Response function of a physical system

A great many physical systems and devices can be described as a "black box" that transforms a quantity supplied "at the input" into a different quantity "at the output".

Some examples:

1. a radio transforms an electromagnetic signal into a sound wave
2. a musical instrument transforms the kinetic energy supplied by the player into a sound wave
3. a transformer turns an oscillating voltage into an oscillating voltage of the same frequency but different amplitude
4. a car transforms the force applied to the accelerator into speed
5. the human body transforms the potential energy of the molecules it eats into other potential energy through chemical reactions, this into electrical energy in nerve impulses, and finally into heat and kinetic energy in movement
6. a photovoltaic cell converts light radiation into an electric potential difference.

The generality of this point of view is evident. The main question the physicist must answer is then: what mathematical relation is there between the input quantity and the output quantity? This relation is a property of the system called the response function.

The "relation" in question can be a very complex function of many properties of the system, and the ultimate aim of physics is to determine this relation starting from the most elementary possible components of matter.

However, in many cases we have to be content with a less detailed, but simpler, form of the response function, which is equally useful for all practical purposes.

In many cases we are not interested in the exact response of the system to an arbitrary stimulus, but in the response in a particular dynamical regime. In each regime we can expect the (perhaps approximate) expression of the response function to be simpler than the most general exact expression. Commonly used cases are summarised in the following table.

• Important features emerge that are common to all oscillating systems subject to a periodic external force in the linear approximation:
  1. the existence of natural frequencies of oscillation
  2. the phenomenon of resonance

Impedance

The most important approximation of the response function is certainly the so-called linear response. Calling I the input signal and O the output signal, the response is linear when the two are related by the simple linear relation

${\displaystyle O=YI}$

• The quantity Y is the linear response of the system, also called admittance. Its inverse ${\displaystyle Z=1/Y}$, that is, the ratio between input and output (${\displaystyle I=ZO}$), is the impedance of the system. (More specifically, the term impedance is often used to indicate the linear response in frequency.)
• Z can be a simple constant, a function of space and time, or of frequency, or even a more complex quantity (such as a tensor or a linear operator).
• Note: the response of an arbitrary system is in general not linear, but it is very often well approximated by a linear response when the intensity of the stimulus I remains small.
• The explicit form of Z depends on the quantities involved and must be examined case by case
• Typically we are interested in defining input and output so that their product has the dimensions of a power (that is, an energy per unit time). In that case, in fact, the impedance of the system tells us how efficiently, per unit time, the system manages to convert energy from the form in which it receives it at the input to the form in which it returns it at the output.
• Using the tool of analogy we can easily find pairs of such quantities for the different classes of systems. In each column of the following table, for example, the product of a quantity in the first row and one in the second always has the dimensions of a power.

Effects of impedance

In the simplest case, that of the linear frequency response, we supply the system from outside with an input oscillating at a certain frequency ν, that is, an input signal oscillating according to a law of the type

${\displaystyle I(t)=I_0\sin (2\pi \nu t)}$,

and we study the behaviour of the output signal for all possible frequencies. The output signal will in general also be an oscillating function, of the type

${\displaystyle O(t)=O_0(\nu )\sin (2\pi \nu t+\varphi (\nu ))}$.

The impedance of the system therefore has, in this case, essentially only two possible effects (which can occur simultaneously):

1. attenuation: the amplitude of the output quantity depends on the frequency of the input signal;
2. phase shift: the output quantity may not be in phase with the input, but may lead (or lag) it by ${\displaystyle \varphi (f)}$, which therefore also depends on frequency.

Attenuation can have two causes: in the presence of dissipative forces (friction), not all the input energy can be converted into the output signal; moreover, even without dissipation, at certain frequencies the system "accepts" little energy from outside (see reactance below). The phase shift corresponds to the fact that every system has its own "reaction time", an inertia corresponding to the fact that part of the energy supplied is stored for a certain time before being given up in another form.

More specific terms are often used to indicate the different "components" of the impedance:

1. resistance: the part of the total impedance responsible for the dissipation of energy.
2. reactance: the part of the total impedance responsible for the transformation of energy without dissipation.

The attenuation in the response of a physical system can be caused both by resistive and by reactive terms. The phase shift, instead, is due only to the reactive terms. For this reason two types of reactance are distinguished:

1. capacitive reactance, which causes attenuation at low frequencies but not at high ones (and a phase shift of the output leading the input by 90°).
2. inductive reactance, which causes attenuation at high frequencies but not at low ones (and a phase shift of the output lagging the input by 90°).

The total impedance (in terms of attenuation and phase shift) of a physical system as a function of the frequency of the input signal can be a very complex quantity. See the examples in the section musical instruments, and the following pages for more elementary examples

• mechanical impedance:
  • impedance of the harmonic oscillator
  • impedance of a vibrating string
• electrical impedance
• acoustic impedance

Impedance and waves

For waves, impedance has a particularly concrete meaning. Every medium in which a mechanical wave propagates has its own characteristic impedance, which is the product of the density of the medium and the speed of the wave, ${\displaystyle Z=\rho c}$ (for a string the linear density is used, see energy and impedance of the string). When a wave passes from one medium into another with a different impedance, part of its energy is reflected, the larger the more different the impedances are; if the impedances are equal the wave passes without reflection. This is why, when energy is to be transferred from one medium to another, one tries to "match" the impedances (see impedance matching).

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