AC circuits, Capacitors, Induction

School Science Lessons
2025-09-03
(UNPh30.1)

Alternating currents and voltages
Contents
30.1.0 Alternating currents and voltages
30.2.0 Inductors
30.3.0 Resonance
30.4.0 RC circuits
30.5.0 RLC circuits
30.6.0 Transformers

30.1.0 Alternating currents and voltages
30.1.1 AC current and voltage
30.1.2 AC circuits, (Table)
30.1.3 AC frequency
30.1.4 Current, AC, root mean square value (RMS)
30.1.5 Voltage, AC, root mean square value (RMS)
30.1.6 Alternators
30.1.7 Circuits in homes
30.1.8 Power, AC
30.1.9 Resistors and Ohm's law
30.1.10 Resistor in simple AC circuit
30.1.11 Root mean square values, RMS values
30.1.12 Saving energy

30.2.0 Inductors
30.2.1 Inductors
30.2.2 Inductor circuits
30.2.3 Inductive reactance
30.2.5 Dimmers
30.2.6 Impedance
30.2.7 RL circuits, impedance
30.2.8 RL series combinations
30.2.9 RMS currents

30.3.0 Resonance
30.3.1 Bandwidth and Q factor
30.3.2 Current vs frequency graph
30.3.3 Impedance vs frequency graph
30.3.4 Metal detectors
30.3.5 Resonance in LC circuits

30.4.0 RC circuits
30.4.1 RC circuits, impedance, phase angle and power factor
30.4.2 Impedance in an RC circuit
30.4.6 Voltage and current in an RC circuit

30.5.0 RLC circuits
30.5.2 RLC circuits, impedance
30.5.3 RLC series combinations
30.5.4 Impedance, RLC circuit
30.5.5 Limit of frequencies
30.5.6 Phase angle

30.6.0 Transformers
30.6.1 Hall effect
30.6.2 Magnetoresistance
30.6.3 Magnetostriction

30.1.1 AC current and voltage
Alternating current reverses polarity at regular intervals.
See diagram 30.1.3: Simple electric motor.
See diagram: AC voltage and current.
In the diagram, V = V_max sin ωt, and I = I_max sin ωt = ( V_max / R) sin ωt, corresponding to the AC voltage.
The voltages of the generator and the current through the resistor are in phase, so their maxima and minima occur simultaneously.

30.1.2 AC circuits
Table 30.1.12: Properties of different AC circuits

Circuit element	Impedance, Z	Average power, P_av	Phase angle, Φ
Resistance, R	Z = R	P_av = I²_RMSR = V²_RMS / R	Φ = 0^o
Capacitor, C	Z = X_C = 1 / ωC	P_av = 0	Φ = -90^o
Inductor, I	Z = X_L = ωL	P_av = 0	Φ = +90^o
RC circuit	Z = (R² + X_c²) = [R² + (1 / ωC)²]	P_av = I_RMSV_RMS cos Φ	-90^o< Φ <0^o
RL circuit	Z = (R² + X_L²) = (R² + ωL)²	P_av= I_RMSV_RMS cos Φ	0^o < Φ < 90^o\
RLC circuit .	Z = R² + ( X_L - X_C)² = R² + (ωL - 1 / ωC)^2 .	P_av = I_RMSV_RMS cos Φ .	-90^o < Φ <0^o (X_C > X_L)
"	"	"	0^o< Φ < 90^o (X_C < X_L)

30.1.3 AC frequency
The frequency of alternating current is the number of complete alternations (cycles) in one second, measured in hertz, Hz (formerly cycles per second, c / s, ) (Heinrich Hertz, Germany, 1857-1894).
In many countries, e.g. UK, the mains supply has frequency 50 Hz, so one alternating current cycle lasts 1 / 50 = 0.02 seconds.

30.1.4 Current, AC, root mean square value (RMS)
The current through a light bulb in a simple AC circuit, I = V / R = ( V_max / R) sin ωt = I_max sin ωt.
Calculate root mean square value, RMS I = I_max sin ωt.
Square both sides so I² is always positive, so I² = I²_max sin² ωt.
A graph of I² against time shows I² fluctuating symmetrically between 0 and I²_max, so the mean of I² = I² / 2.
So I²_av = I²_max.
So I_av, i.e. I _RMS = I_av = (1 / 2) I _max = 0.70710678 I _max = 0.71 I_max= 0.71 x peak value.
So a perfect sine wave has a RMS amplitude = about 0.7071 x the peak amplitude.
When voltage or current is quoted, it may be a maximum value or an RMS value, as is usually quoted by ammeters and voltmeters.
Use the equation: I _max = 2 x I_RMS.

30.1.5 Voltage, AC, root mean square value (RMS)
Voltage = electrical potential difference.
The AC generator produces a voltage that varies with time.
The equation for the voltage delivered by an AC generator, V = V_max sin ωt, where V_max is the maximum value of a voltage during a cycle, ω is the angular frequency (ω = 2πf, where f = 60 Hz in USA).
The graph for the sinusoidal time dependence for an AC circuit is similar to the graph for simple harmonic motion.
RMS value of the voltage in an AC circuit, V RMS = (1 / 2) V _max, V_max / 2 or V_max = 2V RMS.
If RMS of the voltage in a domestic circuit = 120 V, peak value in the circuit, V _max = 2V RMS = V_max = 2 x 120 = 169.7056 = 170 V.

