Electrical Engineering » Notes » Circuit Laws
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Resistance
The property of a substance due to which it opposes (or restricts) the flow of electricity (i.e., electrons) through it.
Metals, acids and salts solutions are good conductors of electricity.
Due to the presence of a large number of free or looselyattached electrons in their atoms. These vagrant electrons assume a directed motion on the application of an electric potential difference.
Those substances which offer relatively greater
difficulty or hindrance to the passage of these electrons
are said to be relatively poor conductors of electricity like
bakelite, mica, glass, rubber and dry wood.
Unit of Resistance
The practical unit of resistance is ohm. The symbol for ohm is Ω.
A conductor is said to have a resistance of one ohm if it permits one ampere current to flow through it when one volt is impressed across its terminals.
For insulators whose resistances are very high, a much bigger unit units are i.e.,
megaohm = 10^{6} ohm
or
kiloohm = 10^{3} ohm
In the case of very small resistances, smaller units are used i.e.,
milliohm = 10^{3} ohm
or
microohm = 10^{6} ohm.
Prefix  Meaning  Abbreviation  Equal to 
Mega  One million  M Ω  10^{6} Ω 
Kilo  One thousand  k Ω  10^{3} Ω 
Centi  One hundredth  –  – 
Milli  One thousandth  m Ω  10^{3} Ω 
Micro  One millionth  μ Ω  10^{6} Ω 
Laws of Resistance
The resistance R offered by a conductor depends on the following factors :
(i) It varies directly as its length, l.
(ii) It varies inversely as the crosssection A of the conductor.
(iii) It depends on the nature of the material.
(iv) It also depends on the temperature of the conductor.
R = ρl / A
where ρ is a constant depending on the nature of the material of the conductor and is known as its specific resistance or resistivity.
Units of Resistivity
ρ = AR/l
In the S.I. system of units,
ρ = A meter^{2} * R ohm / l metre = AR/l ohmmetre
Conductance and Conductivity
Conductance (G) is reciprocal of resistance. Whereas resistance of a conductor measures the opposition which it offers to the flow of current, the conductance measures the inducement which it offers to its flow.
R = ρl / A
G = (1/ρ) * (A/l) = σA/l
where σ is called the conductivity or specific conductance of a conductor. The unit of conductance is siemens (S). Earlier, this unit was called mho.
σ = Gl/A = G siemens * l metre / A meter^{2 }= Gl/A siemens/metre
Hence, the unit of conductivity is siemens/metre (S/m).
Ohm’s Law
This law applies to electric to electric conduction through good conductors and may be stated as follows :
The ratio of potential difference (V) between any two points on a conductor to the current (I) flowing between them, is constant, provided the temperature of the conductor does not change.
V/I = R
where,
I is the current through the conductor in units of amperes,
V is the voltage measured across the conductor in units of volts, and
R is the resistance of the conductor in units of ohms.
1. The relationship of current, voltage, and resistance is given by Ohm’s law.
2. Ohm’s law states that the current in an electric circuit is directly proportional to the voltage applied and inversely proportional to the resistance in the circuit.
I = E/R
3. Ohm’s law applies to all series, parallel, and seriesparallel circuits.
Resistance in Series
When some conductors having resistances R_{1}, R_{2} and R_{3} etc. are joined endonend as in Fig. 1, they are said to be connected in series.
Fig. 1. Resistance in Series
It can be proved that the equivalent resistance or total resistance between points A and D is equal to the sum of the three individual resistances.
Being a series circuit, it should be remembered that
(i) current is the same through all the three conductors
(ii) but voltage drop across each is different due to its different resistance and is given by Ohm’s Law and
(iii) sum of the three voltage drops is equal to the voltage applied across the three conductors. There is a progressive fall in potential as we go from point A to D as shown in Fig. 1.
V = V_{1} + V_{2} + V_{3}
R = R_{1} + R_{2} + R_{3}
Also
1/G = 1/G_{1 }+ 1/G_{2 }+ 1/G_{3}
But, Ohm’s Law :
V = IR = IR_{1} + IR_{2} + IR_{3}
As seen from above, the main characteristics of a series circuit are :
1. same current flows through all parts of the circuit.
2. different resistors have their individual voltage drops.
3. voltage drops are additive.
4. applied voltage equals the sum of different voltage drops.
5. resistances are additive.
6. powers are additive.
Resistances in Parallel
Three resistances, as joined in Fig. 2 are said to be connected in parallel.
Fig. 2. Resistance in Parallel
Being a parallel circuit, it should be remembered that
(i) p.d. across all resistances is the same.
(ii) current in each resistor is different and is given by Ohm’s Law and
(iii) the total current is the sum of the three separate currents.
I = I_{1} + I_{2} + I_{3 }1/R = 1/R_{1} + 1/R_{2} + 1/R_{3}
Also
G = G_{1 }+ G_{2 }+ G_{3}
But, Ohm’s Law :
I = V/R = V/R_{1} + V/R_{2} + V/R_{3}
The main characteristics of a parallel circuit are :
1. same voltage acts across all parts of the circuit
2. different resistors have their individual current.
3. branch currents are additive.
4. conductances are additive.
5. powers are additive.
Short and Open Circuits
When two points of circuit are connected together by a thick metallic wire (Fig. 3), they are said to be shortcircuited. Since ‘short’ has practically zero resistance, it gives rise to two important facts :
(i) no voltage can exist across it because V = IR = I × 0 = 0.
(ii) current through it (called shortcircuit current) is very large (theoretically, infinity).
Fig. 1. Short Circuits
Two points are said to be opencircuited when there is no direct connection between them
(Fig. 4). Obviously, an ‘open’ represents a break in the continuity of the circuit. Due to this break :
(i) resistance between the two points is infinite.
(ii) there is no flow of current between the two points.
Fig. 1. Open Circuits
Kirchhoff’s Current Law and Kirchhoff’s Voltage Law
1. KCL is a law stating that the algebraic sum of the currents flowing into any node in a network must be zero.
2. KVL is a law stating that the algebraic sum of the voltages around any closed path in a network must be zero.
A helpful mnemonic for writing KVL equations is to assign the polarity to a given voltage in accordance with the first sign encountered when traversing that voltage around the loop.
3. The following is the basic method (or fundamental method or KVL/KCL method) of solving networks :
1. Define voltages and currents for each element.
2. Write KVL.
3. Write KCL.
4. Write constituent relations.
5. Solve.
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