Class 11> Unit # 10: DC Circuits > Power Dissipation & Maximum Power Transmission Theorem

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Power Dissipation in Resistors & Maximum Power Transfer

Video Lecture

Watch the complete video lecture below to understand power dissipation, expressions for power, and the maximum power transfer theorem.

Power Dissipation in Resistors

"When an electric current passes through a conductor, electrical energy is dissipated in the form of heat energy due to the collision of moving charges with the atoms of the conductor. The loss of electrical energy per unit time is referred to as power dissipation."

Suppose a battery of voltage $V$ is connected across a resistor $R$. If a current $I$ flows through the resistor for a time $t$, the total charge $Q$ transported between the terminals is given by:

$$Q = I t$$

The charge $Q$ moving through potential difference $V$ loses potential energy equal to $QV$, which is entirely converted into heat energy:

$$\text{Heat Produced} = Q V$$

Substituting $Q = I t$ into the equation:

$$\text{Heat Produced} = (I t) V = V I t$$

According to Ohm's Law, $V = I R$. Substituting this into the heat equation:

$$\text{Heat Produced} = (I R) I t = I^2 R t$$

Derivation of Power Expressions

By definition, power ($P$) is the work done (or energy converted) per unit time:

$$P = \frac{\text{Energy (Heat)}}{\text{Time}} = \frac{I^2 R t}{t} = I^2 R$$

Using Ohm's law ($I = \frac{V}{R}$), we can also express power in terms of voltage and resistance:

$$P = \left(\frac{V}{R}\right)^2 R = \frac{V^2}{R^2} R = \frac{V^2}{R}$$

Alternatively, using $V = I R$ directly in $P = V I$:

$$P = V I$$

SI Unit of Power: The unit of power is the watt ($\text{W}$). Its larger multiples are the kilowatt ($\text{kW}$) and megawatt ($\text{MW}$).

Kilowatt-Hour ($\text{kWh}$)

Usually, electrical energy consumed or supplied by generating stations is measured in kilowatt-hours ($\text{kWh}$), commonly known as a "unit" of electrical energy.

One kilowatt-hour is the electrical energy consumed by a device operating at a power rate of 1000 watts for a duration of 1 hour:

$$1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^6\,\text{J} = 3.6\,\text{MJ}$$

Condition for Maximum Power Transfer

"The maximum power transfer theorem states that maximum external power can be obtained from a source with a finite internal resistance if the resistance of the load is equal to the internal resistance of the source."

Consider a circuit with a DC voltage supply $V_s$, an internal resistance $R_s$, and a variable load resistance $R_L$. According to Ohm's Law, the current in the circuit is:

$$I = \frac{V_s}{R_s + R_L}$$

The power dissipated in the load resistance $R_L$ is:

$$P_L = I^2 R_L = \left(\frac{V_s}{R_s + R_L}\right)^2 R_L$$

Analysis shows that:

• Maximum power transfer occurs when the load resistance equals the internal resistance: $R_L = R_s$ (known as the matched condition).
• Power is zero in an open-circuit condition ($R_L \to \infty$, zero current).
• Power is zero in a short-circuit condition ($R_L = 0$, zero voltage across load).

Fig: Graph of power versus load resistance $R_L$ peaking at the matched condition $R_L = R_s$.

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