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Circuit Theory + Prof Circuit Theory + Prof. Todd Huffman + What is circuit theory? Analysing (electrical) circuits, applying basic rules (Ohm’s law, Kirchoff’s law) to networks of components, to calculate the current and voltage at any point – + 1 To do this: • You need to know the basic rules. – Ohm’s Law – Kirkoff’s Current law (Conservation of Charge) – Kirkoff’s Voltage law (Conservation of Energy) • And you need to know the techniques – Mesh analysis – Node analysis • And I need to show you some tricks • And You need to practice, as with any skill. 2 Reading List • Electronics: Circuits, Amplifiers and Gates, D V Bugg, Taylor and Francis Chapters 1-7 • Basic Electronics for Scientists and Engineers, D L Eggleston, CUP Chapters 1,2,6 • Electromagnetism Principles and Applications, Lorrain and Corson, Freeman Chapters 5,16,17,18 • Practical Course Electronics Manual http://www-teaching.physics.ox.ac.uk/practical_course/ElManToc.html Chapters 1-3 (Very Good one to read!) • Elementary Linear Circuit Analysis, L S Bobrow, HRW Chapters 1-6 • The Art of Electronics, Horowitz and Hill, CUP 3 Ohm’s law Voltage difference current Resistor symbols: L I R A V V IR R=Resistance Ω[ohms] 4 Other Circuit Symbols Sources of Electrical Power Passive Devices or V0 I0 Capacitor Direction of Current or +/- Terminals must be shown. Inductor Ground – Reference Potential V ≡ Zero volts (by definition) 5 Passive Sign Convention Passive devices ONLY - Learn it; Live it; Love it! R=Resistance Ω[ohms] Three deceptively Simple questions: Which way does the current flow, left or right? Voltage has a ‘+’V side andIR a ‘-’ side (you can see it on a battery) on which side should we put the ‘+’? On the left or the right? Given V=IR, does it matter which sides for V or which direction for I? 6 Explain on white board – Trick Kirchoff’s Laws I Kirchoff’s current law: Sum of all currents at a node is zero I1 I 2 I1+I2–I3–I4=0 In 0 I 3 I 4 (conservation of charge) Prof. Huffman – before you get carried away, show them what a ‘node’ is Passive Sign Convention: If you follow it, you can arbitrarily choose whether “incoming” or “outgoing” currents are Positive at each node independently from all other nodes! 7 II Kirchoff’s voltage law: Around a closed loop the net change of potential is zero R 1 -V0+IR1+IR2+IR3=0 I 1kΩ Vn 0 V 0 3kΩ R2 5V Calculate the voltage 4kΩ across R2 5V=I(1+3+4)kΩ R3 5V I 62.5mA Passive Sign Convention 8000 really helps!!! Notice “I” VR2=62.5mA×3kΩ=1.9V8 R1=1kΩ R1 R3 VX? R2=2kΩ R =4kΩ + 3 + 10V I2 R2 I1 Solve example on board Let us use these rules to find Vx. This technique is called “mesh analysis”. Start by labelling currents and using KVL. Then apply KCL at nodes. If you then use Passive Sign Convention, the direction you 9 chose for the currents DOES NOT MATTER! R1=6kΩ R2=2kΩ R R R3=2kΩ 1 VX? 3 R4=2kΩ 10mA + 10V R2 R4 You might think, up to now, that Passive Sign convention is a bit silly. OK…So which is the “right” direction for Vx now? What if we had a circuit with 5 loops and an additional current source? 10 Let’s solve the above circuit and find out what Vx is. Node Analysis • Instead of currents 3. Now at each node around loops – (that isn’t ground) voltages at nodes use KCL 1. Choose one node as a “ground”. The reference There are Tricks you 2. Now label all nodes can use IF you use with a voltage. This passive sign is positive wrt convention at each ground. node! 11 Capacitors Unit – “Farad” C = eA/d Capacitors are also PASSIVE – They too have a kind of “ohm’s law” that + ++ ++ relates voltage and current. – – – – – Q=CV dQ dV I I C dt dt 12 Capacitor types Range Maximum voltage Ceramic 1pF–1μF 30kV Mica 1pF–10nF 500V (very stable) Plastic 100pF–10μF 1kV Electrolytic 0.1μF–0.1F 500V + Polarised Parasitic ~pF 13 1 RC circuits Initially VR=0 VC=0 2 + R Switch in Position “1” for a long time. + 1 V0 Then instantly flips at time t = 0. I + C Capacitor has a derivative! How do we analyze this? There are some tricks… ALWAYS start by asking these three questions: 1.) What does the Circuit do up until the switch flips? (Switch has been at pos. 1 for a VERY long time. easy) 2). What does the circuit do a VERY long time AFTER the switch flips? (easy) 14 3). What can we say about the INSTANT after switch flips? (easy if you know trick) The Trick!! 