## 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

### 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.

*Admittances add in parallel,* so the combined admittance is
, and the impedance is

### 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}

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:

*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

^{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.*,

*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].^{8.4}Each 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.

**Next Section:**

Digitization of Lumped Models

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Impedance