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AC THEORY

When a battery is connected to a circuit the current flows steadily in one direction, this is called a Direct current (d.c).

The use of Direct currents is limited to a few applications e.g charging of batteries, electroplating etc.

Most of electrical energy is generated and used in the form of alternating current due to many reasons including,
i) Alternating voltages can be changed in value very easily by means of transformers.
ii) A.c motors are simpler in construction and cheaper than d.c motors.

ALTERNATING VOLTAGE AND CURRENT
i) Alternating voltage
An alternating voltage is one whose magnitude changes with time and direction reverses periodically. The instantaneous value (i.e value at any time t) of an alternating voltage is given by,
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
where,
E = Value of the Alternating voltage at time t
E0 = Maximum value of the Alternating voltage
ω = Angular frequency of supply
From EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1),where f is the frequency of the alternating voltage, If T is the time period of alternating voltage then
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The voltage varies from zero to a positive peak (+E0) then back via zero to negative peak (-E0) and so on.
In time period T, the wave completely cycle.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
ii) (ii)Alternating current
This is one whose magnitude changes with time and direction reverses periodically.
The Instantaneous value (EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value at any time t) of sinusoidally varying alternating current is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
where
I = value of alternating current at time t
I0 = maximum value (Amplitude) of alternating current.
ω = Angular frequency of supply.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Figure below shows the waveform of alternating current.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Current varies sinusoidally with time. The current increases gradually from zero to a positive peak (+Io), then back via zero to a negative peak (Io) and so on.
In time period T, the wave completes a cycle.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
An Alternating voltage and current also be represented as a cosine function of time.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Both these representations give the same result as is given by the one containing sine functions.

MEASUREMENT OF ALTERNATING CURRENT
Since the average value of sinusoidal alternating current is zero, an ordinary (DC) ammeter or galvanometer will not show any deflection when connected in an AC circuit.
Due to inertia it will not be possible for the needle to oscillate with the frequency of the current.
Therefore to measure AC we use hot wire instruments because the heating effect of current is independent of the direction of current.

MEAN OR AVERAGE VALUE OF ALTERNATING CURRENT
The mean value or average value of alternating current over one complete circle is zero.
It is because the area of positive half cycle is exactly equal to the area of the negative half cycle.
However, we can find the average or mean value of alternating current over any half cycle.

Half Cycle Average Value of a.c
This is that value of steady current (d.c) which would send the same amount of charge through a circuit for half the time period of a.c EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) as it sent by the a.c through the same circuit in the same time.It is represented by Im or EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The instantaneous value of alternating current is given by, EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Suppose current I remains constant for a small timeEcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1). Then small amount of charge sent by alternating current in a small time EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If Q is the total charge sent by the positive half cycle of a.c (EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If the Im is the half cycle average means values of positive half cycle of a.c then by definition
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
From equation (i) and (ii)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Hence half cycle average value of a.c is 0.637 times the peak value of a.c
For positive ½ cycle Im EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) +0.637 I0
For negative half cycle Im EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) -0.637 I0
Obviously, average value of a.c over a complete cycle is zero.

MEAN OR AVERAGE VALUE OF ALTERNATIVE VOLTAGE
Half Cycle Average value of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
This is that value of steadyEcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1), (EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)which would send the same amount charge through a circuit for half the time period of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) as is sent by the alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) through the same circuit in the same time.It is denoted by EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)The instantaneous value of alternating e. m. f is given by EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Suppose this alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is applied to a circuit of resistance R. Then by ohm’s law the instantaneous value of alternating current is
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If this current remains constant a small timeEcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1), then small amount of charge send by alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) in small time EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If Q is the total charge sent by positive half cycle of a alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) then
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is the half cycle average or mean value of the positive half cycle of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) then by definition
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
From equation (i) and equation (ii)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Therefore, half cycle average value of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)is 0.637 times the peak value of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1).
For positive half cycle
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
For negative half cycle
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
A d.c voltmeter or ammeter reads average (or d.c) value. Therefore, they can be used to measure alternating voltage on current.
It is because the average value of alternating voltage or current over a complete cycle is zero.
We use a.c meters to measure alternating voltage/current.

ROOT MEAN SQUARE VALUE OF ALTERNATING CURRENT
The average value cannot be used to specify an alternating current (or voltage).
It is because its value is zero over one cycle and cannot be used for power calculations.
Therefore we must search for more suitable criterion to measure the effectiveness of an alternating current or voltage.
The obvious choice would be to measure it in terms of direct current that would do work (or produce heat) at the same average rate as a.c under similar conditions.
This equivalent direct current is called the root mean square (EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)) or effective value of alternating current.

Effective or value of Alternating Current
The root mean square (r.m.s) of alternating current is that steady current (d.c) which when flowing through a given resistance for given time produces the same amount of heat as produced by the alternating current when flowing through the same resistance for the same time.
– It is also called virtual value of a.c
– It is denoted by called EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
For example, when we say that EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) or effective value of an alternating current is 5A, it means that the alternating current will do the work (or produce heat) at the same rate as 5A direct current under similar conditions.

