More than one electrical resistance can be connected either in series or in parallel in addition to that, more than two resistances can also be connected in combination of series and parallel both. Here we will discuss mainly about series and parallel combination.
Resistances in Series
Suppose you have, three resistors, R1, R2 and R3 and you connect them end to end as shown in the figure below, then it would be referred as resistances in series. In case of series connection, the equivalent resistance of the combination, is sum of these three electrical resistances.
That means, resistance between point A and D in the figure below, is equal to the sum of three individual resistances. The current enters in to the point A of the combination, will also leave from point D as there is no other parallel path provided in the circuit. Now say this current is I. So this current I will pass through the resistance R1, R2 and R3. Applying Ohm’s law , it can be found that voltage drops across the resistances will be V1 = IR1, V2 = IR2 and V3 = IR3. Now, if total voltage applied across the combination of resistances in series, is V.
Then obviously V = IR1 + IR2 + IR3 ………….(1)
Since, sum of voltage drops across the individual resistance is nothing but the equal to applied voltage across the combination.
Now, if we consider the total combination of resistances as a single resistor of electric resistance value R, then according to Ohm’s law ,
V = IR ………….(2)
Now, comparing equation (1) and (2), we get
IR = IR1 + IR2 + IR3
IR = I(R1 + R2 + R3)
R = R1 + R2 + R3 So the above proof shows that equivalent resistance of a combination of resistances in series is equal to the sum of individual resistance. If there were n number of resistances instead of three resistances, the equivalent resistance will be R = R1 + R2 + R3 + ………………..+Rn
Resistances in Parallel
Let’s three resistors of resistance value R1, R2 and R3 are connected in such a manner, that right side terminal of each resistor are connected together as shown in the figure below, and also left side terminal of each resistor are also connected together.
This combination is called resistances in parallel. If electric potential difference is applied across this combination, then it will draw a current I (say).As this current will get three parallel paths through these three electrical resistances, the current will be divided into three parts. Say currents I1, I1 and I1 pass through resistor R1, R2 and R3 respectively. Where total source current I = I1 + I2 + I3 Now, as from the figure it is clear that, each of the resistances in parallel, is connected across the same voltage source, the voltage drops across each resistor is same, and it is same as supply voltage V (say). Hence, according to Ohm’s law ,
V = I1 R1 + I2 R2 + I3 R3 Now, if we consider the equivalent resistance of the combination is R Then, V = IR ⇒ I = V ⁄ R Now putting the values of I, I1, I2 and I3 in equation (1) we get, The above expression represents equivalent resistance of resistor in parallel. If there were n number of resistances connected in parallel, instead of three resistances, the expression of equivalent resistance would be
Bridge circuit is nothing but the electrical circuit configuration which
is used to measure unknown values of the resistance, impedance,
induction, and capacitance. Many bridges like Wheatstone bridge, Maxwell
Bridge, Kelvin Bridge, and many more are very useful to measure
quantities with accuracy and working on the same principle. Here is a
brief description of functioning of some of the bridges given below:
Wheatstone Bridge
A Wheatstone bridge is an electrical circuit developed by Charles Wheatstone, and it is used to determine the value of an unknown electrical resistance in the circuit. Wheatstone bridge is highly capable in calculating very low valued resistances which other instruments like multimeter does not calculate accurately. The Wheatstone bridge circuit is a diamond-shaped arrangement of four resistors. It has two parallel legs and each leg having two resistors in series. A third leg connected between the two parallel legs at some point within the legs, as drawn in figure. Among the four resistors, one resistance value can be determined by balancing the two legs. Out of four resistors, the value of two resistors R1 and R3 are known, the value of R2 is adjustable, and the value of Rx is to be calculated. Then this adjustment is connected to electric supply and a galvanometer between terminal D and terminal B. Now the value of an adjustable resistor is adjusted until the ratio of the two branches resistances become equal i.e. (R1/ R2) = (R3/Rx), and galvanometer reads zero as current stop flowing through the circuit. Now the circuit is balanced and the value of the unknown resistor could be measured easily. The reading of the R3 decides the direction of the flow of current.
