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CP2 Circuit Theory

+. –. +. +. Dr Todd Huffman t.huffman1@ph ysics.ox.ac.uk http:// www-pnp.physics.ox.ac.uk/~huffman/. CP2 Circuit Theory. Aims of this course: Understand basic circuit components (resistors, capacitors, inductors, voltage and current sources, op-amps)

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CP2 Circuit Theory

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  1. + – + + Dr Todd Huffman t.huffman1@physics.ox.ac.uk http://www-pnp.physics.ox.ac.uk/~huffman/

    CP2 Circuit Theory

    Aims of this course: Understand basic circuit components (resistors, capacitors, inductors, voltage and current sources, op-amps) Analyse and design simple linear circuits
  2. Circuit Theory: Synopsis Basics: voltage, current, Ohm’s law… Kirchoff’s laws: mesh currents, node voltages… Thevenin and Norton’s theorem: ideal voltage and current sources… Capacitors: Inductors: AC theory: complex notation, phasor diagrams, RC, RL, LCR circuits, resonance, bridges… Op amps: ideal operational amplifier circuits… Op-amps are now on the exam syllabus Stored energy, RC and RL transient circuits
  3. Reading List Electronics: Circuits, Amplifiers and Gates, D V Bugg, Taylor and FrancisChapters 1-7 Basic Electronics for Scientists and Engineers, D L Eggleston, CUPChapters 1,2,6 Electromagnetism Principles and Applications, Lorrain and Corson, FreemanChapters 5,16,17,18 Practical Course Electronics Manualhttp://www-teaching.physics.ox.ac.uk/practical_course/ElManToc.htmlChapters 1-3 Elementary Linear Circuit Analysis, L S Bobrow, HRWChapters 1-6 The Art of Electronics, Horowitz and Hill, CUP
  4. Why study circuit theory? Foundations of electronics: analogue circuits, digital circuits, computing, communications… Scientific instruments: readout, measurement, data acquisition… Physics of electrical circuits, electromagnetism, transmission lines, particle accelerators, thunderstorms… Not just electrical systems, also thermal, pneumatic, hydraulic circuits, control theory
  5. Mathematics required Differential equations Complex numbers Linear equations V(t)=V0ejωt V0–I1R1–(I1–I2)R3 = 0 (I1–I2)R3–I2R2+2= 0 Covered by Complex Nos & ODEs / Vectors & Matrices lectures
  6. Charge, voltage, current Charge: determines strength of electromagnetic force quantised: e=1.6×10-19C [coulombs] Potential difference: V=VA–VB Energy to move unit charge from A to B in electric field [volts]
  7. Charge Q=e Current: rate of flow of charge Power: rate of change of work Drift velocity No. electrons/unit vol Cross-section area of conductor [amps] [watts]
  8. Ohm’s law Resistor symbols: Voltage difference  current L I R A V R=Resistance Ω[ohms] ρ=ResistivityΩm
  9. Resistivities Silver 1.6×10-8 Ωm Copper 1.7×10-8Ωm Manganin 42×10-8 Ωm Distilled water 5.0×103 Ωm PTFE ~1019 Ωm conductance conductivity Power dissipation by resistor:
  10. Voltage source Ideal voltage source: supplies V0 independent of current V0 +– Rload Real voltage source: V=V0–IRint Rint V0 Rload battery cell
  11. Constant current source Ideal current source: supplies I0 amps independent of voltage Rload I0 Symbol: or Real current source: Rint Rload I0
  12. AC and DC DC (Direct Current): Constant voltage or current Time independent V=V0 AC (Alternating Current): Time dependent Periodic I=I0sin(ωt) +– + - 50Hz power, audio, radio…
  13. RMS values AC Power dissipation Why ? Square root of mean of V(t)2
  14. Passive Sign Convention Passive devices ONLY - Learn it; Live it; Love it! R=Resistance Ω[ohms] Two seemingly Simple questions: Which way does the current flow, left or right? Voltage has a ‘+’ side and 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 whichdirection for I?
