Impedance

In this section, we are interested in understanding the concept of impedance, which relates the frequency-dependent nature of “force” and “motion” in a system. In subsequent sections of this course, we will evaluate the impedance of various parts of musical instruments to gain some understanding of how they vibrate.

An impedance provides a frequency response representation of a system in terms of input and output physical signals, in a way similar to the frequency response of filters. Likewise, the inverse frequency transform of an impedance provides an impulse response characterization.

Note that the input and output signals used to compute impedance can be taken at the same or different points of a system. If the input and output locations are at the input to a system, the resulting frequency-domain ratio is referred to as an input or driving point impedance. If the input and output locations are different, the frequency-domain ratio is referred to as a transfer impedance.

• In AC electrical systems, impedance is defined as voltage divided by current.

• Electrical resistors have constant, frequency-independent impedances.

• Electrical capacitors and inductors, however, have current-to-voltage characteristics that change with respect to the frequency of an applied source.

• For mechanical systems, impedance is defined as the ratio of force to velocity.

• The inverse of impedance is called admittance. One can use the term immittance to refer to either an impedance or an admittance.

• In lossless systems, an immittance is purely imaginary and called a reactance.

• Immittances are steady state characterizations that imply zero initial conditions for elements with “memory” (masses and springs, capacitors and inductances).

• In acoustics, impedance is given by pressure divided by either particle or volume velocity.

• More generally, we can derive (or measure) frequency-domain transfer function representations of vibrating systems in terms of impedances or admittances. As well, we can derive or calculate corresponding time-domain impulse responses from these characterizations.