1• RESISTANCE• KIRKOFFS LAW and LOOP-MESH METHOD• VOLTAGE DIVIDER• INTERNAL RESISTANCE and OUTPUT IMPEDANCE• HOW TO MEASURE OUTPUT IMPEDANCE OF A DEVICE•THEVENIN's THEOREMChapter - 1 Direct Current Circuits2Resistivity ρ• ρ property of amaterial characterizingelectron flow• σ=1/ρ = conductivity •Metal – Energy gap, Eg, betweenValence and Conduction bandsoverlap.•Semiconductor - small Energygap between valence andconduction bands, of order 1 eV.•Insulator – large Energy gapbetween valence and conductionbands, of order 10 eV.conductionvalenceconductionvalenceconductionvalenceEgEgEg3Resistance•Materials have different resistivities ρ• R = ρ (L/a) = resistance (Ohms)L=length a = cross sectional areaLaρaρρa45Ohm's Law• The Voltage drop across a circuit element R is proportional to the current flowing through it. RiVi =V/R or V = i R • The constant of proportionality is the resistance R or impedance.6Series and Parallel Connections• Resistances in series directly add. • Resistances in parallel follow the inverse rule of addition. R1 R2R = R1+R2When a voltage V is placed across the pair inseries we writeV = i(R1+R2) = iRR = R1 + R2When a voltage V is placed across the pair inparallel we writeV = i1 R1 = i2 R2 = iRi = i1 + i21/R =1/R1 + 1/R2R1 R2ViR112VR2i1iR1 R21/R = 1/R1+1/R2VV7Kirchoffs Circuit Laws (by Mesh Currents)+V-i R = 0 (consevation of energy) VRi(1) Choose mesh currents to run CW in each circuit loop (convention).(2) Add the battery emfs when in direction of mesh current, subtract when opposing mesh current.(3) Write Kirkoff’s law for each mesh(4) Subtract the voltage drops (-iR) across resistor in the mesh, add (+iR )the adjacent mesh. (5) Physical currents through components can be determined from i1 and i2. +Va-i1 R1 +i2 R1 = 0+Vb -i2 R2 -i2 R1 +i1 R1 = 0Va = i1 R1 -i2 R1Vb = -i1 R1 +i2 (R1+R2)solveVaR1i1R2i2Vbmeshmesh-1 mesh-2Subtract Voltage dropAdd voltage from adj loopBattery emfIn directio ofMesh current8Example5V10Ωi12Ωi25V - 10 i1 + 10 i2 = 00 - 2 i2 - 10 i2 + 10 i1 = 01) 5V = 10 i1 - 10 i22) 0 = -10 i1 + 12 i2 5V = 0 + 2 i2 adding 1) + 2) --> i2 = 5/2 A i1= (12/10)i2 = 3A i2 = 5/2 A I = i1 = 3A Find I1, I2, I ?I16 - 2 i1 -9 -3 i1 +3 i2 = 0-6 -3 i2 +3 i1 +9 -6 i2 = 07 = 5 i1 -3 i23 = -3 i1 +9 i221 = 15 i1 -9 i2 3 = -3i1 +9 i224 = 12 i1 i1 = 2A7 = 10 -3 i2-3 = -3 i2 i2 = 1AV = 2 1i = 4Vi1 opposes battery polarityi2 with battery polarityExample from web!102LOOP Graphical SolutionRewrite!equaions in terms of straight lines! y= mx + b !y = x !VaR1y =R1R1 + R2( )x +VbR1 + R2( )x(i1)y(i2)(x, y)(i1,i2)11Voltage DividerA simple resistive voltage divider allows us to adjust the input voltage to a lower level. This high voltage probe usesa voltage divider to allow us tomeasure a large voltage bydropping it to a lower range. Vout =R2R1 + R2!"