EE 42 Lecture 41/29/2004Circuit Analysis Basics, Cont. Resistors in Parallel Current Division Realistic Models of Sources Making Measurements Tips and Practice Problems1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Elements in Parallel KVL tells us that any set of elements which 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 = Vb1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Elements in Parallel--ExamplesWhich of these resistors are in parallel?R1R2R3R4R5R6R7R8NoneR4and R5R7and R81/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Resistors in Parallel Resistors in parallel carry the same voltage. 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.R1R2R3+VR_i1i2i3i1= VR/ R1i2= VR/ R2i3= VR/ R31/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Resistors in Parallel If we view the three resistors as one unit, with a current iTOTALgoing 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)-1iTOTAlSo to the outside world, the parallel resistors look like one:R1R2R3+VR_i1i2i3iTOTALREQ+VR_iTOTALREQ= (1/R1+ 1/R2+ 1/R3)-11/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Current Division If we know the current flowing into two parallel resistors, we can find out how the current will divide up in one step. The value of the current through R1isi1= iTOTALR2/ (R1+ R2) The value of the current through R2isi2= iTOTALR1/ (R1+ R2) Note that this differs slightlyfrom the voltage divisionformula for series resistors.R1R2i1i2iTOTAL1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Current Division—Other Cases If more than two resistors are in parallel, one can: Find the voltage over the resistors, VR, by combining the resistors in parallel and computing VR= iTOTALREQ.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.R1R2R3+VR_i1i2i3REQ+VR_iTOTALiTOTAL1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Issues with Series and Parallel Combination Resistors 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.1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Issues with Series and Parallel Combination For 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 -REQ1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Issues with Series and Parallel Combination For 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.R1R2R3+VR_i1i2i3iTOTALREQ+VR_iTOTAL1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Approximating Resistor Combination Suppose we have two resistances, RSMand RLG, where RLGis much larger than RSM. Then:RSMRLG≈RLGRSMRLG≈RSM1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Ideal Voltage Source The 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.++++−−−−Vs1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Realistic Voltage Source A 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”.++++−−−−VsRS1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Realistic Current Source Constant-current sources are much less common than voltage sources. There are a variety of circuits that can produces constant currents, and these circuits are usually composed of transistors. Analogous to realistic voltage sources, the current output of the realistic constant currents source does depend on the voltage. We may investigate this dependence further when we study transistors.1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Taking Measurements To measure voltage, we use a two-terminal device called a voltmeter. To measure current, we use a two-terminal device called a ammeter. To measure resistance, we use a two-terminal device called a ohmmeter. A multimeter can be setup to function as any of these three devices. In lab, you use a DMM to take measurements, which is short for digital multimeter.1/29/2004 EE 42 Lecture 4EE 42 Lecture 41/29/2004Measuring Current To measure current, insert the measuring instrument in series with the device you are measuring. That is, put your measuring instrument in the path of the current flow. The measuring devicewill contribute a verysmall resistance (like wire)when used as an ammeter. It usually does not introduce serious error into your measurement, unlessthe circuit resistance is small.iDMM1/29/2004 EE
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