Circuit Laws – Notes

Electrical Engineering » Notes » Circuit Laws

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 loosely-attached 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.,

In the case of very small resistances, smaller units are used i.e.,

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 cross-section A of the conductor.
(iii) It depends on the nature of the material.
(iv) It also depends on the temperature of the conductor.

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

In the S.I. system of units,

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.

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.

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 series-parallel circuits.

Resistance in Series

When some conductors having resistances R₁, R₂ and R₃ etc. are joined end-on-end 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 = V₁ + V₂ + V₃
R = R₁ + R₂ + R₃

Also

1/G = 1/G₁ + 1/G­₂ + 1/G₃

But, Ohm’s Law :

V = IR = IR₁ + IR₂ + IR₃

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 = I₁ + I₂ + I₃
1/R = 1/R₁ + 1/R₂ + 1/R₃

Also

G = G₁ + G­₂ + G₃

But, Ohm’s Law :

I = V/R = V/R₁ + V/R₂ + V/R₃

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 short-circuited. 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 short-circuit current) is very large (theoretically, infinity).

Fig. 1. Short Circuits

Two points are said to be open-circuited 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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