30.1.6 Alternators
1. In a direct current, DC, circuit the electrons flow only in one direction.
In an alternating current circuit, AC, circuit, the direction of electron flow reverses regularly.
So a moving coil meter in a DC circuit show deflection in one way only, but in an AC circuit the pointer moves about the zero reading, if the current is very slow, or shows no deflection.
AC current can be rectified to DC current for use in certain processes, e.g. electroplating and battery charging.
2. AC generators (alternators), operate on the same principles of electromagnetic induction as DC generators.
Alternating voltage is generated by rotating a coil in a magnetic field or by rotating a magnetic field within a stationary coil.
The value of the voltage generated depends on the number of turns in the coil, the strength of the magnetic field, and the speed of rotation of the coil or magnetic field.
The transmission of high voltage alternating current, AC, over long distances is more efficient than the transmission of direct current, DC.
Currents that vary periodically in their size and direction with the time are called alternating current.
Alternating current, like the wave, has its frequency, period, amplitude and phase.
Classify the alternating current into low frequency and high frequency.
Lighting circuits use the low frequency of a sine wave.
Alternating voltages and currents supplied to a light bulb vary sinusoidally, i.e. a curve having the form of a sine wave.
Alternating current reverses its polarity at regular sinusoidal intervals.
3. Alternating current is produced by an alternator, i.e. a synchronous alternating current generator.
In the AC generator, alternator, mechanical energy turns a coil in a magnetic field, B, and a variable electromagnetic field, emf, is induced across the ends of the coil.
Permanent sliding contacts are made with the ends of the coil.
A given contact will change from positive to negative, depending on the relative direction in which that side of the coil is moving through the magnetic field, slip rings and contacts.
If the coil moves clockwise in the magnetic field, an end-on view of the sides of the loop shows how the emf, is alternating.
An alternator produces an emf, which changes from positive to negative, so the average will be zero emf.
See diagram 30.10: Alternator, AC Generator.
When side BC of the coil is in positions (1.) and (3.), the coil is moving in the same direction as the magnetic field, from N to S, no emf is produced.
When side BC of the coil is in positions (2.) and (4.), the coil is moving perpendicular to the magnetic field from N to S, maximum emf is produced.
Experiment
4. Simple alternator
Use 2 solenoids, an inner and an outer solenoid.
Observe the movement of the pointer of the centre zero galvanometer when the inner solenoid is stationary.
The galvanometer pointer does not move.
Observe the movement of the pointer when the solenoid supported by a light spiral spring moves up and down.
The galvanometer pointer moves each side of the zero mark in a regular manner corresponding to the motion of the coil.
The current passing through the galvanometer is a simple form of alternating current.

30.1.79 Circuits in homes
Circuits in homes, ring circuits, radial circuits, lighting circuits, power sockets and plugs
1. Ring circuits are commonly used in the United Kingdom, Indonesia, Republic of Ireland, Singapore and United Arab Emirates.
Radial circuits are commonly used in Australia, and in many other countries.
The incompatibility in the over current protection of appliance leads between countries using ring and radial circuits has been a major stumbling block on the road to world wide standardization of domestic AC power plugs and sockets.
Plug fuses can be better matched to the maximum current required by an appliance, but some plugs in the UK are fitted with a fuse of the maximum permitted rating of 13 A, because a lower rated device may well operate intermittently due to "surges".
So the fused plug offers no advantage over an unfused plug used on radial circuit with a 13 A fuse, or B16 or lower circuit breaker.
2. Ring circuits (ring mains circuits, ring final circuits), have a cable leaving the consumer unit and travelling to each socket on the main.
When it reaches the last socket, it then returns to the consumer unit, so creating a ring.
The advantage of ring circuits is that power can reach the sockets in the circuit from both directions and so reduce the power load on the cables.
A ring circuit has a 30 amp fuse or 32 amp MCB on the consumer unit.
A house usually has one ring circuit upstairs and one ring circuit downstairs.
Ring circuits can have extra sockets added to them by adding a "spur" onto a ring circuit.
The maximum load of the circuit (30 / 32 amp) still exists).
3. Ring circuits in England
The wiring in a house connects all appliances together in parallel.
This is so that each has the mains supply of 230 volts across it and also so that they can all be used independently.
The consumer unit contains the main switch and the fuses for all of the fixed circuits, e.g. the power ring circuit and the lighting circuit.
The power sockets in a house are connected by means of a ring circuit.
In a ring circuit, the live, neutral and earth wires form a loop of cable going from the consumer unit to all of the sockets in turn and then back to the consumer unit.
The advantages of using a ring circuit are that the cables can be made thinner, because there are two paths for the current, so each part of the cable carries less current.
A ring circuit is more convenient, because sockets can be placed anywhere on the ring.
The principle of the radial circuit, is that the mains cable leaves the consumer unit and passes through each socket until it reaches and ends at the last socket.
Alternatively, on a ring circuit the mains cable leaves the consumer unit passes through every socket and then returns to the consumer unit.
The advantage of the ring circuit is that electricity can reach the sockets from two directions and so reduces the load on the cable.
4. Electricity is usually connected to homes by a supply cable containing two wires, the live wire, L, and the neutral wire, N.
The neutral wire is earthed at the local substation and so is at zero potential.
The supply is AC, so the live wire is alternately positive and negative.
Every circuit is connected in parallel with the supply, i.e. across the live, L (brown wire) and neutral, N (blue wire) and receives the full mains potential difference of 240 V.
Switches and fuses are always in the live, L, wire.
If they were in the neutral, N, wire, lamp and power sockets would be "live" when switches were "off" or fuses "blown".
A fatal shock could then be received, e.g. by touching the element of an electric fire when it was switched off.
In the ring main circuits, the live and neutral wires run in two complete rings around the house.
The power sockets, rated at 13A, are connected off from them.
The ring has a 30 A fuse and total current must not exceed 30 A.
Each fused plugs used in the ring main circuit has its own cartridge fuse.
A ring main circuit has a third earth wire, E (yellow green), which connects to the top sockets on all power points, and is usually earthed by being connected either to a metal water pipe.
This third wire is a safety precaution to prevent electric shock if an appliance develops a fault.
The earth pin on a 3-pin plug is connected to the metal case of an appliance, which is thus joined to earth by a path of almost zero resistance.
So if the element of an electric fire breaks and touches the metal case, a large current flows to earth and "blows" the fuse.
5. In a ring final circuit (ring-main system, ring circuit), the line conductor starts and ends at the same line terminal, and the neutral conductor starts and ends at the same neutral terminal in the consumer unit.
So the line current can flow in two opposite directions to the same load and the current-carrying capacity of the conductors is larger than it would be if the same cross-sectional area conductors were used in a radial system.
Loads connected to the ring main system are individually fused using fused plugs.
A ring circuit goes to every socket and at the last socket it returns to the consumer unit.
It starts at the consumer unit and returns to the consumer unit and is usually protected by a 30 amp fuse or 32 amp circuit breaker.
It can have any number of sockets or fused connection units on it, but the maximum load is 7200 watts.
So usually three ring mains circuits are installed in a house, on the ground floor, on the upstairs floor and one serving the kitchen where most of the high consumption appliances are connected.
Both ends of the ring are connected to the same terminals at the consumer unit, so the current runs in both directions imposing less load on the cables.
Electricity loses power over long lengths of cable and trying to put too much power through a cable, which is not designed for it, is dangerous, so a ring main delivers power from both ends to keep the load as light as possible.
6. In both ring final circuits and radial final circuits each socket outlet is connected from the previous socket, but in ring final circuits the last socket is returned to the same three terminals in the distribution board, so the two ends of each conductor must be clamped under the same terminal screw.
All the outlets should be included in the ring, but a branch connection called a spur may be used to supply permanently connected appliances.
The spur cable must be protected, e.g. by a 5 amp fuse.
7. Radial circuits, radial final circuits, have a cable leaving the consumer unit and travelling to each socket, similar to the ring circuit.
However, when the circuit reaches the last socket the cable ends, unlike a ring main that travels back to the consumer unit.
Radial circuits can therefore only serve a smaller area.
In a similar way to ring circuits, spurs can be added at points along the radial circuit if required.
High-powered appliances (cookers / showers) must have their own radial circuit.
In a "radial circuit" only one end of the circuit is connected to the line and neutral terminals, with opposite ends terminating at a load.
A radial circuit stops at the final socket.
Radial circuit is from the panel and stops at the outlet without going back to the panel.
It is simply a length of appropriately rated cable feeding one power point then going on to the next.
The circuit terminates with the last point on it.
It does not return to the consumer unit or fuse box.
In a radial final circuit supplying sockets, each socket outlet is fed via the previous socket.
Live is connected to live, neutral to neutral and earth to earth at each socket outlet.
The maximum number of socket outlets per circuit is ten.
The radial final circuit should not supply more than two rooms.
A kitchen should be supplied by at least two radial circuits.
8. Radial circuits in Australia
Socket outlet and 3-pin plug, 230 volt male plug.
Australian circuit.
In 1983, Standards Australia adopted a 20 year plan to convert Australia from the nominal 240 volts to 230volts, to align with European Standards - IEC38.
The aim was to align Australian manufactured products with the main trading partners.
Domestic power sockets and plugs in Australia
Contacts, flex conductor colour:
Active A, "live wire" at 230 V with respect to earth, brown flex wire,
Neutral, N, the return to the substation, nearly 0 V, blue flex wire,
Earth, E, connects to the earth near the building, usually via water pipes. green / yellow flex wire.
The 230 volt male plug, when viewed so that you are looking at the pins and the earth pin is at the bottom, has the active pin at the right.
Appliances with a metal case usually have that case connected to the earth pin, so when plugged in this connection provides a low resistance pathway to ground.
An earthed case provides low resistance return path and blows fuse, or preferably triggers an earth leakage detector.
The switches are in the active line so the master switches should be off before changing fuses.
GPO, General Purpose Outlet, it is an electrical term.
The definition for GPO in the AS3000 "Australian Standard for Wiring Rules" is "three pin flat pin 10 amp socket outlet" (or "power point" as commonly known).
However, the definition was removed from the 2000 edition of the wiring rules and "power points" are now defined as "socket outlets".
9. Lighting circuits are basically radial circuits.
There are two distinct types of lighting, circuit the loop-in circuit and the older junction box circuit.
The loop-in circuit has a cable, running from light to light terminating at the last light as in the conventional radial circuits and then single cable runs from the lights to the light switches.
The other type of lighting circuit has a junction box for each light.
The cable runs from the consumer unit to the first junction box and then onto the next terminating at the last junction box.
Then another cable is run from each junction box to its light and another wire from the junction box to that light switch.
A 5 amp fuse or 6 amp MCB is used on the consumer unit for a lighting circuit.