푑푉 • Remember 퐼 = 퐶 푑푡 • Suppose the voltage on a 1 farad Capacitor changes by 1 volt in 1 second. – What is the current? – What if the same change in V happens in 1 microsecond? – So…What if the same change in V happens instantly? • Rule: It is impossible to change the voltage on a capacitor instantly! – Another way to say it: The voltage at t = 0-e is the same as at t = 0+e. 15 RC circuits 1 Initially at t = 0- 2 VR=0 VC=0 I=0 R + 1 V0 I C Then at t = 0+ 2 Must be that V =0 V V V 0 C 0 R C If V =0 then KVL says V V V C 0 R C VR=V0 and I=V0/R. Q Capacitor is acting like IR a short-circuit. C differentiate wrt t Finally at t = +∞ dI I 0 R VR=0 VC=V0 I=0 dt C 16 dI I 0 R dt C I dI RC dt 1 1 dt dI RC I t lnI a RC lnI lnb lnbI t e RC bI t 1 I e RC b t RC I0e 17 R + V V 0 It 0 et R 18 R 1 VX(t)? + 9V R2 C Switch Closes at time t=0. What is the voltage VX as a function of time? a) R1=R2 b) R1=2R2 Work it out with student’s help. 19 Inductance I N2 L L A 0 for solenoid Unit – “Henry” Inductors are also PASSIVE – They too have a kind of “ohm’s law” that relates voltage and current. dI V L dt How should the inductor’s voltage be labelled in the above diagram? 20 Inductors Wire wound coils - air core - ferrite core Wire loops Straight wire 21 RL circuits Initially t=0- 2 VR=V0 VL=0 I=V0/R R + 1 V0 I 20R L This one is going to be fun, I promise! 1.) What does the Circuit do up until the switch flips? (Switch has been at pos. 2 for a VERY long time. easy) 2). What does the circuit do a VERY long time AFTER the switch flips? (easy) 22 3). What can we say about the INSTANT the switch flips? (easy if you know trick) The Next Trick!! 푑퐼 • Remember V = 퐿 푑푡 • Suppose the current on a 1 henry Capacitor changes by 1 amp in 1 second. – What is the voltage? – What if the same change in I happens in 1 microsecond? – So…What if the same change in I happens instantly? • Rule: It is impossible to change the current on an inductor instantly! – Another way to say it: The current at t = 0-e is the same as at t = 0+e. – Does all this seem familiar? It is the concept of “duality”. 23 RL circuits Initially t=0- 2 VR=V0 VL=0 I=V0/R + R + 1 + V0 I 20R L + And at t=∞ (Pos. “1”) VR=0 VL=0 I=0 What about at t=0+? USE A TRICK! 24 RL circuits Initially t=0- VR=V0 VL=0 I=V0/R 2 + R + 1 + V0 I At t=0+ Current in L 20R L MUST be the same + I=V0/R So… VR=V0 VR20=20V0 For times t > 0 and VL = -21V0 !!! 0 VR20 VR VL dI 0 20IR IR L dt dI And at t=∞ (Pos. “1”) 0 21IR L V =0 V =0 I=0 25 dt R L dI 21R I 0 dt L I t t dI dt L I 0 I 0 21R It t ln I0 t It I0exp V It 0 0 I0 0 R V0 t V It 0 exp I t 0 0 R R 26 R V0 t + It exp V0 R L Notice Difference in Scale! 27 I2 R2 + V0 L R1 t<0 switch closed I2=? t=0 switch opened t>0 I2(t)=? voltage across R1=? Write this problem down and DO attempt it at home 28 LC circuit Position shown @ t=0+ V + 0 L I=I0 for t=0 C Return to white board t 0 VL VC 0 dI Q L 0 dt C d2I 1 L I 0 dt2 C 29 d2I 1 I dt2 LC Initial conditions? I(0)=I0 dI 0 = ? dt 30 0 t 0 LCR circuit Vt for V0 t 0 R + V0 L I(t)=0 for t<0 V(t) C t 0 VR VL VC V0 dI Q IR L V dt C 0 d2I R dI 1 I 0 dt2 L dt LC 31 d2I R dI 1 I 0 dt2 L dt LC R=6Ω Initial conditions? L=1H I(0) C=0.2F dI 0 dt 32 2 2 I k e1t k e2t LCR circuit 1 2 0 1,2 2 2 0 R 2L 2 1 V 0 LC It 0 e t e t R 2 L 2 2 C 33 LCR circuit 22 V0 t 0 It 0 e sinRt L R 2 C 2 t 34 .
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