A
linear time-invariant system cannot produce intermodulation. If the input of a linear time-invariant system is a signal of a single frequency, then the output is a signal of the same frequency; only the
amplitude and
phase can differ from the input signal. Non-linear systems generate
harmonics in response to sinusoidal input, meaning that if the input of a non-linear system is a signal of a single frequency, ~f_a, then the output is a signal which includes a number of integer multiples of the input frequency signal; (i.e. some of ~ f_a, 2f_a, 3f_a, 4f_a, \ldots). Intermodulation occurs when the input to a non-linear system is composed of two or more frequencies. Consider an input signal that contains three frequency components at~f_a, ~ f_b, and ~f_c; which may be expressed as :\ x(t) = M_a \sin(2 \pi f_a t + \phi_a) + M_b \sin(2 \pi f_b t + \phi_b) + M_c \sin(2 \pi f_c t + \phi_c) where the \ M and \ \phi are the amplitudes and phases of the three components, respectively. We obtain our output signal, \ y(t), by passing our input through a non-linear function G: :\ y(t) = G\left(x(t)\right)\, \ y(t) will contain the three frequencies of the input signal, ~f_a, ~ f_b, and ~f_c (which are known as the
fundamental frequencies), as well as a number of
linear combinations of the fundamental frequencies, each in the form :\ k_af_a + k_bf_b + k_cf_c where ~k_a, ~ k_b, and ~k_c are arbitrary integers which can assume positive or negative values. These are the
intermodulation products (or
IMPs). In general, each of these frequency components will have a different amplitude and phase, which depends on the specific non-linear function being used, and also on the amplitudes and phases of the original input components. More generally, given an input signal containing an arbitrary number N of frequency components f_a, f_b, \ldots, f_N, the output signal will contain a number of frequency components, each of which may be described by :k_a f_a + k_b f_b + \cdots + k_N f_N,\, where the coefficients k_a, k_b, \ldots, k_N are arbitrary integer values.
Intermodulation order The
order \ O of a given intermodulation product is the sum of the absolute values of the coefficients, :\ O = \left|k_a\right| + \left|k_b\right| + \cdots + \left|k_N\right|, For example, in our original example above, third-order intermodulation products (IMPs) occur where \ |k_a|+|k_b|+|k_c| = 3: • f_a + f_b + f_c • f_a + f_b - f_c • f_a + f_c - f_b • f_b + f_c - f_a • 2f_a - f_b • 2f_a - f_c • 2f_b - f_a • 2f_b - f_c • 2f_c - f_a • 2f_c - f_b In many radio and audio applications, odd-order IMPs are of most interest, as they fall within the vicinity of the original frequency components, and may therefore interfere with the desired behaviour. For example, intermodulation distortion from the third order (
IMD3) of a circuit can be seen by looking at a signal that is made up of two
sine waves, one at f_1 and one at f_2. When you cube the sum of these sine waves you will get sine waves at various
frequencies including 2\times f_2-f_1 and 2\times f_1-f_2. If f_1 and f_2 are large but very close together then 2\times f_2-f_1 and 2\times f_1-f_2 will be very close to f_1 and f_2. ==Passive intermodulation (PIM)== As explained in
a previous section, intermodulation can only occur in non-linear systems. Non-linear systems are generally composed of
active components, meaning that the components must be biased with an external power source which is not the input signal (i.e. the active components must be "turned on"). Passive intermodulation (PIM), however, occurs in passive devices (which may include cables, antennas etc.) that are subjected to two or more high power tones. The PIM product is the result of the two (or more) high power tones mixing at device nonlinearities such as junctions of dissimilar metals or metal-oxide junctions, such as loose corroded connectors. The higher the signal amplitudes, the more pronounced the effect of the nonlinearities, and the more prominent the intermodulation that occurs — even though upon initial inspection, the system would appear to be linear and unable to generate intermodulation. The requirement for "two or more high power tones" need not be discrete tones. Passive intermodulation can also occur between different frequencies (i.e. different "tones") within a single broadband carrier. These PIMs would show up as
sidebands in a telecommunication signal, which interfere with adjacent channels and impede reception. Passive intermodulations are a major concern in modern communication systems in cases when a single antenna is used for both high power transmission signals as well as low power receive signals (or when a transmit antenna is in close proximity to a receive antenna). Although the power in the passive intermodulation signal is typically many orders of magnitude lower than the power of the transmit signal, the power in the passive intermodulation signal is oftentimes on the same order of magnitude (and possibly higher) than the power of the receive signal. Therefore, if a passive intermodulation finds its way to receive path, it cannot be filtered or separated from the receive signal. The receive signal would therefore be clobbered by the passive intermodulation signal.
Sources of passive intermodulation Ferromagnetic materials are the most common materials to avoid and include ferrites, nickel, (including nickel plating) and steels (including some stainless steels). These materials exhibit
hysteresis when exposed to reversing magnetic fields, resulting in PIM generation. Passive intermodulation can also be generated in components with manufacturing or workmanship defects, such as cold or cracked solder joints or poorly made mechanical contacts. If these defects are exposed to high radio frequency currents, passive intermodulation can be generated. As a result, radio frequency equipment manufacturers perform factory PIM tests on components, to eliminate passive intermodulation caused by these design and manufacturing defects. Passive intermodulation can also be inherent in the design of a high power radio frequency component where radio frequency current is forced to narrow channels or restricted. In the field, passive intermodulation can be caused by components that were damaged in transit to the cell site, installation workmanship issues and by external passive intermodulation sources. Some of these include: • Contaminated surfaces or contacts due to dirt, dust, moisture or oxidation. • Loose mechanical junctions due to inadequate torque, poor alignment or poorly prepared contact surfaces. • Loose mechanical junctions caused during transportation, shock or vibration. • Metal flakes or shavings inside radio frequency connections. • Inconsistent metal-to-metal contact between radio frequency connector surfaces caused by any of the following: • Trapped dielectric materials (adhesives, foam, etc.), cracks or distortions at the end of the outer conductor of coaxial cables, often caused by overtightening the back nut during installation, solid inner conductors distorted in the preparation process, hollow inner conductors excessively enlarged or made oval during the preparation process. • Passive intermodulation can also occur in connectors, or when conductors made of two
galvanically unmatched metals come in contact with each other. • Nearby metallic objects in the direct beam and side lobes of the transmit antenna including rusty bolts, roof flashing, vent pipes, guy wires, etc.
Passive intermodulation testing IEC 62037 is the international standard for passive intermodulation testing and gives specific details as to passive intermodulation measurement setups. The standard specifies the use of two +43 dBm (20 W) tones for the test signals for passive intermodulation testing. This power level has been used by radio frequency equipment manufacturers for more than a decade to establish PASS / FAIL specifications for radio frequency components. == Intermodulation in electronic circuits ==