30.1.8 Power, AC
Average power, P_av = V_RMS x I_RMS.
Australian and British power systems operate at 240 volts RMS, at 50 cycles per second (50 Hertz).
The peak values are 340 volt.
V_{alternating current} = V_{peak value} / 2 = 340/ 2 = 240 volts.
American and Japanese systems use 110 volts RMS and 60 cycles per second.
Average power, P_av = V_RMS x I_RMS.

Average power, P_av, consumed by a circuit
1. I_RMS = V_RMS / R.
2. I_max = V_max / R.
3. Instantaneous power consumed by a resistor, P = I²R.
4. Over time I = I _max sin ωt.
So P = I²R = I²_max R sin² ωt.
6. The average of sin² ωt = 1/2.
7. So P_av = I²_max R(sin² ωt)_av.
8. I_RMS = I_max / 2.
9. Average power, P_av (using current) = I²_RMSR.
If the frequency changes, the average power stays the same.

Instantaneous power consumption, P = I² R, in DC and AC circuits.
Replace I with I_RMS to find average power.
Average power, P_av = I²_max x R.
P_av (using voltage) = V² / R = (V²_max / R)sin² ωt.
So average power, P_av = (V²_max / R) 1/2 = V²_RMS / R.

30.1.9 Resistors and Ohm's law
Resistances, (Commercial).
Voltage and current in a resistor as a function of time.
Let v = voltage and i = current when studying their variation with time.
Let peak values, amplitudes, of a sinusoidal variation be V_m and I_m.
Let RMS values be V = Vm / 2 and I = Im / 2.
The voltage, v, across a resistor, R, is proportional to the current, i, travelling through it, so v = Ri.
Diagram 30.1.8 shows the peak value of voltage through a resistor = R x peak value of current, I_m.
They are in phase when the current is a maximum and the voltage is a maximum.

30.1.10 Resistor in simple AC circuit
(In an alternating current circuit diagram, the symbol for a wall socket by an AC generator is a circle enclosing one cycle of a sine wave.)
If a 50 Hz, maximum voltage 24 V generator is connected directly to a 265 ohm resistor in a circuit,
RMS voltage, V_RMS = V_max / 2 = 24.0 / 2 = 16.970 = 17.0 V.
RMS current in the circuit, I _RMS = V_RMS / R = 17.0 / 265 = 0.0641509 = 0.0642 A.
Average power (using the current), P_av = I²_RMS R = (0.0642)² x 265 = 1.0922346 = 1.09 W.
Average power (using the voltage), P_av = V²_RMS / R = (17.0)² / 265 = 1.09056603 = 1.09 W.
Maximum power used by the resistor, P_max = 2 x average power = 2 x 1.09 = 2.18 W.
This calculation does not use the frequency of the generator (50 Hz), because resistance is independent of frequency.
So if the frequency were varied, the average power would still be the same.
If the coil moves clockwise in the magnetic field, the emf is alternating to produce a sine wave pattern of induced emf, characteristic of alternating current, AC.
However, when the coil is moving in the same direction as the magnetic field, no emf is produced.
A coil moving perpendicular to the magnetic field produces maximum emf.

