Continuing the discussion from Electrical wire:
Example 10. Calculate the resistance at the working temperature of 75° C of a 4-pole, lap
connected armature winding from the following data :
Number of slots …100
Conductors per slot …12
Mean length of turn …3 m
Cross-section of each conductor …1.5 cm × 0.2 cm
Specific resistance of copper of 20° C … 1.72 × ohm metre
Temperature co-efficient of resistivity at 0°C … 0.00427/° C.
Solution. Total number of conductors, Z = 100 × 12 = 1200
Length of each conductor, l = = 1.5 m
Cross-sectional area of each conductor = 1.5 cm × 0.2 cm = 0.3 = 0.3 ×
Specific resistance of copper at 20° C, ρ20 = 1.72 × ohm-metre.
Temperature co-efficient of resistivity at 0°C, α0 = 0.00427/°C
If ‘a’ is the number of parallel paths,
Resistance of each parallel path, R0 =
As there are such ‘a’ paths in parallel, hence equivalent resistance,
R0 = …(i)
Also,
1.72 × 10–8 = (1 + 0.00427 × 20) = 1.0854
or = = 1.585
Now by substituting the different values in eqn. (i), we get,
R0 = = 0.0594
Resistance of winding at 75° C,
= (1 + × 75)
= 0.0594 (1 + 0.00427 × 75) = 0.0784 Ω
Hence resistance of winding at 75° C = 0. 0784 Ω. (Ans.)
Conditions for applying Ohm’s law. Ohm’s law is applicable under the following
conditions :
- Ohm’s law can be applied either to the entire circuit or to a part of the circuit.
- When Ohm’s law is applied to a part of a circuit, part resistance and the potential difference
across that part resistance should be used. -
Ohm’s law can be applied to both D.C. and A.C. circuits. However, in case of A.C. circuits,
impedance Z, is used in place of resistance.
Then, I = =
Electrical power. It is the rate of doing work. In other words, the amount of work done in
one second is called ‘‘power’’.
or P = … (8)
where P = The power in watts,
W = The work done in joules, and
t = Time in second.
Power is equal to the product of voltage (V) and current (I) in a particular circuit
i.e., *P = V × I …(9)
The following relations hold good :
(i) *P = VI = R =
(ii) I =
(iii) R =
(iv) V =
Power is expressed in terms of kW (kilowatt = 1000 W) or MW (megawatt = 1000 kW or
106 W).
Electrical energy. It is the total amount of work done in an electric circuit.
In other words, it is measured by the product of power and time.
i.e., W = P × t …(From eqn. (8)]
or W = VI t
= VQ joules
where Q = The quantity of electricity passing through the circuit in coulombs.
The unit of electrical energy is joules or watt-sec.
It is expressed in kWh (kilowatt hours)
1 kWh (commercial unit) = 1 kW × 1 hour = 1000 watt-hours
= 1000 × 60 × 60 watt-sec
= 3.6 × 106 watt-sec. or joules.
Linear and non-linear resistors. A linear resistor is one which obeys ohm’s law.
A circuit which contains only linear components is called a linear circuit.
Such elements in which V/I (volt-ampere) plots are not straight lines but curves are called
non-linear resistors or non-linear elements.
Examples of non-linear elements : Filaments of incandescent lamps, diodes, thermistor and
varistor.
‘‘Varistor (Non-linear resistor)’’ :
It is a voltage-dependent metal-oxide material whose resistance decreases sharply with
increasing voltage.
The zinc oxide-based varistors are primarily used for protecting solid-state power supplies
from low and medium size voltage in the supply line.
Silicon carbide varistors provide protection against high-voltage surges caused by lightning
and by discharge of electromagnetic energy stored in the magnetic fields of large coils.
Limitations of Ohm’s law. Ohm’s law does not apply under the following
conditions :
-
Electrolytes where enormous gases are produced on either electrode.
-
Non-linear resistors like vacuum radio valves, semiconductors, gas filled tubes etc.
-
Arc lamps.
-
Metals which get heated up due to flow of current.
-
Appliances like metal rectifiers, crystal detectors, etc. in which operation depends on the
direction of current.
An Electric Circuit
An ‘‘electric circuit’’ is a conducting path through which either an electric current flows, or is
intended to flow. It can be divided into four categories :
(i) Closed circuit (ii) Open circuit
(iii) Short circuit (iv) Earth or leakage circuit.
(i) Closed circuit. It is the complete path for flow of electric current through the load.
Example. The glowing of a bulb.
(ii) Open circuit. In case any one of the supply wires is disconnected or the fuse burns out,
then the current will not flow through the bulb, which is an example of open circuit.
(iii) Short circuit. If the supply mains are connected directly by a piece of wire without any
load, then the value of current will be much greater than that in closed circuit. Hence, the fuse
gets
blown off and this circuit is known as short circuit.
(iv) Earth or leakage circuit. If any wire of supply mains touches the ‘body of an appliance’,
then it is known as earth or leakage circuit.
Depending upon the type of current flowing in it, an electric circuit may be further classified
as :
(i) D.C. circuit (ii) A.C. circuit.
Resistances in Series
Fig. 9 shows three resistance connected in series. Obviously current flowing through each
resistance will be same but voltage drop across each of them will vary as per value of individual
resistance.
Also the sum of all the voltage drops (V1 + V2 + V3) is equal to the applied voltage (V).
i.e., V = V1 + V2 + V3
IR = IR1 + IR2 + IR3 (Using Ohm’s law : V = IR)
i.e., R = R1 + R2 + R3 …(10)
where R is the equivalent resistance of series combination.
Also = + + …[10(a)]
As seen from above, the main characteristics of a series circuit are :
-
Same current flows through all parts of the circuit.
-
Different resistors have their individual voltage drops.
-
Voltage drops are additive.
-
Applied voltage equals the sum of different voltage drops.
-
Resistances are additive.
-
Powers are additive.
Voltage divider rule. Since in a series circuit, same the same current flows in each
of the given resistors, voltage drop varies directly with its resistance. Fig. 10 shows a 24 V
battery connected across a series combination of three resistors.
4 Ω 8 Ω
Total resistance, R = R1 + R2 + R3 = 4 + 8 + 12 = 24 Ω
= V = 24 8V
= V = 24 12V