Master Degree (LM) in Electronic Engineering MICROWAVES

NOISE AND NON-LINEAR

Prof. Luca Perregrini

Università di Pavia, Facoltà di Ingegneria [email protected] http://microwave.unipv.it/perregrini/

Microwaves, a.a. 2019/20 Prof. Luca Perregrini and non-linear distortion, pag. 1 SUMMARY

Chapter 10

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 2 SUMMARY

• Dynamic range and sources of noise • Noise power, equivalent noise temperature, measurement of noise temperature • of a component and of a cascaded system • Noise figure of a passive two-port network, a mismatched lossy line, a mismatched • Nonlinear distortion • Gain compression • Harmonic and intermodulation distortion • Third-order intercept point • Intercept point of a cascaded system • Linear and spurious free dynamic range

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 3 MOTIVATION

Noise is critical to the performance of most RF and microwave communications, radar, and remote sensing systems. Noise determines the threshold for the minimum signal that can be reliably detected by a receiver. Noise power in a receiver will be introduced from the external environment through the receiving antenna, as well as generated internally by the receiver circuitry. The sources of noise in RF and microwave systems, and the characterization of components in terms of noise temperature and noise figure, including the effect of impedance mismatch, will be studied.

Compression, harmonic distortion, intermodulation distortion, and dynamic range will also be discussed. These can have important limiting effects when large signal levels are present in mixers, , and other components that use nonlinear devices such as diodes and transistors.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 4 DYNAMIC RANGE

Till now, all components were implicitly assumed as linear (the output signal level is directly proportional to the input signal level), and deterministic (the output signal is predictable from the input signal). In reality no component can perform in this way over an unlimited range of input/output signal levels. In practice, however, there is usually a range of signal levels over which such assumptions are approximately valid. This range is called the dynamic range dynamic range of the component.

First we will address the noise that makes the component nonlinear at low power.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 5 NOISE IN MICROWAVE CIRCUITS

Noise power is a result of random processes such as the flow of charges or holes in an electron tube or solid-state device, propagation through the ionosphere or other ionized gas, or, most basic of all, the thermal vibrations in any component at a temperature above absolute zero. Noise can be passed into a microwave system from external sources, or generated within the system itself. In either case the noise level of a system sets the lower limit on the strength of a signal that can be detected in the presence of the noise. It is generally desired to minimize the residual noise level of a radar or communications receiver to achieve the best performance. In some cases, such as radiometers or radio astronomy systems, the desired signal is actually the noise power received by an antenna, and it is necessary to distinguish between the received noise power and the undesired noise generated by the receiver system itself.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 6 SOURCES OF NOISE

Internal noise in a device or component is usually caused by random motions of charge carriers in devices and materials, due to several mechanisms, leading to various types of noise: • Thermal noise, being caused by thermal vibration of bound charges. It is also known as Johnson or Nyquist noise. • is due to random fluctuations of charge carriers in an electron tube or solid-state device. • occurs in solid-state components and vacuum tubes, and varies inversely with frequency (it is often called 1/ -noise). • Plasma noise is caused by random motion of charges in an ionized gas, 𝑓𝑓 such as a plasma, the ionosphere, or sparking electrical contacts. • Quantum noise results from the quantized nature of charge carriers and photons; it is often insignificant relative to other noise sources.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 7 SOURCES OF NOISE

External noise may be introduced into a system either by a receiving antenna or by electromagnetic coupling. Some sources of external RF noise include the following: • Thermal noise from the ground • Cosmic from the sky • Noise from stars (including the sun) • Lightning • Gas discharge lamps • Radio, TV, and cellular stations • Wireless devices • Microwave ovens • Deliberate jamming devices The external noise can be reduced by filtering the signal, thus reducing the noise bandwidth.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 8 SOURCES OF NOISE

The characterization of noise effects in RF and microwave systems in terms of noise temperature and noise figure will apply to all types of noise, regardless of the source, as long as the spectrum of the noise is relatively flat over the bandwidth of the system. Noise with a flat frequency spectrum is called .

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 9 NOISE POWER

The electrons in a at a physical temperature of degrees (K) are in random motion, with a kinetic energy proportional to the temperature. These random motions produce small, random voltage𝑇𝑇 fluctuations at the resistor terminals, with zero average value but a nonzero root mean square (rms). The rms voltage value is given by Planck’s blackbody radiation law:

h = 6.626×10−34 J-sec (Planck’s constant) k = 1.380×10−23 J/K (Boltzmann’s constant) T = temperature in degrees kelvin (K) B = bandwidth of the system in Hz f = center frequency of the bandwidth (Hz) R = resistance in Ω

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 10 NOISE POWER

At microwave frequencies and for not too low temperatures, so that /( ) 1 /( ), and the above result can be simplified: ℎ𝑓𝑓 𝑘𝑘𝑘𝑘 ℎ𝑓𝑓 ≪ 𝑘𝑘𝑘𝑘 𝑒𝑒 − ≈ ℎ𝑓𝑓 𝑘𝑘𝑘𝑘 (Rayleigh–Jeans approximation)

≈

k = 1.380×10−23 J/K (Boltzmann’s constant) T = temperature in degrees kelvin (K) B = bandwidth of the system in Hz R = resistance in Ω

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 11 NOISE POWER

A resistor can be replaced with a Thevenin equivalent circuit consisting of a noiseless resistor and a generator with voltage = 4 :

Connecting a load with value , the maximum power𝑉𝑉𝑛𝑛 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 transfer from the noisy resistor is achieved: 𝑅𝑅

which represents the maximum available noise power from the noisy resistor at temperature .

