Power and Passivity

The total instantaneous power absorbed by a real $N$-port is defined by

$$w_{inst}(t) = \sum_{j=1}^{N}v_{j}(t)i_{j}(t)$$ (2.2)

where $v_{j}(t)$ and $i_{j}(t)$ are the real instantaneous voltage and current at port $j$. In general, for an $N$-port which contains stored energy $E(t)$, which dissipates energy at rate $w_{d}(t)$, and which contains sources which provide energy at rate $w_{s}(t)$, then the energy balance

$$\int_{t_{1}}^{t_{2}}\left(w_{inst}+w_{s}-w_{d}\right)dt = E(t_{2}) - E(t_{1})$$ (2.3)

must hold over any interval $[t_{1},t_{2}]$. Such an $N$-port is called passive if we have

$$\int_{t_{1}}^{t_{2}}w_{inst}dt \geq E(t_{2}) - E(t_{1})$$ (2.4)

over any time interval; the increase in stored energy must be less than the energy delivered through the ports. The $N$-port is called lossless if (2.4) holds with equality over any interval.

For a linear time-invariant $N$-port, in an exponential state of complex frequency $s$, we can define the total complex power absorbed to be the inner product

$$w = {\bf\hat{i}^{*}\hat{v}}$$

and the average or active power as

$$\bar{w} = {\rm Re}({\bf\hat{i}^{*}\hat{v}})$$

where $^{*}$ denotes transpose conjugation. For an $N$-port defined by an impedance relationship, we may immediately write, in terms of the voltage and current amplitudes,

$$\bar{w} = {\rm Re}\left({\bf\hat{i}^{*}\hat{v}}\right) = \frac{1}... ...}}\right) = \frac{1}{2}\left({\bf\hat{i}^{*}\left(Z+Z^{*}\right)\hat{i}}\right)$$

For such a real LTI $N$-port, passivity may be defined in the following way. If the total active power absorbed by an $N$-port is always greater than or equal to zero for frequencies $s$ such that Re$(s)\geq 0$, then it is called passive. This implies that

$${\bf Z}+{\bf Z^{*}}\geq {\bf0}\hspace{0.5in}{\rm for}\hspace{0.5in}{\rm Re}(s)\geq 0$$ (2.5)

A matrix Z with such a property is called a positive matrix. In the present case of a real $N$-port, ${\bf Z}$ is called positive real (though in general, positivity is all that is required for passivity). If the average power absorbed is identically zero for Re$(s)=0$, or, in terms of impedances, if

$${\bf Z}+{\bf Z^{*}} = {\bf0}\hspace{0.5in}{\rm for}\hspace{0.5in}{\rm Re}(s) = 0$$

then the $N$-port is called lossless.