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Classical network theory [12] is partly concerned with the properties of connections of N-port devices. In the abstract, an $ N$-port is a mathematical entity whose internal behavior is only accessible through its $ N$ ports. With the $ j$th port is associated a current $ i_{j}$, a voltage $ v_{j}$, and two terminals (see Figure 2.1). The two terminals of any port must always be connected to the terminals of another port. A network is simply a collection of $ N$-ports connected such that no port is left free% latex2html id marker 78686

Figure 2.1: N-port.
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For lumped networks, the voltages, currents and possibly element values in the networks are allowed to be real-valued functions of a sole real parameter $ t$ which is usually interpreted as physical time. Multidimensional networks [208] are more general in the sense that the voltages, currents and port resistances may be functions of one or many other parameters, which may represent spatial dimensions. In this introductory chapter, we will be concerned only with lumped networks, but it should be kept in mind that Chapter 3 and parts of Chapters 4 and 5 are devoted chiefly to a particular class of multidimensional network which can represent the behavior of a distributed physical system.

If an $ N$-port is linear and time-invariant (LTI), then the port quantities may exhibit a purely exponential time-dependence at a single complex frequency $ s$. For such an exponential state [12], it is also useful to define, for any port with voltage $ v(t)$ and current $ i(t)$, the complex amplitudes $ \hat{v}$ and $ \hat{i}$. We can then write

$\displaystyle v(t) = \hat{v}e^{st}\hspace{1.0in}i(t) = \hat{i}e^{st}$    

Under certain conditions [12], a LTI network will possess an $ N\times N$ impedance matrix% latex2html id marker 78709
\setcounter{footnote}{2}\fnsymbol{footnote} $ {\bf Z}$, so that the steady-state voltages and currents are related by

$\displaystyle {\bf\hat{v}} = {\bf Z\hat{i}}$ (2.1)

where $ {\bf\hat{v}}$ and $ {\bf\hat{i}}$ are the column $ N$-vectors containing the amplitudes $ \hat{v}_{1},\hdots,\hat{v}_{N}$ and $ \hat{i}_{1},\hdots,\hat{i}_{N}$ respectively. In general, if the $ N$-port contains elements which behave as differential or integral operators, then we will have $ {\bf Z} = {\bf Z}(s)$. The admittance of such an $ N$-port is defined as

$\displaystyle {\bf Y} = {\bf Z}^{-1}$    

at frequencies $ s$ for which $ {\bf Z}$ is invertible, and as infinity otherwise.

In most cases of interest, the entries of $ {\bf Z}(s)$ will be rational functions of $ s$. An $ N$-port so defined is called real if the coefficients of these rational functions are real numbers. In this case, there is no loss in generality [12] in considering the port voltages and currents to be real-valued functions of $ t$, in which case we may write, for an exponential state,

$\displaystyle v(t) = {\rm Re}\left(\hat{v}e^{st}\right)\hspace{1.0in}i(t) = {\rm Re}\left(\hat{i}e^{st}\right)$    

Now $ v(t)$ and $ i(t)$ are referred to as the real instantaneous port voltage and current respectively.
next up previous
Next: Power and Passivity Up: Classical Network Theory Previous: Classical Network Theory
Stefan Bilbao 2002-01-22