Maxwell’s Bridge
The working principle of the Maxwell’s inductance bridge is same as the Wheatstone bridge. Only little modifications have been done in Wheatstone bridge. In this bridge, the four branches consist of unknown inductance (L1), a variable capacitor (C4), four resistors and detector instead of galvanometer as shown in the figure. It is used to measure the value of inductance by comparing the unknown value with the standard variable capacitance.
The basic principle of the bridge is to compensate the positive angle phase of the unknown impedance with the negative phase of a capacitance by putting it in opposite branch. By doing so, the potential difference across the detector will become zero and no current will flow through it. The capacitor C4 and resistor R4 are connected in parallel and the value of both are adjusted so that bridge get balanced.
Kelvin Bridge
Kelvin Bridge is another modification of the Wheatstone bridge which is used to measure low resistance in the range of 1mΩ to 1kΩ with great accuracy. For precise measurement of low resistance, high voltage supply and a sensitive galvanometer are required in Kelvin Bridge. While measuring low resistance, the resistance of connecting wires plays an important role. Wheatstone bridge is used which has two additional resistors as shown in the figure. The resistors R1 and R2 are connected to the second set of ratio-arm and constructed four terminal resistors. Here R is unknown and S is the standard resistor. A galvanometer is placed between c and d so that resistance of connecting wire r can be neglected and does not affect the measurement value. Under the balance condition, galvanometer shows zero and no current flows through the circuit. The equation at balance condition is:
Hay’s Bridge Circuit
Hay’s bridge is another variation of Maxwell’s bridge circuit. In Maxwell’s circuit resistance is kept parallel to the capacitor where as, in Hay’s circuit, the resistor is connected in series with the standard capacitor as shown in the figure. It is very useful if the phase angle of inductive impedance is very large, which could be overcome by taking a low resistance in series.
Anderson’s Bridge
Anderson Bridge is modified version of Maxwell’s inductor capacitance bridge. It is mainly used for measuring self-inductance in a coil by using standard capacitor and resistors. The main advantage of this bridge is that it does not require the frequent balancing of the bridge.
To balance the bridge by steady current, variable resistance r is adjusted and AC source is replaced by battery and headphone by moving coil galvanometer. Once the bridge is balanced the potential at the terminal D is similar to the potential at E. the flow of current in respective branches are denoted by I1, I2,and I3 as shown in the figure.
Diode Bridge Circuit
It is a bridge circuit having an arrangement of four diodes that gives the same output polarity for every input polarity. Diode bridge circuit which also called bridge rectifier is used where ever there is a need to change alternating current into direct current. It is also used to detect the amplitude of radio signals. When the positive terminal of the input is connected to the upper left and negative to the lower right, the current flows from upper supply terminal to the output flowed by red path and returns back to the lower supply terminal through the blue path as shown in the figure.
There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e the current
will flow through only one wire and there will be one return path
called neutral line to complete the circuit. So in single phase minimum
amount of power can be transported. Here the generating station and load
station will also be single phase. This is an old system using from
previous time.
In 1882, new invention has been done on polyphase system, that more than one phase can be used for generating, transmitting and for load system. Three phase circuit is the polyphase system where three phases are send together from the generator to the load. Each phase are having a phase difference of 120°, i.e 120° angle electrically. So from the total of 360°, three phases are equally divided into 120° each. The power in three phase system is continuous as all the three phases are involved in generating the total power. The sinusoidal waves for 3 phase system is shown below
The three phases can be used as single phase each. So if the load is single phase, then one phase can be taken from the three phase circuit and the neutral can be used as ground to complete the circuit.
Why Three Phase is preferred Over Single Phase?
There are various reasons for this question because there are numbers of advantages over single phase circuit. The three phase system can be used as three single phase line so it can act as three single phase system. The three phase generation and single phase generation is same in the generator except the arrangement of coil in the generator to get 120° phase difference. The conductor needed in three phase circuit is 75% that of conductor needed in single phase circuit. And also the instantaneous power in single phase system falls down to zero as in single phase we can see from the sinusoidal curve but in three phase system the net power from all the phases gives a continuous power to the load.
Till now we can say that there are three voltage source connected together to form a three phase circuit. And actually it is inside the generator. The generator is having three voltage source s which are acting together in 120° phase difference. If we can arrange three single phase circuit with 120° phase difference, then it will become a three phase circuit. So 120° phase difference is must otherwise the circuit will not work, the three phase load will not be able to get active and it may also cause damage to the system.
