Power and energy

Power

Instantaneous power delivered to a device

p (t) = e_{1} (t) i_{1} (t) + \dots e_{n} (t) i_{n} (t)

Instantaneous power generated by a device

p_{g e n} (t) = - p (t)

Important

Total sum of instantaneous power delivered to all devices in a circuit equals 0.

One-ports

Device

P = U \times I

Resistor

P = U \times I = I^{2} R = \frac{U^{2}}{R}

Capacitor

Becomes a short-circuit, so $P = 0$

Coil

Becomes a wire, so $L = 0$

Maximal Transfer Theorem

If $E_{g e n} \neq_{0}$ and $R_{g e n} > 0$ are const, then max power that can be transferred is equal to

P_{m a x} = \frac{E_{g e n}^{2}}{4 R_{g e n}}

and is delivered to $R_{l o a d}$ iff $R_{l o a d} = R_{g e n}$

Energy

Energy delivered in period $(t_{0}, t_{1})$ is a quantity $w (t_{0}, t_{1}) = \int_{t_{0}}^{t_{1}} p (t) d t$

Resistor

w (t_{0}, t_{1}) = \int_{t_{0}}^{t_{1}} p (t) d t = \int_{t_{0}}^{t_{1}} i^{2} (t) d t

Inductor

w (t_{0}, t_{1}) = \int_{t_{0}}^{t_{1}} p (t) d t = \int_{t_{0}}^{t_{1}} \underset{u (t)}{\underset{⏟}{L i^{'} (t)}} i (t) d t = \int_{t_{0}}^{t_{1}} {(\frac{1}{2} L i^{2} (t))}^{'} d t = \frac{1}{2} L i^{2} (t_{1}) - \frac{1}{2} L i^{2} (t_{0})

Energy stored

w_{L} (i) = \frac{1}{2} L i^{2}

Capacitor

w (t_{0}, t_{1}) = \int_{t_{0}}^{t_{1}} p (t) d t = \int_{t_{0}}^{t_{1}} \underset{i (t)}{\underset{⏟}{C u^{'} (t)}} u (t) d t = \int_{t_{0}}^{t_{1}} {(\frac{1}{2} C u^{2} (t))}^{'} d t = \frac{1}{2} C u^{2} (t_{1}) - \frac{1}{2} C u^{2} (t_{0})

Energy stored

w_{C} (u) = \frac{1}{2} C u^{2}

Transformer

p (t) = 0 \to w = 0