RELATION BETWEEN R.M.S. VALUE AND PEAK VALUE OF A.C
Let the alternating current be represented by
I EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If this alternating current flows through a resistance R for a small timeEcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1), then small amount of heat produced is given.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
In one complete cycle EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) for time 0 to T of alternating current the total amount of heat produced in R is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If Irms is the virtual or EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of the alternating current, then heat produced in R in the same time EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) (0 to T) is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
From equation (i) and (ii), we have
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Hence the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)value of effective value or virtual value of alternating current is 0.707 times the peak value of alternating current.
The EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)value is the same whether calculated for
half cycle or full cycle.

Alternative Method
Let the alternating current be represented by
I EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If this current is passed through a resistance R, then power delivered at any instant is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Because the current is squared, power is always positive since the value of EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) varies
between 0 and 1
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Average power delivered
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is the virtual or EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of the alternating current then by definition
Power delivery P
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
From equation (I) and (ii)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

ROOT MEAN SQUARE VALUE OF ALTERNATING E.M.F
The root mean square (r.m.s) value of alternating voltage is that steady voltage (d.c voltage when applied to a given resistance for a given time produces the same amount of heat as is produced by the alternative EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) when applied to the same resistance for the same time.
It is also called virtual value of alternating e.m.f and is donated by EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) of EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) or EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The instantaneous value of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is given by
E EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
When this alternating voltage is applied to a resistance R, small amount of heat produced in a small time EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
But
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Then,
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
In one complete cycle EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) for time 0 to T the total amount of heat produced in resistance R is
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If Erms is the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of the alternatingEcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1), then heat produced in the same resistance R for the same time T is
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
From the equation (i) and equation (ii)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Hence the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is 0.707 times the peak value of the alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Therefore, EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)value is the same whether calculated for half cycle or full cycle.

Importance of r.m.s values
An alternating voltage or current always specified in terms of EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)values.
T
hus an alternating current of 10A is the one which has the same heat effect as 10A d.c under similar conditions. The following points may be noted carefully.
i) The domestic a.c supply is 230V, 50H. It is the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) or effectively value. It means that the alternative voltage available has the same effect as 230 V d.c under similar conditions.
The equation of this alternative voltage is
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
ii) When we say that alternating current in a circuit is 5A, we specifying the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value.
– It means that the alternating current flowing in the circuit has the same heating effect as 5A d.c under similar conditions.
iii) A.C ammeters and voltmeters record EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) values of current and voltage respectively.
The alternating voltage/current can be measured by utilizing the heating effect of electric current.
Such Instruments are called hot wire instruments and measure the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of the voltage/current since r.m.s value is the same for half cycle or complete cycle.

WORKED EXAMPLES
1. An a.c main supply is given to be 220V what is the average EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) during a positive half cycle?
Solution
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Average EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) during positive half is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

2. An alternating current I is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Find
(i) The maximum value
(ii) Frequency
(iii) Time period
(iv) The instantaneous value when t= 3ms
Solution
Comparing the given equation of the alternating current with the standard form
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
(i) Maximum value, EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
(ii) Frequency f
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
f = 50 HZ
(iii) Time period T
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
(iv) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
When EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
3. Calculate the EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) value of the current shown in figure below
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)


Solution
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

4. An a.c voltmeter records 50V when connected across the terminals of sinusoidal power source with frequency 50Hz. Write down the equation for the instantaneous voltage provided by the source.
Solution
An a.c voltmeter records EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) values
Now
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Here
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
5. An alternating voltage of 50 Hz has maximum value of 200EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) volts. At what time measured from a positive maximum value will the instantaneous voltage be 141.4 volts.
Solution
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
This equation is valid when time is measured from the instant the voltage is zeroi.e point O. Since the time is measured from the positive maximum value at point A.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The above equation is modified to
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Let the value of voltage become 141.4 volts t second after passing then the maximum positive value.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
6. An alternating current of frequency 60 Hz has a maximum value of 120 A.
i) Write down the equation for the instantaneous value.
ii) Recording time from the instant the current is zero and becoming positive. Find the instantaneous value after 1/360 second.

iii) Time taken to reach 96 A for the first time.
Solution
i) The instantaneous value of alternating current is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
ii) Since point 0 has been taken as the reference the current equation is

EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
When
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
iii) To reach the current 96A for the first time
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
7. Find the r.m.s value of the current after current I shown in figure below
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Solution
Consider the current variation over one complete cycle.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
A.C CIRCUIT
An A.C circuit is the closed path followed by alternating current.
When a sinusoidal alternating voltage is applied in a circuit, the resulting alternating current is also sinusoidal and has the same frequency as that of applied voltage.
However there is generally a phase difference between the applied voltage and the resulting current.
As we shall see, this phase difference is introduced due to the presence of inductance (L) and capacitance (C) in circuit.
While discussing A.C circuits our main points of interest are;
(i) Phase difference between the applied voltage and circuit current
(ii) Phasor diagram. It is the diagram representation of the phase difference between the applied voltage and the result circuit current.
(iii) Wave diagram
(iv) Power consumed.