The size or metal quantity of three phase devices is not having much difference. Now if we consider the transformer, it will be almost same size for both single phase and three phase because transformer will make only the linkage of flux. So the three phase system will have higher efficiency compared to single phase because for the same or little difference in mass of transformer, three phase line will be out whereas in single phase it will be only one. And losses will be minimum in three phase circuit. So overall in conclusion the three phase system will have better and higher efficiency compared to the single phase system.
In three phase circuit, connections can be given in two types:
Star connection
Delta connection
Star Connection
In star connection, there is four wire, three wires are phase wire and fourth is neutral which is taken from the star point. Star connection is preferred for long distance power transmission because it is having the neutral point. In this we need to come to the concept of balanced and unbalanced current in power system.
When equal current will flow through all the three phases, then it is called as balanced current. And when the current will not be equal in any of the phase, then it is unbalanced current. In this case, during balanced condition there will be no current flowing through the neutral line and hence there is no use of the neutral terminal. But when there will be unbalanced current flowing in the three phase circuit, neutral is having a vital role. It will take the unbalanced current through to the ground and protect the transformer. Unbalanced current affects transformer and it may also cause damage to the transformer and for this star connection is preferred for long distance transmission.
The star connection is shown below-
In star connection, the line voltage is √3 times of phase voltage. Line voltage is the voltage between two phases in three phase circuit and phase voltage is the voltage between one phase to the neutral line. And the current is same for both line and phase. It is shown as expression below
Delta Connection
In delta connection, there is three wires alone and no neutral terminal is taken. Normally delta connection is preferred for short distance due to the problem of unbalanced current in the circuit. The figure is shown below for delta connection. In the load station, ground can be used as neutral path if required.
In delta connection, the line voltage is same with that of phase voltaage. And the line current is √3 times of phase current. It is shown as expression below,
In three phase circuit, star and delta connection can be arranged in four different ways-
Star-Star connection
Star-Delta connection
Delta-Star connection
Delta-Delta connection
But the power is independent of the circuit arrangement of the three phase system. The net power in the circuit will be same in both star and delta connection. The power in three phase circuit can be calculated from the equation below,
Since there is three phases, so the multiple of 3 is made in the normal power equation and the PF is power factor. Power factor is a very important factor in three phase system and some times due to certain error, it is corrected by using capacitors.
Electrical DC Series and Parallel Circuit
Electrical DC Circuit
Definition of Electrical Circuit
An electrical circuit
is a combination of two or more electrical components which are
interconnected by conducting paths. The components may be active or
inactive or both. This is a very basic definition of electrical circuit.
DC Circuit
There are two types of electricity - direct current and alternating current, i.e, DC and AC. The circuit that deals with direct current or DC, is referred as DC circuit and the circuit that deals with alternating current or AC, is generally referred as AC Circuit. The components of the electrical DC circuit are mainly resistive, whereas components of the AC circuit may be reactive as well as resistive. Any electrical circuit can be categorized into three different groups - series, parallel and series parallel. So for example, in the case of DC, the circuits can also be divided into three groups, such as series DC circuit, parallel DC circuit and series and parallel circuit.
Series DC Circuit
When all the resistive components of a DC circuit are connected end to end to form a single path for flowing current , then the circuit is referred as series DC circuit. The manner of connecting components end to end is known as series connection.
Suppose we have n number of resistors R1, R2, R3............Rn and they are connected in end to end manner, means they are series connected. If this series combination is connected across a voltage source, the current starts flowing through that single path. As the resistors are connected in end to end manner, the current first enters in to R1, then this same current comes in R2, then R3 and at last it reaches Rn from which the current enters into the negative terminals of the voltage source . In this way, the same current circulates through every resistor connected in series. Hence, it can be concluded that in a series DC circuit, the same current flows through all parts of the electrical circuithttp://www.electrical4u.com/electric-circuit-and-electrical-circuit-element/. Again according to Ohm’s law , the voltage drop across a resistor is the product of its electrical resistance and the current flow through it. Here, current through every resistor is the same, hence the voltage drop across each resistor's proportional to its electrical resistance value. If the resistances of the resistors are not equal then the voltage drop across them would also not be equal. Thus, every resistor has its individual voltage drop in a series DC circuit.