  15. Kirchoff’s Laws I Kirchoff’s current law: Sum of all currents at a node is zero I1 I2 I1+I2–I3–I4=0 I3 I4 (conservation of charge) Here is a cute trick:It does not matter whether you pick “entering”or “leaving” currents as positive. BUT keep the same convention for all currents on one node!
  16. II Kirchoff’s voltage law: Around a closed loop the net change of potential is zero (Conservation of Energy) R1 1kΩ I V0 R2 3kΩ 5V 4kΩ Calculate the voltage across R2 R3
  17. +IR1 + – –V0 +IR2 + + – – +IR1 + – Kirchoff’s voltage law: -V0+IR1+IR2+IR3=0 5V=I(1+3+4)kΩ 1kW I 3kW V0 4kW VR2=0.625mA×3kΩ=1.9V
  18. Series / parallel circuits R1 R2 R3 Resistors in series: RTotal=R1+R2+R3… Resistors in parallel R1 R2 R3 Two parallel resistors:
  19. Potential divider R1 V0 R2 USE PASSIVE SIGN CONVENTION!!! Show on blackboard
  20. I1 I2 Mesh currents R1=6kΩR2=2kΩR3=1kΩ R1 R2 + + I1 I3 I2 + + + 9V R3 2V First job: Label loop currents in all interior loops Second job: USE PASSIVE SIGN CONVENTION!!! Third job: Apply KCL to elements that share loop currents Define: Currents Entering Node are positive I1–I2–I3 = 0  I3 = I1-I2
  21. I1 I2 Mesh currents R1=6kΩR2=2kΩR3=1kΩ R1 R2 + + I1 I3 I2 + + + 9V R3 2V Fourth job: Apply Kirchoff’s Voltage law around each loop. Last job: USE Ohm’s law and solve equations.
  22. I1 I2 Mesh currents R1=6kΩR2=2kΩR3=1kΩ R1 R2 + + I1 I3 I2 + + + 9V R3 2V -9V+I1R1+I3R3 = 0 I3 =I1–I2 –I3R3+I2R2+2V= 0 Solve simultaneous equations 9V/kΩ = 7I1–I2 -2V/kΩ = I1–3I2 -23mA=20I2 I2=–1.15mA
  23. Node voltages R1=3kΩR2=2kΩR3=6kΩ R1 VX R2 + I1 I3 I2 + - 9V R3 2V - + - 0V Step 1: Choose a ground node! Step 2: Label voltages on all unlabled nodes Step 3: Apply KCL and ohms law using the tricks
  24. Node voltages R1=3kΩR2=2kΩR3=6kΩ R1 VX R2 + I1 I3 I2 + - 9V R3 2V - + - 0V All currents leave all labeled nodesAnd apply DV/R to each current. Only one equation,Mesh analysis would give two. USE PASSIVE SIGN CONVENTION!!!
  25. R1 VX R2 I1 I3 I2 + 9V R3 2V + 0V
  26. Thevenin’s theorem Any linear network of voltage/current sources and resistors  Req Veq Equivalent circuit
  27. + V0 I2 I2 R2 R2 In Practice, to find Veq, Req… RL (open circuit) IL0 Veq=VOS RL0 (short circuit) VL0 Req= resistance between terminals when all voltages sources shorted – Warning! This is not always obvious! I1 R1 Iss VL RL
  28. Norton’s theorem Any linear network of voltage/current sources and resistors  Req I0 Equivalent circuit
  29. R V0  R Rload Rload R1 V0 Req VL=V0–ILR1
  30. A R1=3kΩ R2=2kΩ + R3=6kΩ 9V 2V + B 1kΩ A + 2V B using Thevenin’s theorem:from Prev. Already know VAB =Veq= 2V A R2=2kΩ I1 I2 + 2V Iss + B
  31. A R1=3kΩ R2=2kΩ + R3=6kΩ 9V 2V + B A 1kΩ 2mA B using Norton’s theorem: Same procedure: Find ISS and VOC IEQ = ISS and REQ = VOC/ISS
  32. Superposition R1 R2 I1 I3 I2 + 9V R3 2V Important: label Everything the same directions! + = R2 R1 R2 R1 IE IG IF IA IC IB + + 9V R3 2V R3 + I1=IA+IE I2=IB+IF I3=IC+IG
  33. R2 R1=3kΩR2=2kΩR3=6kΩ R1 IA IC IB + Example: Superposition 9V R3 IA R1 + 9V IB IC R2 R3
  34. R1=3kΩR2=2kΩR3=6kΩ R1 R2 IE IG IF R3 2V + IE IG R3 R1 + 2V IF R2
  35. R1 R2 I1 I3 I2 + 9V R3 2V + = R2 R1 R2 R1 IE IG IF IA IC IB + + 9V R3 2V R3 + I1=IA+IE I2=IB+IF I3=IC+IG
  36. Rload Matching: maximum power transfer Find RL to give maximum power in load Rin V0 Maximum power transfer when RL=Rin Note – power dissipated half in RL and half in Rin
  37. Circuits have Consequences Problem: My old speakers are 60W speakers. Special 2-4-1 deal at El-Cheap-0 Acoustics on 120W speakers!! (“Offer not seen on TV!”) Do I buy them? Depends! 4W, 8W, or 16W speakers? Why does this matter?
  38. And Now for Something Completely Different
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