30.1.11 Root mean square values, RMS values
Root mean square values, RMS values, peak value, electrical power systems
Sinusoidal time dependence, Instantaneous current, I, varies with time: I = I_o.
As alternating current fluctuates from positive to negative vales it is measured by its peak value or by its root mean square value.
RMS = (peak value / 2).
This is the steady direct voltage or current that would give the same heating effect.
The effective emf, is calculated by squaring the maximum positive emf, (peak value), and the maximum negative emf (peak value), adding them together, dividing by two, and taking the square root.
The root mean square value, RMS = (peak value / 2) = (peak value is approx. 0.7071). RMS values for alternating voltage and current, AC, are equivalent to the same values for direct voltage and current, DC.
The waveform is sinusoidal.
As an alternator produces an emf that changes from positive to negative, the average will be zero emf.
The effective emf, is measured by squaring the maximum positive emf (peak value), and the maximum negative emf, (peak value), adding them together, dividing by two, and taking the square root.
The root mean square value (RMS) = (1 / 2) = peak value = 0.71 x peak value.
RMS values for alternating voltage and alternating current, AC, are equivalent to the same values for direct voltage and direct current, DC.
RMS values for AC in the British and Australian electrical power systems use 240 volts, RMS, at 50 cycles per second (50 Hz).
If root mean square value, RMS, voltage of the mains supply = 240 V, the peak value is 240 / 0.7 = 340 V.
American and Japanese electrical power systems use 110 volts RMS and 60 cycles per second (60 Hz). Peak value = 120 / 2 = 170 V.

30.1.12 Saving energy
1. Turn incandescent lights off when they are not needed.
Incandescent lighting accounts for about 85% of household illumination.
Incandescent light bulbs are the least expensive to buy, they are more expensive to operate, because of their short life spans and relative inefficiency.
Only 10 to 15% of the electricity that incandescent lights use results in light and the rest turns into heat.
The value of the energy saved by not having such lights on is far greater than the cost of replacing these bulbs.
The power surge from turning on a light uses as much power as leaving it on for a fraction of a second.
However, for fluorescent tube lights, the start-up uses about 23 seconds worth of power.
Turning the light on and off repeatedly does not reduce the total life expectancy of the light bulb enough to offset the increased electricity usage.
Turning a light off is more economical than leaving it on.
Fluorescent light tubes are more expensive than incandescent light bulbs and their operating life is affected by the number of times they are switched off and on.
Turn fluorescent lights off if leaving a room for more than five minutes.
2. Determine how much energy the light consumes when it is turned on by reading the watt rating printed on the light.
Check the cost of electricity per kWh from the utility bill.
The cost may be different during peak versus general periods.
So leaving a 40-watt bulb on for one hour consumes 0.04 kWh.
If your electric rate is 15 cents per kWh, your energy savings is 0.6 cents.
Compact florescent lights (CFLs) use 1 / 6th of the energy that a standard incandescent bulb uses, while producing the same amount of light.
They are more expensive, but they last about 10 times longer than standard incandescent bulbs.
3. When using a computer, turn off the monitor if away for 20 minutes, and shut down the main computer if away for more than two hours.
A computer may have a "stand by" mode that uses 70% less power than normal, and a "shut down" option that turns the computer fully off.
A screen saver that shows any image on the screen does not save any energy.
Laptop computers use much less energy than desktop computers.

30.2.1 Inductors
Inductors, inductance, inductive reactance, self induction, mutual induction
Induction is the production of an electrical or magnetic state in an object, caused by its nearness to an electrified or magnetic object, but not touching it.
1. An inductor is a conductor in which an electromotive force or current is induced.
The inductance of a conductor is when any variation in current through it induces an electromotive force (voltage) in it (self-inductance), or in any other conductor (mutual inductance).
In self-induction, a current is induced in a circuit caused by a variation in current in that circuit.
In mutual induction, the variation in current flowing through one circuit causes an electromotive force in another nearby circuit.
The SI unit of inductance, the henry, H, is the inductance of a circuit where an electromotive force of one volt is formed by a current changing at the rate of one ampere per second.
2. In accordance with Ohm's law, current passing through a resistance, e.g. a straight wire, decreases as resistance increases, whether the current is DC or AC.
So when a straight wire or a coil of wire is connected to DC, it draws an electric current, I = V / R, and it gets hot as energy is dissipated in the wire until the wire melts.
However, if a coil of wire is connected to AC and the frequency increased, the coil shows inductive reactance, called "electrical sluggishness", because it takes some time for the current to form in the coil as each voltage cycle occurs.
As the AC frequency is increased, at some frequency the current cannot form in the coil before the polarity of the voltage reverses.
So almost no current forms in the coil and it remains cool.
If the supply delivered the same power as a DC source it would melt the wire.
3. Inductive reactance X_L (ohms) = 2 π fL, where f = frequency of the current (hertz), and L = inductance of the coil (henry).
If inductance is constant in a coil, inductive reactance increases with AC frequency.
However, if AC frequency is constant, inductive reactance increases with increasing inductance.
4. When electric current flows through the coil from the left to the right, a magnetic field is generated in a clockwise direction.
The inductor stores electrical energy in the form of magnetic energy.
The more turns of the coil, the stronger the magnetic field generated.
A stronger magnetic field can be generated by increasing the cross-sectional area of the inductor, or by changing the core of the inductor.
If AC current is flowing through the inductor, the magnetic field generated by that current cuts across the other windings, causing an induced voltage and preventing any changes in the current level.
If the current is about to rise suddenly, an electromotive force is generated in the opposite direction to the current, preventing any increase in the current.
If the current is about to drop, an electromotive force is generated in the direction in which the current is increased.
So the coil allows DC, but not AC, to flow through it.
The emf always opposes the change in current (c.f. Lenz's law).
5. The largest voltage across the inductor occurs when the current is changing most rapidly, i.e. when the current is instantaneously zero.
The voltage across the ideal conductor is 90^o ahead of the current, i.e. it reaches its peak one quarter cycle before the current does.
6. An inductor, choke, is usually a coil of wire, with an air or iron core, that ideally has negligible resistance and capacitance.
The voltage that appears across an inductor is caused by its own magnetic field and Faraday's law of electromagnetic induction.
The current passing through the coil creates a magnetic field whose magnetic flux is proportional to the field strength, which is proportional to the current flowing.