A.C CIRCUIT CONTAINING RESISTANCE ONLY
When an alternating voltage is applied across a pure resistance, then from electrons EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) current flow in one direction for the first half cycle of the supply and then flow in the opposite direction during the next half cycle, thus constitute alternate current in the circuit.
Consider a pure resistor of resistance R connected across an alternating source of EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Suppose the instantaneous value of the alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) is given by
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
If I is the circuit current at that instant, then by ohm’s law
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The value of I will be maximum Io when
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Therefore equation (ii) becomes
From EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Since sin EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)t = 1 then I = Io
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

1) Phase Angle
It is clear from equation (i) and (iii) the applied EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) and circuit current are in phase with each other EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) they pass through their zero values at the same instant and attain their peak value both positive and negative peaks at the same instant.
This is indicated in the wave diagram shown in figure below.
The EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) diagram shown in figure below also reveals that current is in phase with the applied voltage.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Hence in an a.c circuit, current through R is in phase with voltage across R.
This means that current in R varies in step with voltage across R. If voltage across R is maximum current in R is also maximum, if voltage across R is zero, current in R is also zero and so on.

2) Power Absorbed
In a.c circuit, voltage and current vary from instant to instant. Therefore power at any instant is equal to the product of voltage and current at that instant.
Instantaneous power P
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Since power varies from instant to instant the average power over a complete cycle is to be considered.
This is found by integrating equation………..(iv) with respect to time for 1 cycle and dividing by the time of 1 cycle. The time per one cycle is T.

Average power P

EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

Therefore, average power absorbed by a resistor in an a.c circuit is equal to the product of virtual voltage ( Erms) across it and virtual current ( Irms) through it.
Obviously, this power is supplied by the source of alternating EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1).

Since
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
WORKED EXAMPLES

1. An a.c circuit consists of a pure resistance of 10Ω and is connected across an a.c supply of 230V, 50 Hz.
Calculate
i) Circuit current
ii) Power dissipated and
iii) Equations for voltage and current

Solution
Erms = 230V R = 10Ω f = 50HZ
i) Circuit current
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
ii) Power dissipated P
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
= 230 x 23
P = 5290 W
iii)Equations for voltage and current
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
= EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) x 230
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) = 325. 27V
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
= 2 x 23
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) = 32.52A
ω = 2EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
= 2EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
ω = 314EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
The equations of voltage and current
E = 325.27 sin 314t and I = 3.52 sin 314t

2. In a pure resistive circuit, the instantaneous voltage and current are given
E = 250 sin 314t
I = 10 sin 314t
Determine
i) Peak power
ii) Average power

Solution
In a pure resistive circuit
i) Peak power = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
= 250 x 10
Peak power = 2500W
ii) Average power P
P = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
P = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
P = 1250 W
3. Calculate the resistance and peak current in a 1000 W hair dryer connected to 120V, 60Hz supply. What happens if it is connected to 240V line?

Solution

EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Peak current Io
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

Resistance of hair dryer R
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

When connected to 240V line, the average power delivered would be
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
This would undoubtedly melt the heating element or the coils of the motor.

4. A voltage E = 60sin 314t is applied across a 20 Ω resistor. What will;
i) An a.c ammeter
ii) Ordinary moving coil ammeter in series with resistor read?

Solution
i) E = 60 sin 314t
An a.c ammeter will read the r.m.s value.
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)=EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Therefore a.c a meter will read 2.12A

ii) An ordinary moving coil ammeter will read average value of alternating current. Since the average value of a.c over one cycle is zero, this meter will record zero reading.

5. What is the peak value of an alternating current which produces three times the heat per second as a direct current of 2A in a resistor R?
Solution
Heat per second by 2A
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)= EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)R
H = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)R
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) 4R
Three times heat per second
3EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
3EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) = 12R
If Iv is the r .m .s value of the a.c heat per second in R
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)R
12R = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)R
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)= 12
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1) EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Peak value is given by, EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
Peak value = 24 = 4.9A

6. An a.c voltage of 4V peak (maximum) is connected to a 100Ω resistor R
a) What is the phase of the current and voltage?
b) Calculate the current in R in mA
c) What is the power in R in mW
Solution
a) The current and voltage are in phase
b) Current is R

EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)
c) Power in R
P = EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)R
P = 0.0282 x 100
P = 0.078W
P = 78mW





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1 Comment

  • EcoleBooks | PHYSICS A LEVEL(FORM SIX) NOTES - AC THEORY(1)

    Willing, January 24, 2024 @ 8:14 am Reply

    So good sofar

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