Electrical DC Series Circuit with Three Resistors
The flow of current is shown here by a moving point. This is just a conceptual representation.
An Example of Series DC Circuit
Suppose three resistors R1, R2 and R3 are connected in series across a voltage source of V (quantified as volts) as shown in the figure. Let current I (quantified as Ampere) flow through the series circuit. Now according to Ohm’s law ,Voltage drop across resistor R1, V1 = IR1 Voltage drop across resistor R2, V2 = IR2 Voltage drop across resistor R3, V3 = IR3
Voltage drop across whole series DC circuit, V = Voltage drop across resistor R1 + voltage drop across resistor R2 + voltage drop across resistor R3 According to Ohm’s law , the electrical resistance of an electrical circuit is given by V ⁄ I and that is R. Therefore, So, effective resistance of the series DC circuit is R = R1 + R2 + R3. From the above expression it can be concluded, that when a number of resistors are connected in series, the equivalent resistance of the series combination is the arithmetic sum of their individual resistances. From the above discussion, the following points come out:
When a number of electrical components are connected in series, the same current flows through all the components of the circuit.
The applied voltage across a series circuit is equal to the sum total of voltage drops across each component.
The voltage drop across individual components is directly proportional to its resistance value.
Parallel DC Circuit
When two or more electrical components are connected in a way that one end of each component is connected to a common point and the other end is connected to another common point, then the electrical components are said to be connected in parallel, and such an electrical DC circuit is referred as a parallel DC circuit. In this circuit every component will have the same voltage drop across them, and it will be exactly equal to the voltage which occurs between the two common points where the components are connected. Also in a parallel DC circuit, the current has several parallel paths through these parallel connected components, so the circuit current will be divided into as many paths as the number of components.Here in this electrical circuit, the voltage drop across each component is equal. Again as per Ohm’s law , voltage drop across any resistive component is equal to the product of its electrical resistance and current through it. As the voltage drop across every component connected in parallel is the same, the current through them is inversely proportional to its resistance value.
Electrical DC Parallel Circuit with Three Resistors
The flow of current is shown here by a moving point. This is just a conceptual representation.
An Example of Parallel DC Circuit
Suppose three resistors R1, R2 and R3 are connected in parallel across a voltage source of V (volt) as shown in the figure. Let I (Ampere) be the total circuit current which is divided into current I1, I2 and I3 flowing through R1, R2 and R3 respectively. Now according to Ohm’s law :Voltage drop across resistor R1, V = I1.R1 Voltage drop across resistor R2, V = I2.R2 Voltage drop across resistor R3, V = I3.R3 Voltage drop across whole parallel DC circuit,
V = Voltage drop across resistor R1 = voltage drop across resistor R2 = voltage drop across resistor R3
⇒ V = I1.R1 = I2.R2 = I3.R3 I = I1 + I2 + I3 and as per Ohm’s law , I = V ⁄ R hence, Thus when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is given by the arithmetic sum of the reciprocals of their individual resistances. From the above discussion of parallel DC circuit, we can come to the following conclusion:
Voltage drops are the same across all the components connected in parallel.
Current through individual components connected in parallel, is inversely proportional to their resistances.
Total circuit current is the arithmetic sum of the currents passing through individual components connected in parallel.
The reciprocal of equivalent resistance is equal to the sum of the reciprocals of the resistances of individual components connected in parallel.
Series and Parallel Circuit
So far we have discussed series DC circuit and parallel DC circuit separately, but in practice, the electrical circuit is generally a combination of both series circuits and parallel circuits. Such combined series and parallel circuits can be solved by proper application of Ohm’s law and the rules for series and parallel circuits to the various parts of the complex circuit.
Battery is an electrical element where electrical potential is produced due to chemical reaction. Every electrochemical reaction has its limit of producing electric potential difference between two electrodes.
