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Circuit Theory Dr 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 + – + +

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To do this: You need to know the basic rules. – Ohm’s Law – Current law (Conservation of Charge) – 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.

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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

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Ohm’s law Voltage difference current I L A V R=Resistance Ω[ohms] Resistor symbols: R

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Other Circuit Symbols I0I0 V0V0 Ground – Reference Potential V ≡ Zero volts (by definition) Passive Devices Sources of Electrical Power Direction of Current or +/- Terminals must be shown. or Capacitor Inductor

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Passive Sign Convention Passive devices ONLY - Learn it; Live it; Love it! R=Resistance Ω[ohms] Which way does the current flow, left or right? Three deceptively Simple questions: 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 which direction for I? Explain on white board – Trick

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Kirchoff’s Laws I Kirchoff’s current law: Sum of all currents at a node is zero I1I1 I3I3 I2I2 I4I4 I 1 +I 2 –I 3 –I 4 =0 (conservation of charge) 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!

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II Kirchoff’s voltage law:Around a closed loop the net change of potential is zero V0V0 I R1R1 R2R2 R3R3 -V 0 +IR 1 +IR 2 +IR 3 =0 5V 3kΩ 4kΩ Calculate the voltage across R 2 5V=I(1+3+4)kΩ V R2 =62.5mA×3kΩ=1.9V 1kΩ Passive Sign Convention really helps!!! Notice “I”

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R 1 =1kΩ R 2 =2kΩ R 3 =4kΩ 10V + R1R1 R3R3 R2R2 VX?VX? Let us use these rules to find V x. I1I1 I2I2 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 chose for the currents DOES NOT MATTER! Solve example on white board +

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10V + R1R1 R3R3 R2R2 R 1 =6kΩ R 2 =2kΩ R 3 =0.5kΩ R 4 =2kΩ VX?VX? R4R4 10mA OK…So which is the “right” direction for V x now? You might think, up to now, that Passive Sign convention is a bit silly. What if we had a circuit with 5 loops and an additional current source? Let’s solve the above circuit and find out what V x is.

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Node Analysis Instead of currents around loops – voltages at nodes 1.Choose one node as a “ground”. The reference 2.Now label all nodes with a voltage. This is positive wrt ground. 3.Now at each node (that isn’t ground) use KCL There are Tricks you can use IF you use passive sign convention at each node!

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Capacitors + ++ – – – Q=CV Capacitors are also PASSIVE – They too have a kind of “ohm’s law” that relates voltage and current. Unit – “Farad” C = A/d

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RC circuits + V0V0 R C 1 2 Initially V R =0 V C =0 1 Switch in Position “1” for a long time. Then instantly flips at time t = 0. Capacitor has a derivative! How do we analyze this? There are some tricks… 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) 3). What can we say about the INSTANT after switch flips? (easy if you know trick) I + + ALWAYS start by asking these three questions:

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The Trick!!

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RC circuits + V0V0 R C 1 2 Initially at t = 0 - V R =0 V C =0 I=0 1 2 differentiate wrt t Then at t = 0 + Must be that V C =0 If V C =0 then KVL says V R =V 0 and I=V 0 /R. Capacitor is acting like a short-circuit. Finally at t = +∞ V R =0 V C =V 0 I=0 I

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+ V0V0 R

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9V + R1R1 R2R2 V X (t)? C a)R 1 =R 2 b)R 1 =2R 2 Switch Closes at time t=0. What is the voltage V X as a function of time? Work it out with student’s help.

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Inductance I for solenoid Unit – “Henry” Inductors are also PASSIVE – They too have a kind of “ohm’s law” that relates voltage and current. How should the inductor’s voltage be labelled in the above diagram? L

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RL circuits + V0V0 R L 1 2 Initially t=0 - V R =V 0 V L =0 I=V 0 /R 20R 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) 3). What can we say about the INSTANT the switch flips? (easy if you know trick) I This one is going to be fun, I promise!

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RL circuits + V0V0 R L 1 2 Initially t=0 - V R =V 0 V L =0 I=V 0 /R 20R I And at t=∞ (Pos. “1”) V R =0 V L =0 I=0 What about at t=0 + ? A TRICK! + + +

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The Next Trick!!

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RL circuits + V0V0 R L 1 2 20R Initially t=0 - V R =V 0 V L =0 I=V 0 /R And at t=∞ (Pos. “1”) V R =0 V L =0 I=0 At t=0 + Current in L MUST be the same I=V 0 /R So… V R =V 0 V R20 =20V 0 and V L = -21V 0 !!! I + + + For times t > 0

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+ V0V0 R L Notice Difference in Scale!

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+ V0V0 R2R2 R1R1 L I2I2 t<0 switch closed I 2 =? t=0 switch opened t>0 I 2 (t)=? voltage across R 1 =? Write this problem down and DO attempt it at home

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+ V0V0 LC circuit C LI=I 0 for t=0 Return to white board Position shown @ t=0 +

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Initial conditions? I(0)=I 0 = ?

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LCR circuit + V0V0 R C L V(t) I(t)=0 for t<0

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R=6Ω L=1H C=0.2F Initial conditions? I(0)

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LCR circuit

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t

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