## One-Port Network Theory

The basic idea of a one-port network [524] is shown in Fig. 7.5. The one-port is a black box'' with a single pair of input/output terminals, referred to as a port. A force is applied at the terminals and a velocity flows'' in the direction shown. The admittance seen'' at the port is called the driving point admittance. Network theory is normally described in terms of circuit theory elements, in which case a voltage is applied at the terminals and a current flows as shown. However, in our context, mechanical elements are preferable.

### Series Combination of One-Ports

Figure 7.6 shows the series combination of two one-ports.

Impedances add in series, so the aggregate impedance is , and the admittance is

The latter expression is the handy product-over-sum rule for combining admittances in series.

### Mass-Spring-Wall System

In a physical situation, if two elements are connected in such a way that they share a common velocity, then they are in series. An example is a mass connected to one end of a spring, where the other end is attached to a rigid support, and the force is applied to the mass, as shown in Fig. 7.7.

Figure 7.8 shows the electrical equivalent circuit corresponding to Fig.7.7.

### Parallel Combination of One-Ports

Figure Fig.7.10 shows the parallel combination of two one-ports.

which is the familiar product-over-sum rule for combining impedances in parallel. This operation is often denoted by

### Spring-Mass System

When two physical elements are driven by a common force (yet have independent velocities, as we'll soon see is quite possible), they are formally in parallel. An example is a mass connected to a spring in which the driving force is applied to one end of the spring, and the mass is attached to the other end, as shown in Fig.7.11. The compression force on the spring is equal at all times to the rightward force on the mass. However, the spring compression velocity does not always equal the mass velocity . We do have that the sum of the mass velocity and spring compression velocity gives the velocity of the driving point, i.e., . Thus, in a parallel connection, forces are equal and velocities sum.

Figure 7.12 shows the electrical equivalent circuit corresponding to Fig.7.11.

### Mechanical Impedance Analysis

Impedance analysis is commonly used to analyze electrical circuits [110]. By means of equivalent circuits, we can use the same analysis methods for mechanical systems.

For example, referring to Fig.7.9, the Laplace transform of the force on the spring is given by the so-called voltage divider relation:8.2

Similarly, the Laplace transform of the force on the mass is given by

 (8.1)

As a simple application, let's find the motion of the mass , after time zero, given that the input force is an impulse at time 0:

Then, by the voltage divider'' relation Eq.(7.1), the Laplace transform of the mass force after time 0 is given by

where we have defined . The mass velocity Laplace transform is then

Thus, the impulse response of the mass oscillates sinusoidally with radian frequency , and amplitude . The velocity starts out maximum at time , which makes physical sense. Also, the momentum transferred to the mass at time 0 is ; this is also expected physically because the time-integral of the applied force is 1 (the area under any impulse is 1).

### General One-Ports

An arbitrary interconnection of impedances and admittances, with input and output force and/or velocities defined, results in a one-port with admittance expressible as

In any mechanical situation we have , in principle, since at sufficiently high frequencies, every mechanical system must look like a mass.''8.3 However, for purposes of approximation to a real physical system, it may well be best to allow and consider the above expression to be a rational approximation to the true admittance function.

### Passive One-Ports

It is well known that the impedance of every passive one-port is positive real (see §C.11.2). The reciprocal of a positive real function is positive real, so every passive impedance corresponds also to a passive admittance.

A complex-valued function of a complex variable is said to be positive real (PR) if

1)
is real whenever is real
2)
whenever .

A particularly important property of positive real functions is that the phase is bounded between plus and minus degrees, i.e.,

This is a significant constraint on the rational function . One implication is that in the lossless case (no dashpots, only masses and springs--a reactance) all poles and zeros interlace along the axis, as depicted in Fig.7.14.

Referring to Fig.7.14, consider the graphical method for computing phase response of a reactance from the pole zero diagram [449].8.4Each zero on the positive axis contributes a net 90 degrees of phase at frequencies above the zero. As frequency crosses the zero going up, there is a switch from to degrees. For each pole, the phase contribution switches from to degrees as it is passed going up in frequency. In order to keep phase in , it is clear that the poles and zeros must strictly alternate. Moreover, all poles and zeros must be simple, since a multiple poles or zero would swing the phase by more than degrees, and the reactance could not be positive real.

The positive real property is fundamental to passive immittances and comes up often in the study of measured resonant systems. A practical modeling example (passive digital modeling of a guitar bridge) is discussed in §9.2.1.

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Impedance