04 · Introductory

Energy and electrical power

Power tells us how quickly electrical energy is transferred; sign conventions decide whether a component absorbs or delivers it.

80–110 minIntroductory

In this lesson

Distinguish energy from power and their SI units.
Connect voltage, charge and energy.
Derive P=VI for constant quantities under the receiver convention.
Interpret positive and negative power physically.

Starting question

A battery and a resistor can carry the same current in one loop. Why can one deliver energy while the other absorbs it?

Current alone cannot answer that question. Power depends on the relationship between voltage and current references because power describes the rate of energy transfer through a component boundary.

04.1Energy and power are different quantities

Energy is an amount transferred or stored, measured in joules. Power is the rate of energy transfer:

p(t) = dE/dt

For constant power over an interval Δt,

ΔE = PΔt

1 W = 1 J/s

A 2 W process sustained for 3 s transfers 6 J.

04.2Derive electrical power from voltage and current

Voltage measures energy change per unit charge and current measures charge transfer per unit time. Under a consistent branch reference,

v(t) = dE/dq

i(t) = dq/dt

Multiplying gives

p(t) = dE/dt = (dE/dq)(dq/dt) = v(t)i(t)

For constant/DC quantities,

P = VI

So P=VI is not an isolated formula to memorize; it follows directly from the meanings of voltage and current.

04.3The sign requires a reference convention

Figure 4.1 — Passive/receiver conventionI enters the positive terminal. The AoN voltage arrow runs from − to +, so the two arrows point in opposite directions.

Under receiver/passive convention, the current reference enters the terminal defined positive by the voltage reference. Then

P = VI

P>0: the element absorbs energy.
P<0: the element delivers energy to the rest of the circuit.

A component symbol does not permanently determine its role. A rechargeable battery may deliver power while discharging and absorb power while charging.

04.4Worked example: same magnitudes, opposite energy flow

Suppose |V|=9 V and |I|=20 mA. The magnitude is

|P| = 9 V × 20 mA = 180 mW

If receiver-convention I=+20 mA, P=+180 mW and the element absorbs power. If I=−20 mA with the same voltage reference,

P = 9 V × (−20 mA) = −180 mW

The negative result means the element delivers 180 mW.

Independent check

The magnitudes are identical because |V| and |I| are unchanged. Only the direction of energy transfer reverses.

04.5Time-varying power and total energy

For arbitrary waveforms the instantaneous relation remains

p(t)=v(t)i(t)

Energy transferred from t₁ to t₂ is

ΔE = ∫_t₁^t₂ p(t)dt = ∫_t₁^t₂v(t)i(t)dt

This will matter later for capacitors, inductors and AC circuits, where instantaneous power can reverse direction within a cycle.

04.6Practice for mastery

V=5 V and receiver-convention I=−30 mA. Find P and interpret it.
A device absorbs 250 mW for 8 s. How much energy is transferred to it?
An element delivers 2.4 W at 12 V under receiver convention. What is the signed current?
Why is “a battery always has negative power” an unsafe statement?

Solutions

1. P=−150 mW, so the element delivers 150 mW.

2. ΔE=2.0 J.

3. I=P/V=−2.4/12=−0.20 A.

4. A rechargeable battery can absorb or deliver energy depending on operating state and chosen references.

The definitions are now precise enough to discuss measurement: how do real instruments interrogate voltage and current without becoming invisible to the circuit?