Additional: Power Transfer in Circuits

Power Delivered by a Source

An electrical energy source, such as a battery or a generator, converts non-electrical energy (chemical, mechanical, etc.) into electrical energy. This electrical energy is then supplied to an external circuit to do work (e.g., light a bulb, run a motor) or be dissipated as heat in components like resistors. Understanding the power delivered by a source is crucial for circuit analysis.

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Ideal vs. Real Sources

An ideal voltage source is conceptualized as a source that provides a constant potential difference across its terminals, regardless of the current drawn from it. It has zero internal resistance. Its terminal voltage is always equal to its EMF ($\mathcal{E}$).

A real voltage source, like a battery or a power supply, always has some internal resistance ($r$). As discussed earlier, its terminal voltage ($V$) decreases as the current ($I$) drawn from it increases, according to the relation $V = \mathcal{E} - Ir$, where $\mathcal{E}$ is the EMF (voltage in open circuit).

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Power Generated by the Source

The total power generated by the EMF source within the real cell is the rate at which it converts non-electrical energy into electrical energy. This power is associated with the action of the source itself, independent of the external circuit resistance.

Consider a real source with EMF $\mathcal{E}$ and internal resistance $r$, connected to an external circuit drawing a current $I$. The total potential difference across which the source acts is its EMF, $\mathcal{E}$.

The total power generated by the source is given by:

$ P_{generated} = \mathcal{E} I $

This is the rate at which the source produces electrical energy.

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Power Delivered to the External Circuit (Output Power)

The power delivered by the source to the external circuit is the rate at which energy is available at the terminals of the source. This is given by the product of the terminal voltage ($V$) and the current ($I$) flowing through the external circuit.

$ P_{delivered} = V I $

Since $V = \mathcal{E} - Ir$, we can substitute this into the equation for delivered power:

$ P_{delivered} = (\mathcal{E} - Ir) I $

$ P_{delivered} = \mathcal{E}I - I^2 r $

This formula shows that the power delivered to the external circuit is the total power generated ($\mathcal{E}I$) minus the power dissipated within the source itself ($I^2r$).

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Power Dissipated within the Source

The internal resistance ($r$) of the source dissipates electrical energy as heat. The power dissipated internally is given by the Joule heating formula, using the current $I$ flowing through the internal resistance $r$:

$ P_{internal} = I^2 r $

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Energy Conservation

The total power generated by the source is equal to the power delivered to the external circuit plus the power dissipated within the source's internal resistance. This is consistent with the conservation of energy:

$ P_{generated} = P_{delivered} + P_{internal} $

$ \mathcal{E}I = (\mathcal{E}I - I^2r) + I^2r $

$ \mathcal{E}I = \mathcal{E}I $

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Power Delivered to an External Resistor

If the external circuit is purely resistive with resistance $R$, the terminal voltage across the source is $V=IR$. In this case, the power delivered to the external circuit is entirely dissipated in the external resistor $R$.

Using Ohm's Law for the entire circuit, $I = \frac{\mathcal{E}}{R+r}$.

The power delivered to the external resistor $R$ (which is $P_{delivered}$) can be expressed in terms of $\mathcal{E}$, $r$, and $R$:

$ P_{delivered} = I^2 R $

Substitute the expression for $I$:

$ P_{delivered} = \left(\frac{\mathcal{E}}{R+r}\right)^2 R $

$ P_{delivered} = \frac{\mathcal{E}^2 R}{(R+r)^2} $

This formula gives the power delivered by a source of EMF $\mathcal{E}$ and internal resistance $r$ to an external load resistor $R$. This is the power that is actually used by the external circuit.

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Maximum Power Transfer Theorem

When designing circuits, especially power supplies, amplifiers, or communication systems, it is often desirable to transfer the maximum possible power from a source to a load. The Maximum Power Transfer Theorem provides the condition under which this occurs.

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Statement of the Theorem

The theorem states that:

"For a source with a fixed internal resistance, maximum power is transferred to an external load when the resistance of the external load is equal to the internal resistance of the source."

That is, maximum power is transferred to the load $R$ when $R = r$.

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Derivation of the Maximum Power Transfer Condition

Consider a source with constant EMF $\mathcal{E}$ and constant internal resistance $r$, connected to a variable external load resistance $R$. The power delivered to the external load is given by the formula we derived earlier:

$ P_{delivered} = \frac{\mathcal{E}^2 R}{(R+r)^2} $

To find the value of $R$ for which $P_{delivered}$ is maximum, we need to differentiate $P_{delivered}$ with respect to $R$ and set the derivative equal to zero. Assume $\mathcal{E}$ and $r$ are constants.