Battery cells are those where these electro-chemical reactions take place to produce the limited electric potential difference. For achieving desired electric potential difference across the battery terminals multiple numbers of cells are to be connected in series. Hence it can be concluded like that, a battery is a combination of several cells where a cell is a unit of a battery . For example, Nickel-cadmium battery cells normally develop about 1.2 V per cell while lead acid battery develop about 2 V per cell. So a 12 volt battery will have total 6 number of cells connected in series.
Terminal voltage of battery is the potential difference across its terminals when the current is being drawn from it. Actually when load is connected with the battery , there will be load current flowing through it. As a battery is an electrical equipment, it must have some electrical resistance inside it. Because of this internal resistance of battery, there will be some voltage drops across it. So, if any one measures the terminal voltage of the load i.e. terminal voltage of battery when load is connected, he or she will get the voltage which is less than emf of the battery by internal voltage drop of the battery.
If E is the emf or no – load voltage of the battery and V is the terminal voltage of load voltage of the battery , then E – V = internal voltage drop of the battery.
As per Ohm’s law , this internal voltage drop is nothing but the product of electrical resistance offered by the battery and the current flows through it.
Internal Resistance of Battery
The entire resistance encountered by a current as if it flows through a battery from the negative terminal to the positive terminal is known as internal resistance of battery.
Series Parallel Batteries
Battery cells can be connected in series, in parallel and as well as a mixture of both the series and parallel.
Series Batteries
When in a battery , positive terminal of one cell is connected with the negative terminal of succeeding cell, then the cells are said to be series connected or simply series battery. Here, overall emf of the battery is algebraic sum of all individual cells connected in series. But overall discharged current of the battery does not exceed the discharged current of individual cells.
If E is the overall emf of the battery combined by n number cells and E1, E2, E3, …………… En are the emfs of individual cells.
Similarly, if r1, r2, r3, …………… rn are the internal resistances of
individual cells, then the internal resistance of the battery will be equal to the sum of the internal resistance of the individual cells i.e.
Parallel Batteries
When positive terminals of all cells are connected together and similarly negative terminals of these cells are connected together in a battery , then the cells are said to be connected in parallel. These combinations are also referred as parallel batteries. If emf of each cell is identical, then the emf of the battery combined by n numbers of cells connected in parallel, is equal to the emf of each cell. The resultant internal resistance of the combination is, The current delivered by the battery is sum of currents delivered by individual cells.
Mixed Grouping of Batteries or Series Parallel Batteries
As we said earlier, the cells in a battery can also be connected in mixture of both series and parallel. These combinations are some time referred as series parallel battery. A load can require both voltage and current more than that of an individual battery cell. For achieving the required load voltage, the desired numbers of battery cells can be combined in series and for achieving the required load current, desired numbers of these series combinations are connected in parallel. Let m, numbers of series, each containing n numbers of identical cells, are connected in parallel. Again assume emf of each cell is E and internal resistance of each cell is r. As n numbers of cells are connected in each series, the emf of each series as well as the battery will be nE. The equivalent resistance of the series is nr. As, m number of series connected in parallel equivalent internal resistance of that series and parallel battery is nr / m.
Video Presentation about Series Parallel Batteries
Electrical resistance may be defined as the basic property of any substance due to which it opposes the flow of current through it. While a voltage is applied across any substance, current starts flowing through it. But if we observe carefully, the current flows through the all substances are not equal even when the same voltage is applied across each of the substances. This is because current carrying capacity of all substances is not equal. The current depends upon the number of electrons' crosses the cross-section per unit time. Again this number of electrons crossing the cross-section is dependable on the free electrons available in the substances. If free electrons are plenty in a substance, the amount of current is more for same applied voltage across the substances. The current through a substance not only depends upon the number of free electrons in it, but also depends upon the length of path an electron has to travel to reach from lower potential end to higher potential end of the substance. In addition to that, every electron has to collide randomly with other atoms and electrons in numbers of times during its traveling.
So, every substance has a property to resist current through it and this property is known as electric resistance. If one volt across a conductor produces one ampere of current through it, then the resistance of the conductor is said to be one ohm (Ω).
Laws of Resistance
There are mainly two laws of resistance from which the resistivity or specific resistance of any substance can easily be determined. One law is related to cross-sectional area of the conductor and other law is related with its length.
