voltage source explicitly defines the voltage between its terminals.The ideal voltage source could have any amount of current flowing through it—even a really large amount of current.This would result in high power generation or absorption (remember P = vi), which is unrealistic.VsEE 42 Lecture 3 14Spring 2005Realistic Voltage SourceA real-life voltage source, like a battery or the function generator in lab, cannot sustain a very high current. Either a fuse blows to shut off the device, or something melts…Additionally, the voltage output of a realistic source is not constant. The voltage decreases slightly as the current increases.We usually model realistic sources considering the second of these two phenomena. A realistic source is modeled by an ideal voltage source in series with an “internal resistance”, RS.VsRSEE 42 Lecture 3 15Spring 2005Realistic Current SourceConstant-current sources are much less common than voltage sources.There are a variety of circuits that can produce constant currents, and these circuits are usually composed of transistors.Analogous to realistic
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