Spring 2018

ECE 110

Course Notes

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The power at a circuit element (such as a source or a resistor) is a measure of how quickly it converts energy. In a closed system, some elements deliver power (i.e. convert energy into electrical form) and other elements absorb power (i.e. convert energy from electrical energy into some other form). Let us revisit the battery connected to a light bulb.

Figure 1

Battery connected to a light bulb
Fig. 1: Battery connected to a light bulb. The battery delivers power because it converts chemical potential energy to electrical energy. The light bulb absorbs power because it converts electrical energy to light and heat energy. Note that the power absorbed by the light bulb matches the power delivered by the battery in accordance with the conservation of energy.

We define the power $P$ at a circuit element to be the amount of electrical energy it absorbs per second. So, power has units of joules/second (J/s), which are called watts, with symbol W. By this definition, if $P>0 \text{ W}$, the element absorbs power (like the light bulb in Fig. 1), but if $P<0 \text{ W}$, the element delivers power (like the battery in Fig. 1).

Type of Energy Conversion Behavior of Power Sign of $P$
Energy is converted from electrical energy
into some other form
Power is absorbed $+$
Energy is converted from some other form
into electrical energy
Power is delivered $-$
Table 1: Sign convention for power $P$.

The power $P$ absorbed at a circuit element with voltage drop $V$ and current $I$ in standard labeling is given by:
P &= VI \label{POW-PVI}
Equation $\eqref{POW-PVI}$ is a consequence of the definitions of voltage drop and current. Voltage drop is the amount of electrical energy absorbed per $+1\text{ C}$ of charge as it moves from the $+$ label to the $-$ label. Current is the amount of positive charge that flows per second in the direction of the arrow label. Multiplying these two quantities cancels out charge, and produces the amount of electrical energy absorbed per second, which is power.

Another way to see this relationship is to multiply $1\text{ V}$ by $1\text{ A}$ and thus obtain $1\text{ W}$:
\left(1\text{ V}\right) \left(1\text{ A}\right) = \left(1\frac{\text{J}}{\text{C}}\right) \left(1\frac{\text{C}}{\text{s}}\right) = 1\frac{\text{J}}{\text{s}} = 1\text{ W}
If a circuit element is in nonstandard labeling, the sign of equation $\eqref{POW-PVI}$ must be flipped to be consistent with the sign convention for $P$.

Labelings Labeling System Power
Standard labeling
"arrow goes from $+$ to $-$"
Nonstandard labeling
"arrow goes from $-$ to $+$"
Table 2: Power formulas for different labelings of voltage drop polarity and current direction.

Given that power delivered is a negative quantity and power absorbed is a positive quantity, the law of conservation of energy implies that the total power of all elements in a closed system is zero. At any point in time, there cannot be more power absorbed than delivered or more power delivered than absorbed. For the battery connected to a light bulb, the power balance is expressed as:
P_{\text{battery}}+P_{\text{light bulb}}=0

We apply Ohm's law to equation $\eqref{POW-PVI}$ to produce two additional formulas for the power of a resistor.
P &= VI = \frac{V^2}{R} = I^2R \label{POW-PRS}
Equation $\eqref{POW-PRS}$ implies that power $P \geq 0$ because $R$, $V^2$ and $I^2$ are always greater than or equal to zero. In words, a resistor can absorb power (by converting electrical energy into heat energy), but can never deliver power.

The results in equation $\eqref{POW-PRS}$ may be confusing because $R$ appears as a multiplicative factor of $I^2R$, but in the denominator of $V^2/R$. Does $P$ scale linearly or inversely with $R$? The answer is neither because changes in $R$ also affect $V$ and $I$ in ways that depend on what is connected to the resistor.

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