30.2.2 Inductor circuits
At time zero the current is zero, but increasing at it maximum time rate.
Voltage across an inductor depends on the rate of change of current, so the voltage of the inductor is a maximum at t = 0.
When the current reaches a maximum at time ωt = π / 2, the rate of change of the current becomes zero.
So at that point the voltage across the inductor falls to zero.
The current and the voltage of the inductor are a quarter of a cycle, 90^o, out of phase.
The voltage reaches a maximum before the current reaches a maximum, so in an inductor the voltage leads the current by 90^o.
An inductor behaves in the opposite way to a capacitor.

30.2.3 Inductive reactance.br> Inductive reactance, X_L = ωL
1. Just as voltage across a capacitor, V = IX_C, and X_C = 1 / ωC, where X_C is the capacitative reactance, similarly voltage across an inductor, V = IX_L, where XL is the inductive reactance.
Inductive reactance, X_L = ωL.
Diagram 30.2.5 shows that the capacitative reactance, X_C, becomes large with decreasing frequency, but the inductive reactance, X_L, becomes large with increasing frequency.
Note the resistance, R, is independent of frequency.
2. Inductive reactance, X_L is the ratio of the voltage and the current, X_L = ωL.
For peak voltage and current, V_m = X_L I_m.
The largest voltage across the inductor occurs when the current is changing most rapidly, i.e. when the current is instantaneously zero amps.
The voltage across the ideal conductor is 90^o ahead of the current, i.e. it reaches its peak one quarter cycle before the current reaches its peak.
3. Diagram 30.2.01b also show that the reactance is frequency dependent, XL = ωL.
When the frequency is halved, but the current amplitude kept constant, the current is varying only half as quickly, as is the Faraday emf.
For an inductor, the ratio of voltage to current increases with frequency.
4. Inductive reactance, X_L, increases with frequency, unlike capacitive reactance, X _c, because the higher the frequency the more rapidly current changes with time, so the greater voltage across the inductor.
RMS current in an inductor, I_RMS = V_RMS / X_L.
The voltage across an inductor leads the current by 90^o.
The characteristic time over which the current changes (time constant), τ = L / R.
Increasing current with time, the switch is closed in the simplest RL circuit at time t = 0, the current increases with time, I = emf / R (1 -e^{-tR / L}).
Total voltage in the circuit, Vmax = (Imax R)² + (ImaxXL)² = Imax (R² + XL²) = Imax Z.
Total voltage in the circuit, V_max = (I_max R)² + (I_max X_L)² = I_max (R² + X_L²) = I_max Z.

30.2.5 Dimmers
Observe the brightness of a light bulb, resistance R, in a circuit powered by an AC generator.
The current through the light bulb, I_RMS = V_RMS / R.
Add an inductor, L, to the circuit.
The impedance, Z = (R² + X_L²) > R.
So the current in the circuit, I_RMS = V_RMS / Z.
The current is less, so the light bulb is glows dimmer.
In a light dimmer, an iron rod is moved into or out of the coil of an inductor.
When the iron rod is deeper in the coil, the inductance is increased, so the current is decreased in the circuit and the light glows dimmer.

30.2.6 RMS currents in RC circuits and R.
If both circuits have the same resistance, the current in RC circuits and RL circuits decreases with increasing frequency.
In the RC circuit, the current is low when the capacitive reactance is high, at low frequency.
In the RL circuit, the current is low where the inductive reactance is high, high frequency.
If the circuit contained only a resistor, the current has the same RMS current with increasing frequency, so the RMS current does not change.

30.2.7 RL circuits, impedance
RL circuits, (resistance-inductance circuits)
The simplest RL circuit contains a battery (emf ), a switch, a resistor, R, and an inductor, L, connected in series.
Maximum current, I_max = V_max / Z, where V_max is the maximum voltage of the AC generator.
Z is the RL circuit impedance, Z = (R² + X_L²) = (R² + ω L)².
The impedance for an RL circuit has the same form as for an RC circuit, (except X_L instead of X_C.)
Power factor for an RL circuit, cos Φ = R / Z = R / (R²+ ω L)².
1. In an AC circuit, impedance refers to the combined opposition to movement of alternating current caused by the resistance (R), and the reactance (X), measured by the ratio of the electromotive force to the resulting current.
Impedance Z = (R² + X²).
Only the ohmic resistance (R), dissipates electrical energy as heat.
2. Total impedance, Z, is the total opposition to current flow from :
2.1. inductance (L),
2.2. resistance (R), and
2.3. capacitance (C), in an AC circuit.
Impedance Z = (R² + [XL - XC]²).
3. Resonance in series AC
A circuit produces the largest possible response to an applied oscillating signal when its inductive and capacitative reactances are balanced.
Maximum current will flow in a circuit affected by inductance, resistance, and capacitance, when the inductive reactance (X_L) cancels the capacitive reactance (X_c).
Impedance is the combined opposition to the passage of AC exerted by the resistance (R).
Reactance (X_c), measures the ratio of the electromotive force to the resulting current, so Z = (R² + X_L²).
Impedance is the general term for the ratio of voltage to current.
Resistance is the special case of impedance when Φ (phi) = 0.
Reactance is a special case of impedance when Φ (phi), = 90^o.
The voltage on the capacitor is behind the current, because the charge does not build up until after the current has been flowing for some time.

Resistor, R	Capacitor, C	Inductor, L
Resistance	Capacitive reactance, X_c	Inductive reactance, X_L
V_R / I = R	V_c / I = X_c = 1 / ω C	VL / I = X_L = ω L
V and I in phase	V lags I by π / 2	V leads I by π / 2

"30.2.8 RL series combinations
The voltage across the inductor is ahead of the current by 90^o.
The inductive reactance, X_L = ω L.
RL circuits are not used as much as RC circuits, except at high frequencies, because finding suitable inductors is difficult.

