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Circuit Elements

The most commonly encountered linear one-ports are the inductor of inductance $ L$, the resistor of resistance $ R_{0}$ and capacitor of capacitance $ C$; their schematic representations are shown in Figure 2.3.

Figure 2.3: One-port elements-- (a) an inductor of inductance $ L$, (b) a resistor of resistance $ R_{0}$ and (c) a capacitor of capacitance $ C$.
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The equations relating voltage and current in the three one-ports, as well as their associated impedances are as follows:

  Inductor$\displaystyle :$ $\displaystyle v$ $\displaystyle =L\frac{di}{dt}$ $\displaystyle Z$ $\displaystyle =Ls$ (2.6)
  Resistor$\displaystyle :$ $\displaystyle v$ $\displaystyle =R_{0}i$ $\displaystyle Z$ $\displaystyle =R_{0}$ (2.7)
  Capacitor$\displaystyle :$ $\displaystyle i$ $\displaystyle =C\frac{dv}{dt}$ $\displaystyle Z$ $\displaystyle =\frac{1}{Cs}$ (2.8)

Each of these circuit elements is passive as long as its element value ($ L$, $ C$ or $ R_{0}$) is positive% latex2html id marker 78885
\setcounter{footnote}{2}\fnsymbol{footnote}; the inductor and capacitor are easily shown to be lossless as well. The inductor and capacitor are examples of reactive circuits elements--all power instantaneously absorbed by either one will be stored and eventually be returned to the network to which it is connected. The resistor is passive, but not lossless.

In addition to the one-ports mentioned above, we can also define the short-circuit, open-circuit, current source and voltage source (see Figure 2.4) by

  Short-circuit$\displaystyle :$ $\displaystyle v$ $\displaystyle =0$    
  Open-circuit$\displaystyle :$ $\displaystyle i$ $\displaystyle =0$    
  Voltage source$\displaystyle :$ $\displaystyle v$ $\displaystyle =e(t)$    
  Current source$\displaystyle :$ $\displaystyle i$ $\displaystyle =f(t)$    

Figure 2.4: Other one-ports-- (a) short-circuit, (b) open-circuit, (c) voltage source and (d) current source. Dots adjacent to the sources indicate polarity.
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The impedances of the short- and open-circuit one-ports are zero and infinity, respectively. Both are lossless.

The two-ports which will occur most frequently in this thesis are the transformer and gyrator, both shown in Figure 2.5. Each of these two-ports has two voltage/current pairs, one for each port. The transformer has associated with it one free parameter $ n$, called the turns ratio, and the gyrator is defined with respect to a parameter $ R_{G}>0$, as well as a direction, represented graphically by an arrow. The relation among the port variables in each case is given by

  Transformer$\displaystyle :$ $\displaystyle v_{2}$ $\displaystyle =nv_{1}$ $\displaystyle i_{1}$ $\displaystyle =-ni_{2}$ (2.9)
  Gyrator$\displaystyle :$ $\displaystyle v_{1}$ $\displaystyle =-R_{G}i_{2}$ $\displaystyle v_{2}$ $\displaystyle =R_{G}i_{1}$ (2.10)

It is easily checked that both the transformer and gyrator are lossless two-ports. The gyrator is the first example we have seen so far of a non-reciprocal element--that is, its impedance matrix is not Hermitian; while we will not make nearly as much use of it here as the other elements, it will find a place in certain parts of this work, especially in dealing with physical systems which have a certain type of asymmetric coupling (see Chapter 5), in optimizing certain wave digital structures for simulation (see §3.12), and will play a pivotal role in linking digital waveguide networks to wave digital networks (see §4.10).

Figure 2.5: Two-ports-- (a) a transformer, of turns ratio $ n$ and (b) a gyrator, of gyration coefficient $ R_{G}$.
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There are other $ N$-ports of interest in network theory, many of which have been applied successfully in wave digital filter designs, such as circulators as well as time-varying [178] and non-linear elements [36,39,64,151], which have been used to study the propagation of nonlinear waves in lumped circuits [126]. For numerical integration purposes, however, the above set of elements proves to be an amply sufficient set of basic tools. An exception will be the non-linear distributed elements which appear in the circuit-based approach to fluid-dynamical problems; we mention these elements briefly in Appendix B.

next up previous
Next: Wave Digital Elements and Up: Classical Network Theory Previous: Kirchoff's Laws
Stefan Bilbao 2002-01-22