Aim of the lecture Appreciation of Current Density Drift Velocity Kirchhoff’s Laws

1. Aim of the lecture • Appreciation of • Current Density • Drift Velocity • Kirchhoff’s Laws • Voltage Loop Law • Current Node Law • RC Circuits • Response to step Voltages • Charge and discharge •   Main learning outcomes • familiarity with • Kirchhoff’s Laws and application to circuits • Typical current densities and drift velocities • Calculation of RC time constant • Charging and discharging Lecture 6

2. Reminder: • Electrons are made to drift in an electric field caused by an external voltage. • They loose energy in collisions with the fixed atoms • They therefore do not accelerate • They drift at constant speed

3. Consider a wire with a voltage across it. Remember: atoms fixed in place electrons move • Current density is • the current per unit cross-sectional area of the wire • if this is too great the wire can melt • as the current density goes up, the wire will get hot • this makes its resistance higher. • bigger currents need bigger wires

4. Current = i Area = A Current density, r = i/A

5. Current density, r = i/A For example, up to 5A currents in household wires use 1mm2 copper wires r = 5/0.0012 = 5 x 106 A/m2 This is not the maximum the wire could take, but it isa safe limit for use in houses.

6. r = 5/0.0012 = 5 x 106 A/m2 At 5A in a 1mm2 wire, there is 5 x 106 A per m2 in the wire This sounds large, so how fast are the electrons moving? 5 x 106 A = 5 x 106 electrons/second 1.6 x 10-19 = 3.1 x 1025 per second That’s a lot, 3 x 1025 peas would cover the earth to a depth of ~1km

7. To work out electron speed, need density of electrons in the wire. Copper has density = 8.9 g/cm3 Copper atom has molecular weight = 63.546 g/mol Avogadros number is 6.02 x 1026 atoms/mol So there are 6.02 x 1023 x 146085.5 = 8.5 x 1028 atoms/m3 And the drift speed = current/(density x area x charge/electron) = 5/(8.5 x 1028 x 1 x 10-6 x 1.6 x 10-19) = 0.4 mm/sec which is about 1.3 x 10-3 km/hr Which makes this guy look fast!

8. This is actuallya VERY thin wire wound in a spiral • The electrons in a wire don’t move far normally • In an incandescent light bulb (one with a wire) • the wire is very thin • the electrons are drifting fast • about walking pace (!) • which is why the wire gets very hot • once the electrons get through the bulb • they move slowly again This is one place where the water in a pipe analogy is a little weak – the water is hardly moving at all to be a good picture

9. Kirchhoff’s Laws • These are effectively • energy conservation • charge conservation • Applied to circuits Current Law Charge cannot be destroyed, so the sum of currents flowing into a node is equal to the sum of currents flowing out ( hence it is vital to understand that capacitors do NOT store charge)

10. Analogy: If water flows into a junction Then the volume of water flowing in equals the total flowing out So ID+IC+IB = IA

11. Note that IIN = Ia + Ib capacitors do NOT store charge. Ia IIN Ib This is the reason we have been so ‘determined’ that capacitors should not be thought of as ‘storing’ charge - if they could then IIN would not necessarily be Ia+Ib.

12. Voltage Law The sum of the voltage drops round a closed loop is zero Recall that voltage is a measure of potential And remember that gravitational potential behaves in a similar way If a mass is moved round a closed path in a gravity field the sum of Dmgh round its path must be zero. Just says that if you start at one height and end up at that sameheight then the sum of all the changes in height must be zero Electric potential is the same, if you move round a closed loopthen the sum of the changes in voltage must be zero.

13. Example This is the old fashionedsymbol for a resistor,it is still used a lot Be careful defining the sign. You MUST measure the voltages in the same direction On all voltages round the loop

14. Svi = 0 round loop SIi = 0 into node Prof. Kirchhoff

15. Note that the loop laws are true for all the loops in a circuit Altogether there are 7 loops in the circuit above (3 shown) Find the others.

16. Kirchhoff’s Laws are used in working out what thecurrents and voltages are in a network of components. Each loop and each node yields an equation These then form a set of simultaneous equations which canbe solved to find the currents and voltages. Not necessarily an easy way to do it,but formulaic and usually gives the right answer.

17. With this definition ofdirections, all the currentswill be positive. Sometimes called Kirchhoff’s First and Second Laws Note that in this diagram,voltage ‘a’ will have the oppositesign to all the others

18. RC Circuits Switch Vr=0 VC=0 Q=CV I = dQ/dt = CdV/dt Svi = 0 round loop SIi = 0 into node

19. RC Circuits A ‘long’ time after the switch is closed. Vr=0 Vc=emf Q=CV I = dQ/dt = CdV/dt Svi = 0 round loop SIi = 0 into node

20. RC Circuits Now open switch again Vr=0 Vc=emf Q=CV I = dQ/dt = CdV/dt Svi = 0 round loop SIi = 0 into node

21. RC Circuits Then close the other way Vr=0 Vr=emf But now thereis a circuit witha resistor acrossan energised capacitor(‘charged’ capacitor) Vc=emf Q=CV I = dQ/dt = CdV/dt Svi = 0 round loop SIi = 0 into node

22. RC Circuits Vr Svi = 0 round loop So Vr = -Vc Vc The current flowinground the circuitis the same everywhere So, using then I = Vr/R = -CdVr/dt I = dQ/dt = CdV/dt The solution is Vr = Ae-t/RC Where A is a constant and is the voltage at time = 0, say V0 (in this case V0 is what was called ‘emf’ earlier) so Vr/R + CdVr/dt = 0

23. RC Circuits Vr Vc A ‘discharging’ capacitor obeys the equation Vr = V0e-t/RC RC is called the time constant for the circuit The voltage drops from V0 to V0/e in a time RC

24. 0 Having made the point that capacitors do not store charge, we will now adopt the usual convention of talking about charging and discharging.

25. Kirchhoff’s voltage law now states Vbattery + Vr + Vc = 0 and again the current is the same round the loop students will be able to show: V = Vbattery (1 – e-t/RC)

26. The bottom axis is shown here in terms of multiples of RC, so it isa ‘universal’ plot.

27. Note that the current and voltage both have exponential forms, as the voltage increases, the current decreases (or vice versa)

28. Finally, these are differential equations. • To find particular solution requires boundary conditions • For step voltages (switches for example) • determined from the conditions at t=0 • Need to evaluate voltages and currents just after switch moved • Important: • The voltage across a capacitor cannot change instantaneously • Because E=CV2/2 so if the voltage change instant, implies infinite power • The voltage across a capacitor CANNOT change instantly • The current through a capacitor CAN change instantly • The voltage and the currents for a resistor can BOTH change instantly

29. Recipe for analysing RC circuits • Develop differential equation using • Kirchhoff’s Laws • V=IR for resistors • V=Q/C for capacitors • Establish boundary conditions by • Working out conditions just before switch moved • Evaluating what changes will occur just after using • V across capacitors unchanged • I through capacitor can change • V and I can both change in a resistor This may be easy, or it may not depending on circuit. It takes practice. Practice!

30. Analysis of RC networks is not an academic exercise, nearly allelectronics will contain them. • They are used as • filters • noise suppressors • voltage smoothers • integrators and MANY more BUT they are mostly used withAC signals, which we will not talk about in this course Something for you to look forward to later!