As stated earlier, the current through any conductor depends upon number of electrons passes through a cross-section per unit time. So if cross section of any conductor is large then more electrons can cross it that means more current can flow through the conductor. For fixed voltage, more current means less electrical resistance. So it can be concluded like that resistance of any conductor is inversely proportional to its cross-sectional area.
If the length of the conductor is increased, the path traveled by the electrons is also increased. If electrons travel long, they collide more and consequently the number of electron passing through the conductor becomes less; hence current through the conductor is reduced. In other word, resistance of the conductor increases with increase in length of the conductor.
The laws of resistance state that,
Electrical resistance R of a conductor or wire is
directly proportional to its length, l i.e. R ∝ l,
inversely proportional to its area of cross-section, a i.e.
Combining these two laws we get, Where ρ (rho) is the proportionality constant and known as resistivity or specific resistance of the material of the conductor or wire. Now if we put, l = 1 and a = 1 in the equation, We get, R = ρ. That means resistance of a material of unit length having unit cross - sectional area is equal to its resistivity or specific resistance. Resistivity of a material can be alliteratively defined as the electrical resistance between opposite faces of a unit cube of that material. Hence we have seen that laws of resistance are very simple.
Unit of Resistivity
The unit of resistivity can be easily determined form its equation The unit of resistivity is Ω-m in MKS system and Ω-cm in CGS system and
1 Ω-m = 100 Ω-cm.
Temperature Coefficient of Resistance and Inferred Zero Resistance Temperature
Materials
Resistivity in μ Ω-cm at 20oC
Temperature Coefficient of Resistance in Ω per oC at 20oC
Inferred Zerro Resistance Temperature in oC
Aluminium
2.82
0.0039
− 236
Brass
6 to 8
0.0020
− 480
Carbon
3k to 7k
0.00005
Constantan
49
0.000008
− 125000
Copper
1.72
0.00393
− 234.5
Gold
2.44
0.0034
− 274
Iron
12.0
0.005
− 180
Lead
22.0
0.0039
− 236
Manganin
42 to 74
0.00003
− 236
Mercury
96
0.00089
− 1100
Nickel
7.8
0.006
− 147
Silver
1.6
0.0038
− 310
Tungsten
5.51
0.0045
− 200
Zinc
6.3
0.004
− 230
Resistance Variation with Temperature
There are some materials mainly metals, such as silver, copper, aluminum, which have plenty of free electrons. Hence this type of materials can conduct current easily that means they are least resistive. But the resistivity of these materials is highly dependable upon their temperature. Generally metals offer more electrical resistance if temperature is increased. On the other hand the resistance offered by a non - metallic substance normally decreases with increase of temperature.
If we take a piece of pure metal and make its temperature 0° by means of ice and then increase its temperature from gradually from 0°C to to 100°C by heating it. During increasing of temperature if we take its resistance at a regular interval, we will find that electrical resistance of the metal piece is gradually increased with increase in temperature. If we plot the resistance variation with temperature i.e. resistance Vs temperature graph, we will get a straight line as shown in the figure below. If this straight line is extended behind the resistance axis, it will cut the temperature axis at some temperature, − t0°C. From the graph it is clear that, at this temperature the electrical resistance of the metal becomes zero. This temperature is referred as inferred zero resistance temperature. Although zero resistance of any substance cannot be possible practically. Actually rate of resistance variation with temperature is not constant throughout all range of temperature. Actual graph is also shown in the figure below.
Let's R1 and R2 are the measured resistances at temperature t1°C and t2°C respectively. Then we can write the equation below, From the above equation we can calculate resistance of any material at different temperature. Suppose we have measured resistance of a metal at t1oC and this is R1. If we know the inferred zero resistance temperature i.e. t0 of that particular metal, then we can easily calculate any unknown resistance R2 at any temperature t2°C from the above equation. The resistance variation with temperature is often used for determining temperature variation of any electrical machine. For example, in temperature rise test of transformer, for determining winding temperature rise, the above equation is applied. This is impossible to access winding inside the an electrical power transformer insulation system for measurement of temperature but we are lucky enough that we have resistance variation with temperature graph in our hand. After measuring electrical resistance of the winding both at the beginning and end of the test run of the transformer, we can easily determine the temperature rise in the transformer winding during test run. 20°C is adopted as standard reference temperature for mentioning resistance. That means if we say resistance of any substance is 20 Ω that means this resistance is measured at the temperature of 20°C.