#$%&Vin12Voltage DividerV2VR1iR2V-i R1 -i R2 = 0V=i(R1+R2)i = V/(R1+R2)V2 = i R2 = R2/(R1+R2) V = (R2/Rtot) VVoltage drop on ith resistor is proportional to ratioof Ri to Rtot! Vi = { Ri/Rtot } VV1 = R1/Rtot x VV2 = R2/Rtot x VV113Examplegroundi2Ω5V5Ω3Ω•What is the voltage drop across the 5Ω resistor? Ans: V3Ω = (5/10) 5V = 2.5V14Current DivisionCurrent will take the path of least resistance, dividingItself by inverse proportion with i = i1 + i2i1 =!Vr1=i!rr1=!rtotr1ii2 =!Vr2=i!rr2=!rtotr2iii1i2r1r2i15Power Dissipated by a Circuit ElementVIIP = I VP = d/dt U U=qVP = d/dt qVP = dq/dt V = I V•The power dissipated by a circuit element is given by P=IV, I = the current passing through the element. V = voltage drop across the element. •For Ohmic circuit elements, V=I R, we can also write: P=I2R P=V2/R•Light bulbs (nonOhmic) Resistors(Ohmic)9V20kJHow!long!does!a!9volt!last!under!1ma!load?Let!U = 20kJ !!!!!V = 9V !!! I = 1mAP = dU / dtdt =1IVdU !!!!!t =!19mW20kJ = 2.2e6s = 620hcircuit16Internal Resistance and Output Impedance• Every Source of Emf has some small internal resistance.• A signal generator has an internal resistance related to its output Impedance r or z. ( z~50 Ohm ). • A voltage divider circuit can be used to measure r and z. Adjust R until Vout=1/2 V ! Then R = internal resistance !batteryViRrr ~ 0.1 - 1 ΩSine WaveGeneratorr+-RVoutV17Input Impedance of Voltmeter and Ammeter• All input devices has some small internal resistance to the currentflowing into it. (Impedance to ground or negative terminal)• A voltmeter has a high input impedance to limit the current flowing in to the measuring device.• An ammeter wants to divert all the current into it and therefore has a very low input impedance. + -1MΩV+ -1ΩAfuse18Input and Output Impedance• Consider an I/O circuit to be an element which transforms an input voltage waveform to and output voltage waveform. • The input voltage source VIN sees an effective input impedance ZIN wrt to the input current IIN=VIN/ZiIN . • The output current IOUT is driven through an effective series resistanceZout with VOUT=IOUTZOUT• All I/O devices can be characterized by an input and output impedance.VoutVinZINεZoutIINIOUT19Consider a complex circuit of which we are dealing with a small part. 1) to calculate the current through (or voltage across) a component in any circuit.2) or develop a constant voltage equivalent circuit which may be used to simplify the analysis of a complex circuitThevenin's Theorem(1)VthRthababThevenin Equivalent CircitV20Thevenin's Theorem(2)21Thevenin's Theorem(2)Thevenin Equivalent Circuithttp://esdstudent.gcal.ac.uk/Thevenin3.htm222 Port Network (extra)V1 = z11! I1 + z12! I 2V 2 = z21! I1 + z22! I 2z's!are!the!open!circuit!impedance!parameters and can be evaluated zero current or no load conditions. z11=V1I1!I 2 = 0!!!!z12=V1I 2!I1= 0!!z21=V 2I1!I 2 = 0!!!!z22=V 2I 2!I1= 0black boxcircuiti1 i2V1 V2A!two!port!network is a linear mathematical model that!can!be!used to analyse a circuit and other systems, if the input and output voltages and currents can be isolated. This linear model can be generalized to other flows (I) and potentials (V). z1 z2232 Port Network (extra)z11=V1I1I 2 = 0!= R1 || R2 + R3 =1R1+1R2 + R3!"#$%&'1=R1(R2 + R3)R1 + R2 + R3!!!!z22=V 2I 2I1= 0!= R2 || R1 + R3!=1R2+1R1 + R3!"#$%&'1=R2(R1 + R3)R1 + R2 + R3!z12=V1I 2!I1= 0!!= !IR1( R1I 2= !(R1 + R3R1 + R2 + R3) I 2 !(R1I 2!!!!!by!current!division!z21=V 2I1!I 2 =