30.3.1 Bandwidth and Q factor
At resonance, the voltages across the capacitor and the pure inductance cancel.
So the series impedance has its minimum value: Z_o = R.
So if the voltage is kept constant, the current is a maximum at resonance.
The current goes down to zero at low frequency, because X_C becomes infinite.
The current falls to zero at high frequency, because X_L increases with ω when the inductor opposes rapid changes in the current.
Diagram 30.3.1 shows I(ω) for a circuit with a large resistor, the lower curve, and with a small resistor, the upper curve.
A circuit with low R, for a given L and C, has a sharp resonance.
Increasing the resistance makes the resonance less sharp.
The former circuit is more selective.
It produces high currents only for a narrow bandwidth, i.e. a small range of ω or f.
The circuit with higher R responds to a wider range of frequencies and so has a larger bandwidth.
The bandwidth Aω (see the horizontal bars on the curves), is defined as the difference between the two frequencies ω and ω at which the circuit converts power at half the maximum rate.
The electrical power converted to heat in this circuit = I²R, so the maximum power is converted at resonance, ω = ω_o.
The circuit converts power at half this rate when the current is I_o 2.
The Q value is defined as the ratio Q = ω_o / ω.

30.3.2 Current versus frequency graph
Diagram 30.3.3 shows current plotted against frequency in an RLC circuit.
The peak current is at the resonance frequency, 1 / LC.
Larger resistance reduces the maximum current in the circuit, but does not change the natural frequency.
Larger resistance does make the resonance peak lower and broader.
So resonance can occur over a wide range of frequencies with only a small increase of current.
Smaller resistance increases the maximum current in the circuit, but does not change the natural frequency.
Smaller resistance does make the resonance peak higher and sharper.
So resonance can occur over a narrow range of frequencies with a large increase in current.

30.3.3 Impedance versus frequency graph
Series impedance rises to infinity at high frequency, because the inductor produces a large emf if the current varies rapidly.
Similarly series impedance is large at very low frequency, because the capacitor has a long time in each half cycle to get charged.
The minimum value of series impedance occurs when the voltages across the capacitor and inductor are equal and opposite, i.e. v_L(t) = - VC(t), so VLt = VC.
ωL = 1 / ωC, and this occurs at frequency ω₀ = 1 / LC.
f₀ = 1 / 2π 1 / LC, where ω₀ is the angular frequency of resonance and f₀ is the cyclic frequency of resonance.
At resonance, series impedance is a minimum, so the voltage for a given current is a minimum, or the current for a given voltage is a minimum.
So in an RLC circuit where the inductor has a low internal resistance, r, a large voltage can occur across the inductor and an equally large voltage can occur across the capacitor.
However, their voltages can cancel, because the two voltages are 180^o out of phase.
So a large voltage oscillation can be formed with a small voltage input from the power source.
The energy stored in the oscillations is gradually supplied by the AC source and is then exchanged between the capacitor and inductor in each cycle.

30.3.4 Metal detectors
Metal detectors have RLC circuits that vary the inductance.
Metal objects carried by airline passengers change the inductance of a large coil inductor that the passengers pass through.

30.3.5 Resonance in LC circuits
1. AC circuits have a natural frequency of oscillation, similar to a pendulum, or mass supported by a spring.
So an LC circuit containing only an inductor and a capacitor has the nature frequency, ω_o = 1 / LC = 2πf_o.
At ω = 1 / LC, an RLC circuit has maximum current, resonance.
2. The expression for the series impedance goes to infinity at high frequency, because the inductor produces a large emf if the current varies rapidly.
Similarly, the expression is large at very low frequencies, because of the capacitor, which has a long time in each half cycle to charge up.
In the plot of Z_seriesω, there is a minimum value of the series impedance, when the voltages across capacitor and inductor are equal and opposite, i.e. v_L(t) = v_C(t) = va(t).
So V_L(t) = V_C.
So ωL = l / ωC, and the frequency at which this occurs is ω_o = 1 / LC, where ω_o is the angular frequency of resonance and f_o is the cyclic frequency of resonance.
At resonance, series impedance is a minimum, so the voltage for a given current is a minimum or the current for a given voltage is a minimum.
3. In an RLC series circuit in which the conductor has relatively low internal resistance, r, it is possible to have a large voltage across the inductor, and an almost equally large voltage across the capacitor, but the inductor and the capacitor are nearly 180^o out of phase, so their voltages almost cancel, giving a small total series voltage.
This is one way to produce a large voltage oscillation with only a small voltage source.
In the circuit diagram at right, the coil corresponds to both the inductance L and the resistance R, which is why they are drawn inside a box representing the physical component, the coil.
They are in series, because the current flows through the coil and thus passes through both the inductance of the coil and its resistance.
A big voltage occurs in the circuit for only a small voltage input from the power source.
The energy stored in the large oscillations is gradually supplied by the AC source when the circuit is closed and the energy is then exchanged between capacitor and inductor in each cycle.

30.4.1 RC circuits, impedance, phase angle and power factor
1. An RC circuit has an AC generator, a resistor, R, and a capacitor C, connected in series.
The values of R, C, V_max and ω are known.
2. The maximum voltage across the resistor, R, V_max = I_max R.
3. The maximum voltage for the capacitor C, V_max = I_max Xc = I_max / ωC.
4. The total voltage for the circuit is not (2.) + (3.), because the resistor and the capacitor do not have maximum voltage at the same time.
They are not in phase.
The resistor and capacitor have a phase difference, Φ, of 90^o.

