1Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003EECS 42 Introduction to Electronics for Computer ScienceAndrew R. NeureutherLecture #3 KCL, KVL, Circuit Elements• Kirchhoff Current Law (and Bag case)• Kirchhoff Voltage Law• Circuit elements symbols and I vs. V graphsOldham and Schwarz: 2.1-2.2http://inst.EECS.Berkeley.EDU/~ee42/Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KIRCHHOFF’S CURRENT LAWCircuit with several branches connected at a node: branch (circuit element)iiii(Sum of currents entering node) − (Sum of currents leaving node) = 0Alternative statements of KCL1 “Algebraic sum” of currents enteringnode = 0where “algebraic sum” means currents leaving are included with a minus sign2 “Algebraic sum” of currents leaving node = 0where currents entering are included with a minus signCopyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KIRCHHOFF”S CURRENT LAW WITH A CAPACITOR AT A NODECircuit with several branches, including a capacitor(Sum of currents enteringnode) − (Sum of currents leaving node) = 0q = charge stored at node is zero. If charge is stored, for example in the capacitor shown as branch 3, the charge is accounted for as the time-integral of i3. Thus the charge is not over at the node; it is on the capacitor.iiiiCopyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003Suppose imbalance in currents is 1µA = 1 µC/s (net current entering node)Assuming that q = 0 at t = 0, the charge increase is 10−6C each second or charge carriers each second WHAT IF THE NET CURRENT WERE NOT ZERO?12196106106.1/10 ×=×−−But by definition, the capacitance of a node to ground is ZERO because we show any capacitance as an explicit circuit element (branch). Thus, the voltage would be infinite (Q = CV).Something has to give! In the limit of zero capacitance the accumulation of charge would result in infinite electric fields … there would be a spark as the air around the node broke down.Charge is transported around the circuit branches (even stored in some branches), but it doesn’t pile up at the nodes!Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KIRCHHOFF’S CURRENT LAW EXAMPLECurrents entering the node: 24 µACurrents leaving the node: −4 µA + 10 µA + i Three statements of KCLA 18i 0i10424 0iA 18i 0i10)4(24 0iA 18i i10424 iiALLIN OUToutinoutinµµµ=⇒=++−−==⇒=−−−−==⇒++−==∑∑∑∑EQUIVALENT i 10 µA 24 µA -4 µA24 = 10 + (−4) + ii = 18 µA}Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KIRCHHOFF’S CURRENT LAW USING SURFACESExampleiA2A5entering leaving5 µA2 µAi = ?Closed surface50 mAi?Another example2Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003Example of the use of KCLR1R2V1X+-At node X:Current into X from the left:(V1-vX)/R1Current out of X to the right:vX/R2KCL:(V1-vX)/R1 = vX/R2Given V1, This equation can be solved for vXOf course we just get the same result as we obtained from our series resistor formulation. (Find the current and multiply by R2)vX= V1R2 /(R1 + R2)5VR1 = 1 kΩ R2 = 2kΩCopyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003BRANCH AND NODE VOLTAGESThe voltage across a circuit element is defined as the difference between the node voltages at its terminalsSpecifying node voltages: Use one node as the implicit reference(the “common” node … attach special symbol to label it)Now single subscripts can label voltages:e.g., vbmeans vb− ve, vameans va− ve, etc.cev2v1dab+−−+0(since it’s the reference)select as ref. ⇒ “ground”Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KIRCHHOFF’S VOLTAGE LAW (KVL)The algebraic sum of the “voltage drops” around any “closed loop” is zero.Voltage drop → defined as the branch voltage if the + sign is encountered first; it is (-) the branch voltage if the − sign is encountered first … important bookkeepingWhy? We must return to the same potential (conservation of energy).+-V2Path“rise” or “step up”(negative drop)+-V1Path“drop”Closed loop: Path beginning and ending on the same nodeCopyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003KVL EXAMPLEPath 1:0vvvb2a=++−↑va− vbYEP!Path 2:0vvvc3b=+−−Path 3:0vvvvc32a=+−+−vbvcva+−+−}|{+ −vcvavbv3v2+ −ref. node+-Examples of Three closed paths:{, |, }Va= 5V Vb= 3V V3= 1Vv2= va-vbv3= vc-vbNote that:Copyright 2003, Regents of University of CaliforniaLecture 3: 09/02/03 A.R. NeureutherVersion Date 08/31/03EECS 42 Intro. Digital Electronics Fall 2003BASIC CIRCUIT ELEMENTS• Voltage Source• Current Source• Resistor•Wire• Capacitor• Inductor(always supplies some constant given voltage - like ideal battery)(always supplies some constant given current)(Ohm’s law)(“short” – no voltage drop
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