o 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

o  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

o 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

Reminder:

Consider a wire with a voltage across it.

oCurrent 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

Remember: atoms fixed in place electrons move

Area = A

Current density, = i/A

Current = i

Current density, = i/A

For example, up to 5A currents in household wires use 1mm2 copper wires

= 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.

= 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

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/molAvogadros 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!

The electrons in a wire don’t move far normally

o 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

This is actuallya VERY thin wire wound in a spiral

Kirchhoff’s Laws

o These are effectively energy conservation charge conservation

o 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)

Analogy:

If water flows into a junctionThen the volume of water flowing in equals the total flowing out

So ID+IC+IB = IA

Note that IIN = Ia + Ib

capacitors do NOT store charge.

This is the reason we have been so ‘determined’ thatcapacitors should not be thought of as ‘storing’ charge - if they could then IIN would not necessarily be Ia+Ib.

IINIa

Ib

Voltage Law

The sum of the voltage drops round a closed loop is zero

Recall that voltage is a measure of potentialAnd remember that gravitational potential behaves in a similar way

If a mass is moved round a closed path in a gravity fieldthe sum of mgh 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.

Be careful defining the sign.

You MUST measure thevoltages in the same directionOn all voltages round the loop

This is the old fashionedsymbol for a resistor,it is still used a lot

Example

Prof. Kirchhoff

vi = 0 round loopIi = 0 into node

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

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 equationThese 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.

Sometimes called Kirchhoff’sFirst and Second Laws

Note that in this diagram,voltage ‘a’ will have the oppositesign to all the others

With this definition ofdirections, all the currentswill be positive.

Switch

RC Circuits

Q=CVI = dQ/dt = CdV/dt

vi = 0 round loopIi = 0 into node

VC=0

Vr=0

RC Circuits

Q=CVI = dQ/dt = CdV/dt

vi = 0 round loopIi = 0 into node

Vc=emf

Vr=0

A ‘long’ time after the switch is closed.

RC Circuits

Q=CVI = dQ/dt = CdV/dt

vi = 0 round loopIi = 0 into node

Vc=emf

Vr=0

Now open switch again

Vr=emf

RC Circuits

Q=CVI = dQ/dt = CdV/dt

vi = 0 round loopIi = 0 into node

Vc=emf

Vr=0

Then close the other way

But now thereis a circuit witha resistor acrossan energised capacitor(‘charged’ capacitor)

RC Circuits

I = dQ/dt = CdV/dt

vi = 0 round loopSo Vr = -Vc

Vc

Vr

The current flowinground the circuitis the same everywhereSo, using

then I = Vr/R = -CdVr/dt

so Vr/R + CdVr/dt = 0

The solution is Vr = Ae-t/RC

Where A is a constant and is the voltageat time = 0, say V0

(in this case V0 is what was called ‘emf’ earlier)

RC Circuits

Vc

Vr

Vr = V0e-t/RC

A ‘discharging’ capacitor obeys the equation

RC is called the time constant for the circuitThe voltage drops from V0 to V0/e in a time RC

0

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

V = Vbattery (1 – e-t/RC)

Kirchhoff’s voltage law now states Vbattery + Vr + Vc = 0and again the current is the same round the loop students will be able to show:

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

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

o Finally, these are differential equations.o To find particular solution requires boundary conditionso For step voltages (switches for example)

determined from the conditions at t=0Need to evaluate voltages and currents just after switch movedImportant:

The voltage across a capacitor cannot change instantaneously Because E=CV2/2 so if the voltage change instant, implies infinite power

o The voltage across a capacitor CANNOT change instantlyo The current through a capacitor CAN change instantlyo The voltage and the currents for a resistor can BOTH change instantly

Recipe for analysing RC circuits

o Develop differential equation using Kirchhoff’s Laws V=IR for resistors V=Q/C for capacitors

o 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!

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!