30.4.21 Impedance in an RC circuit
1. Impedance, Z, in an RC circuit is like resistance in a resistor circuit.
However, resistance is the ratio of voltage to current, and resistance does not depend on frequency.
Impedance is the general name given to the ratio of voltage to current.
So resistance is a kind of special case of impedance.
Impedance is a measure of the combined opposition to the passage of alternating current by the resistance, R, and the capacitive reactance, X_c.
2. Impedance in an RC circuit, the ratio of the emf to the resulting current, Z = (R2 + Xc²) = [R² + (1 / ωC)²] ohm.
So in an RC circuit with R = 135, C = 28 F, and f = 60 HZ, Z = [R² + (1 / ωC)²], [ ω = 2πf].
= 1352 + (1 / 2π X 60 X 30 X 10^-6)² = 164.899 = 16

30.4.6 Voltage and current in an RC circuit
Voltage and current in an RC circuit, VRMS = IRMS R² + Xc² = IRMSZ.
So Z is a ratio of voltage to current and resistance is a special case of impedance.
To find the maximum voltage across the capacitor, C, as Vmax = Imax (R² + Xc²) = Imax Z.
So Imax = Vmax / Z

30.01 RLC circuits, RLC circuit impedance, limit of large and small frequencies
1. The AC RLC circuit has a resistor, inductor and capacitor, connected in series.
RLC circuits are also called LRC and LCR circuits.
2. The voltage of the resistor is in phase with the current.
The voltage of the inductor is 90^o ahead of the current.
The voltage of the capacitor is 90^o behind the current.

30.2 Impedance of an RLC circuit
Impedance of an RLC circuit, Z = R² + ( X_L - X_C)² = R² + (ωL - 1 / ωC)² ohm.
This is the same as for the impedance of the RC circuit and the RL circuit.

30.3 Phase angle, Φ, between the total voltage and the current
tan Φ = I_max(X_L - X_C) / I_max R = (X_L - X_C) / R.
If X_L> X_C, Φ is positive, the voltage leads the current.
If X_L< X_C, Φ is negative, the voltage lags the current.
If X_L= X_C, Φ = 0, the voltage and the current are in phase.
Note that cos Φ = R / Z, as in the RC and RL circuits.

30.5.3 RLC series combinations
1. R, L and C in series
At a given time, the voltage across a resistor, R, an inductor, L, and a capacitor, C, in series, v_series(t) = v_R(t) + v_L(t) + v_C(t).
The current i(t) is sinusoidal.
The voltage across the resistor, v_R(t), is in phase with the current.
The voltage across the inductor, v_L(t), is 90^o ahead of the current.
The voltage across the capacitor, v_C(t), is 90^o behind the current.
2. Time dependent voltages, v(t), add up at any time, RMS voltages do not add up.

30.5.5 Limit of frequencies
1. At very high frequencies, the reactance of an inductor becomes very large and behaves like a large resistor with almost no current flowing, and behaves like an open circuit.
However, the reactance of the capacitor becomes very small, and acts like a wire with no resistance.
2. At very low frequencies, e.g. an AC generator replaced by a battery, the reactance of an inductor becomes almost zero, and acts like a wire with no resistance.
However, the reactance of the capacitor becomes very large, and behaves like an open circuit.
I_RMS approaches zero.

30.6.0 Transformers
Transformer, Dissectible (Commercial).
Electric Motors, transformer, Dissectable, (Commercial).
1. Transformers convert the voltage and current in one circuit to a different voltage and current in another circuit, according to the equation: I_s / I_p = V_p / V_s = N_p / N_s, where I = current, V = voltage, and N = number of turns in the primary, _p, and secondary, _s_, circuits of the transformer.
A primary coil and a secondary coil of insulated wire are wound on a soft iron coil, either one on top of the other or on separate limbs of the core, the primary coil and the secondary coil.
When AC passes through the primary coil an alternating magnetic flux is produced in the iron core that passes through the secondary coil.
An induced emf is produced in the secondary coil.
The output voltage in the secondary coil Vs, and the input voltage in the primary coil Vp, are related by the number of turns on the secondary, Ns compared to the number of turns on the primary, Np.
Vs / Vp = Ns / Np.
A step-up transformer has Ns > Np, and a step-down transformer has Ns < Np.
In an ideal transformer, power input is equal to power output.
Experiments
2. Wind 2 coils of insulated wire on the same iron core, the primary coil and the secondary coil.
If you pass AC through the primary coil, you produce an alternating magnetic flux in the iron core that passes through the secondary coil and an induced emf in the secondary coil.
The output voltage in the secondary coil and the input voltage in the primary coil are related by the number of turns on the secondary compared with the number of turns on the primary.
A step-up transformer, higher output voltage, has the number of turns of the secondary coil greater than the number of turns of the primary coil.
A step-down transformer, lower output voltage, has the number of turns of the secondary coil less than the number of turns of the primary coil.
In theory, power input is equal to power output.
Direct current cannot be stepped up or stepped down with a transformer, because in a transformer a changing magnetic field cuts a conductor, but there is no changing magnetic field from steady DC electric current.
3. Examine a step-up and step-down transformer.
Use a step-up transformer to supply 6 volts AC to DC and light a 12 volt lamp at AB.
Use a step-down transformer to supply 12 volts AC to AB, measuring the current in the AB turns.
Take off 6 volts AC from CD to light up a large 6 volt lamp.
Measure current with an ammeter to show that if you drop the voltage by half you double the current, and if you double the voltage you halve the current.
4. Electrical energy is transmitted at high voltage, because the energy loss is given by I²Rt, i.e. it is proportional to the current, squared.
The fall of potential along the wires is RI with P the potential at the generating end.
The potential at the other end is P-RI.
The power available is I(P-RI), and the power wasted in the leads is I²R.
So I(P-RI) should be kept constant and I²R should be as small as possible.
This may be done by making R small, which means a great expense in providing leads, or by making I small in which case P must be large.
Current is produced at comparatively low voltage and is required at a fairly low voltage for use in households.
So the low tension current is converted into high tension current for long distance transmission by using a step-up transformer at the generating end and a step-down transformer at the suburban or town end.
A transformer consists of two coils, a primary, P, and a secondary S, wound, on an iron core.
An alternating current in P produces an alternating magnetic effect in the iron core, and so also an alternating emf in S.
If N_p and N_s are the number of turns in the primary and secondary respectively, and P and S the emf in the primary and secondary respectively, then the primary emf can be changed to a secondary at either a higher or lower potential, and a step-up or step-down transformer is used according to whether the secondary contains a greater or smaller number of turns than the primary.
PN_s = SN_s, S / P = N_s / N_p.