$ \frac{dP_{delivered}}{dR} = \frac{d}{dR} \left[ \mathcal{E}^2 \frac{R}{(R+r)^2} \right] $

Using the quotient rule for differentiation, $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$, where $u = R$ ($u'=1$) and $v = (R+r)^2$ ($v' = 2(R+r)$):

$ \frac{dP_{delivered}}{dR} = \mathcal{E}^2 \left[ \frac{(1)(R+r)^2 - R \cdot 2(R+r)}{((R+r)^2)^2} \right] $

$ \frac{dP_{delivered}}{dR} = \mathcal{E}^2 \left[ \frac{(R+r)^2 - 2R(R+r)}{(R+r)^4} \right] $

Factor out $(R+r)$ from the numerator:

$ \frac{dP_{delivered}}{dR} = \mathcal{E}^2 \left[ \frac{(R+r)[(R+r) - 2R]}{(R+r)^4} \right] $

$ \frac{dP_{delivered}}{dR} = \mathcal{E}^2 \left[ \frac{(R+r - 2R)}{(R+r)^3} \right] $

$ \frac{dP_{delivered}}{dR} = \mathcal{E}^2 \frac{r - R}{(R+r)^3} $

For maximum power, set the derivative equal to zero:

$ \mathcal{E}^2 \frac{r - R}{(R+r)^3} = 0 $

Since $\mathcal{E}$ is typically non-zero, and $(R+r)^3$ is finite (assuming $R \ge 0, r > 0$), the condition for the derivative to be zero is:

$ r - R = 0 $

$ R = r $

This confirms the Maximum Power Transfer Theorem: maximum power is delivered to the load when the load resistance ($R$) equals the source's internal resistance ($r$).

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Value of Maximum Power

To find the maximum power transferred, substitute the condition $R=r$ into the formula for delivered power:

$ P_{delivered, max} = \frac{\mathcal{E}^2 R}{(R+r)^2} $

Substitute $R=r$:

$ P_{delivered, max} = \frac{\mathcal{E}^2 r}{(r+r)^2} = \frac{\mathcal{E}^2 r}{(2r)^2} = \frac{\mathcal{E}^2 r}{4r^2} $

$ P_{delivered, max} = \frac{\mathcal{E}^2}{4r} $

The maximum power delivered to the load is $\mathcal{E}^2 / (4r)$.

When $R=r$, the current in the circuit is $I = \frac{\mathcal{E}}{R+r} = \frac{\mathcal{E}}{r+r} = \frac{\mathcal{E}}{2r}$.

The terminal voltage is $V = IR = \frac{\mathcal{E}}{2r} \times r = \frac{\mathcal{E}}{2}$.

The power delivered to the load is $P_{delivered} = VI = \left(\frac{\mathcal{E}}{2}\right) \left(\frac{\mathcal{E}}{2r}\right) = \frac{\mathcal{E}^2}{4r}$, which matches the formula above.

Also, the power dissipated internally is $P_{internal} = I^2r = \left(\frac{\mathcal{E}}{2r}\right)^2 r = \frac{\mathcal{E}^2}{4r^2} r = \frac{\mathcal{E}^2}{4r}$.

This shows that at maximum power transfer, the power delivered to the load is equal to the power dissipated internally. The total power generated is $\mathcal{E}I = \mathcal{E} \left(\frac{\mathcal{E}}{2r}\right) = \frac{\mathcal{E}^2}{2r}$, which is twice the power delivered to the load.

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Efficiency at Maximum Power Transfer

The efficiency ($\eta$) of power transfer is defined as the ratio of power delivered to the load to the total power generated by the source:

$ \eta = \frac{P_{delivered}}{P_{generated}} = \frac{I^2 R}{I^2 (R+r)} = \frac{R}{R+r} $

At the condition for maximum power transfer, $R=r$:

$ \eta_{max\_power} = \frac{r}{r+r} = \frac{r}{2r} = \frac{1}{2} $

So, at maximum power transfer, the efficiency is only 50%. Half of the total power generated by the source is wasted as heat within the source itself. This is why, in applications where energy efficiency is critical (like power transmission over long distances), maximum power transfer is generally not the operating goal. Maximum power transfer is more relevant in applications like audio amplifiers delivering power to speakers, or radio transmitters feeding power to antennas, where transferring maximum signal power is important, even at the cost of some efficiency.

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Power Delivered vs. Load Resistance Graph

The graph of power delivered to the load ($P_{delivered}$) as a function of the load resistance ($R$) for a source with fixed EMF and internal resistance would look like this:

Graph of power delivered to the load ($R$) from a source with internal resistance $r$. Maximum power occurs when $R=r$.

The curve starts at $P=0$ when $R=0$ (short circuit, current is $\mathcal{E}/r$, but power $I^2R=0$), increases to a maximum value at $R=r$, and then decreases towards $P=0$ as $R \to \infty$ (open circuit, current is 0, so power $I^2R = 0$).

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