Video Resistance Variation with Temperature
As we discussed in the page under title resistance variation with temperature that electrical resistance of every substance changes with change in its temperature. Temperature coefficient of resistance is the measure of change in electrical resistance of any substance per degree of temperature rise. Let a conductor having a resistance of R0 at 0°C and Rt at t°C respectively. From the equation of resistance variation with temperature we get
This αo is called temperature coefficient of resistance of that substance at 0°C. From the above equation, it is clear that the change in electrical resistance of any substance due to temperature rise mainly depends upon three factors-
The value of resistance at initial temperature,
Rise of temperature and
the αo.
This αo is different for different materials, so effect on resistance at different temperature are different in different materials.
So the temperature coefficient of resistance at 0°C of any substance is the reciprocal of the inferred zero resistance temperature of that substance.
So far we have discussed about the materials that resistance increases with increase in temperature, but there are many materials that's electrical resistance decreases with decrease in temperature. Actually in metal if temperature is increased, the random motion of charged particles inside the materials increases which results to more collisions. More collision resist smooth flow of electrons through the metal, hence the resistance of the metal increases with the increase in temperature. So, temperature coefficient of resistance is considered as positive for metal. But in case of semiconductor or other non - metal, the number of free electrons increases with increase in temperature. That means if temperature increases, more number of electrons comes to the conduction bands from valance band by crossing the forbidden energy gap. As the number of free electrons increases, the resistance of this type of non-metallic substance decreases with increase of temperature. Hence temperature coefficient of resistance is negative for non-metallic substances and semiconductors. If there is approximately no change in resistance with temperature, the value of this coefficient is considered as zero. Such as alloys like constantan and manganin that's temperature coefficient of resistance is nearly zero. The value of this coefficient is not constant, it depends upon the initial temperature on which the increment of resistance is based. When the increment is based on initial temperature of 0°C, the value of this coefficient is αo - which is nothing but the reciprocal of the respective inferred zero resistance temperature of the substance. But at any other temperature , temperature coefficient of electrical resistance is not same as this αo. Actually for any material, the value of this coefficient is maximum at 0°C temperature. Say the value of this coefficient of any material at any t°C is αt, then its value can be determined by the following equation, The value of this coefficient at a temperature of t2°C in the term of the same at t1°C is given as,
Since, sum of voltage drops across the individual resistance is nothing but the equal to applied voltage across the combination.
Now, if we consider the total combination of resistances as a single resistor of electric resistance value R, then according to Ohm’s law ,
V = IR ………….(2)
Now, comparing equation (1) and (2), we get
IR = IR1 + IR2 + IR3
IR = I(R1 + R2 + R3)
R = R1 + R2 + R3 
This combination is called resistances in parallel. If electric potential difference is applied across this combination, then it will draw a current I (say).As this current will get three parallel paths through these three electrical resistances, the current will be divided into three parts. Say currents I1, I1 and I1 pass through resistor R1, R2 and R3 respectively.
In star connection, the line 
In delta connection, the line 
The flow of 
The flow of 
If E is the overall emf of the 
The laws of resistance state that,
Electrical resistance R of a conductor or wire is
If we take a piece of pure metal and make its temperature 0° by means of ice and then increase its temperature from gradually from 0°C to to 100°C by heating it. During increasing of temperature if we take its resistance at a regular interval, we will find that electrical resistance of the metal piece is gradually increased with increase in temperature. If we plot the resistance variation with temperature i.e. resistance Vs temperature graph, we will get a straight line as shown in the figure below. If this straight line is extended behind the resistance axis, it will cut the temperature axis at some temperature, − t0°C. From the graph it is clear that, at this temperature the electrical resistance of the metal becomes zero. This temperature is referred as inferred zero resistance temperature. Although zero resistance of any substance cannot be possible practically. Actually rate of resistance variation with temperature is not constant throughout all range of temperature. Actual graph is also shown in the figure below.
Let's R1 and R2 are the measured resistances at temperature t1°C and t2°C respectively. Then we can write the equation below, 
