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Power and Passivity

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

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

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

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

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

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

and the average or active power as

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

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,

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

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

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

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

then the $ N$-port is called lossless.

next up previous
Next: Kirchoff's Laws Up: Classical Network Theory Previous: N-ports
Stefan Bilbao 2002-01-22