See diagram 30.2.02 "Step-up" transformer 2.
In the diagram, the coil on the left has fewer coils than that at right.
The sketch and circuit show a step-up transformer.
For a step-down transformer, the source is on the right and the load is on the left.
The core has high magnetic permeability.
It forms a magnetic field more easily than free space does, caused by the orientation of atomic dipoles.
So the field is concentrated inside the core and almost no field lines leave the core.
The magnetic flux through the primary and secondary coil are equal, as shown.
From Faraday's law, the emf in each turn of the primary or secondary coil, is -dp / dt.
Neglecting resistance and other losses in the transformer, the terminal voltage equals the emf.
For the Np turns of the primary coil, Vp = - N_p.dΦ / dt.
For the N, turns of the secondary coil, V_s = NsΦ / dt.
Dividing these equations gives the transformer equation:
V_s / V_p = N_s / N_p = r, where r is the turns ratio.
Assuming that the voltage and the current have similar phase relationships in the primary and secondary coils, from the principal of conservation of energy, power in = power out, so V_pI_p = V_sI_s, I_s / I_p = N_p / N_s = i / r.
If the voltage is increased, the current is decreased by at least the same factor.
In the diagram, the coil with more turns has thinner wire, because it is designed to carry less current than the coil with fewer turns.
In power transmission lines, the power lost in heating the wires caused by their resistance is proportional to the square of the current.
To save energy, power is transmitted from power station to city at very high voltages, so that the currents are very low.
For the resistor in the secondary circuit, in the primary circuit:
V_p = V_s / t and I_p = I_sr.
So V_p / I_p = V_s / r²I_s = R / r², the reflected resistance.
If the frequency is not too high, and if there is a load resistance, the inductive reactance of the primary is much smaller than the reflected resistance, so the primary circuit behaves as though the source were driving a resistor of value R / r2.
This allows transformers to be used as impedance matchers.
A load with low input impedance can be matched to a circuit with high output impedance by using a step-down transformer.
6. Transformers are less than 100% efficient.
6.1 Resistive losses in the coils, losing power I²r
The resistance of the coils can be reduced by making their cross section large and by using high purity copper.
6.2 Eddy current losses in the core
Eddy currents can be reduced by laminating the core, because laminations reduce the area of circuits in the core, and so reduce the Faraday emf, and the current flowing in the core, and the energy thus lost.
6.3 Hysteresis losses in the core
The magnetization and demagnetization curves for magnetic materials are often different, being hysteresis or history dependence, so the energy required to magnetize the core while the current is increasing is not entirely recovered during demagnetization.
The difference in energy is lost as heat in the core.
6.4 Design and material
The geometric design and the choice of material for the core must be optimized to ensure that the magnetic flux in each coil of the secondary is nearly the same as that in each coil of the primary CC vs DC generators.
Transformers only work on AC, so they allow 240 V to be stepped down to convenient levels for low power applications, e.g. 12V.
Transformers are used to step up the voltage for transmission and step down the voltage for safe distribution.
It is possible to convert voltages in DC, but more complicated than with AC, and such conversions are often inefficient and expensive.

30.6.1 Hall effect
Hall Effect, (Commercial).
See diagram 30.2.3: Hall effect.
1. In 1879, the U.S. physicist Edwin Herbert Hall discovered that when a current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving carriers of negative charge, which tends to push them to one side of the conductor.
So charge build-up at the sides of the conductor balances the magnetic effect to produce voltage between the sides of the conductor.
The strength of magnetic fields can be measured with a probe using the Hall effect.
2. Hall-effect sensors track small changes in magnetic flux density, too small to operate Hall-effect switches.
As motion detectors, gear tooth sensors, and proximity detectors, they are magnetically-driven mirrors of mechanical events.
As sensitive monitors of electromagnets, they can effectively measure a system's performance with negligible system loading.
Each Hall-effect integrated circuit includes a Hall sensing element, linear amplifier, and emitter-follower output stage.
The Hall cell and amplifier is on a single chip.
Three package styles provide a magnetically optimized package for most applications.
They are rated for continuous operation over the temperature range of -20C to +85C and 4.5 V to 6 V operation.

30.6.2 Magnetoresistance
In 1852, Lord Kelvin discovered that the resistance of iron and nickel increases when the current is in the same direction as a magnetic force and decreases when the current is at 90 to the magnetic force, a phenomenon called magnetoresistance.
So "magnetoresistivity" is the fractional change in resistivity due to the application of a magnetic field.
The electrical resistance and conductance of a body depends on the external magnetic field.
The magnetic domains in a metal, e.g. iron, have magnetic fields scattered in different directions.
When the metal is placed in a strong external magnetic field, the magnetic domains line up to produce a strong magnetic field.
The realignment may cause a slight decrease in length of the iron.
The so-called "giant magnetoresistance" (GMR) occurs in a thin metallic film of alternating layers of iron and chromium, where the electrical resistivity of this film can be halved in a magnetic field.
This phenomenon is being used to improve the efficiency of magnetic data storage systems.

30.6.3 Magnetostriction
Magnetostriction, humming sound of transformers, fluorescent light ballast
Transformer, Dissectible (Commercial).
1. Magnetostriction, a magnetic property of ferromagnetism, refers to how the state of strain in a body and its dimensions depends on its state of magnetization.
Magnetic sheet steel extends itself when magnetized and returns to its original size when the magnetization is taken away.
2. A transformer is magnetically excited by an alternating voltage and current so that it becomes extended and contracted twice during a full cycle of magnetization.
A transformer core contains sheets of special steel to reduce losses and moderate the ensuing heating effect, with each sheet behaving erratically with respect to its neighbour.
The extensions are not normally visible to the naked eye, but they cause vibration and noise, transformer hum.
In a transformer, the change in flow of electric current causes vibrations in the air and a corresponding humming sound.
The degree of magnetic flux determines the amount of magnetostriction and hence, the noise level.
The ratio of transformer voltages to the number of turns in the winding determines the amount of magnetization.
So the amount of magnetic flux at the normal voltage is fixed as is the level of noise and vibration.
The relationship of degree of magnetization and the magnetostriction is not linear.
3. For US circuits, a mains voltage of 60 hertz is associated with maximum length change twice per cycle expansion and contraction of 120 Hz and musical notes and harmonics at 120 Hz.
In Europe, with mains supply at 50 Hz, musical notes about 100 Hz occur.
Nickel constricts and cobalt steel lengthens when magnetized.