This is a white noise source𝑇𝑇 (frequency independent), and the noise power is proportional to the bandwidth, usually limited by the passband of the RF or microwave system. Independent white noise sources can be treated as Gaussian-distributed random variables, so their noise powers (variances) are additive.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 12 EQUIVALENT NOISE TEMPERATURE

If an arbitrary source of noise (thermal or nonthermal) is “white” (i.e., does not change significantly over the bandwidth of interest), it can be modeled as an equivalent thermal noise source, and characterized with an equivalent noise temperature . If the noise source delivers a noise power to a load resistor , it can be 𝑇𝑇𝑒𝑒 replaced by a noisy resistor of value at temperature 𝑁𝑁0 𝑅𝑅 𝑅𝑅 = 𝑁𝑁0 𝑇𝑇𝑒𝑒 𝑘𝑘𝑘𝑘

Components and systems can then be characterized by saying that they have an equivalent noise temperature ; this implies some fixed bandwidth

, which is generally the operational𝑒𝑒 bandwidth of the component or system. 𝑇𝑇 𝐵𝐵 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 13 EQUIVALENT NOISE TEMPERATURE

Example: consider a noisy amplifier with a bandwidth and gain . Let the amplifier be matched to noiseless source and load . 𝐵𝐵 𝐺𝐺 If the source resistor is at = 0 K , then the input power𝑠𝑠 to the amplifier will be =𝑇𝑇 0,

and the output noise power𝑖𝑖 is due only to the𝑁𝑁noise generated by the amplifier. 𝑁𝑁𝑜𝑜 We can obtain the same load noise power by driving an ideal noiseless amplifier with a resistor at the temperature

Then is the equivalent noise temperature of the amplifier.

Microwaves, a.a. 𝑇𝑇2019/20𝑒𝑒 Prof. Luca Perregrini Noise and non-linear distortion, pag. 14 CALIBRATED NOISE SOURCE

It is useful for measurement purposes to have a calibrated noise source. A passive noise source may simply consist of a resistor held at a constant temperature in a temperature-controlled oven, or in a cryogenic flask. Active noise sources may use a diode, transistor, or tube to provide a calibrated noise power output. Can be characterized by an equivalent noise temperature, but a more common measure of noise power for such components is the excess noise ratio (ENR), defined as Power supply 𝑇𝑇0 ENR = 10 log = 10 log 𝑔𝑔 0 𝑔𝑔 0 Power dB 10 𝑁𝑁 − 𝑁𝑁 10 𝑇𝑇 − 𝑇𝑇 supply 𝑇𝑇𝑔𝑔 : noise power at temperature𝑁𝑁0 = 290𝑇𝑇0 K 𝑁𝑁0 , : noise power and equivalent𝑇𝑇0 noise temperature of the powered generator 𝑁𝑁𝑔𝑔 𝑇𝑇𝑔𝑔 Solid-state noise generators typically have ENRs ranging from 20 to 40 dB.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 15 MEASUREMENT OF NOISE TEMPERATURE

Since 0 K source cannot be obtained, a direct measurement of the noise is not feasible. The Y-factor method can be applied. The component under test (e.g., an amplifier) is subsequently connected to two matched loads at different temperatures, and the output power is measured for each case. The output noise power consists of noise power generated by the amplifier as well as noise power from the source resistor:

Taking the ratio we have

The more different the temperatures, the more accurate the results. Usually one source is at room temperature ( = 290 K), the other can be cryogenic (e.g., resistor in liquid nitrogen = 77 K or liquid helium = 4 K). 𝑇𝑇0 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 16 𝑇𝑇 𝑇𝑇 NOISE FIGURE

A component can be characterized by the noise figure, which is a measure of the degradation in the signal-to-noise ratio between the input and output. A noiseless network attenuates/amplifies both noise and signal by the same factor, so that the signal-to-noise ratio / (ratio of desired signal power to undesired noise power) will be unchanged. 𝑆𝑆 𝑁𝑁 In a noisy network, the output noise power will be increased more than the output signal power, so that the output signal-to-noise ratio will be reduced.

The noise figure is a measure of this reduction: 𝐹𝐹

By definition, the input noise power is assumed to be the noise power resulting from a matched resistor at = 290 K (i.e., = ).

Microwaves, a.a. 2019/20 Prof.𝑇𝑇 0Luca Perregrini 𝑁𝑁𝑖𝑖 𝑘𝑘𝑇𝑇Noise0𝐵𝐵and non-linear distortion, pag. 17 NOISE FIGURE

Consider a network characterized by a gain , bandwidth , and equivalent noise temperature . 𝐺𝐺 𝐵𝐵 = 𝑇𝑇𝑒𝑒 (input noise) = (output signal) 𝑁𝑁𝑖𝑖 𝑘𝑘𝑇𝑇0𝐵𝐵 = ( + ) (output noise) 𝑆𝑆𝑜𝑜 𝐺𝐺𝑆𝑆𝑖𝑖 The𝑁𝑁𝑜𝑜 noise𝑘𝑘𝑘𝑘𝑘𝑘 figure𝑇𝑇0 𝑇𝑇is𝑒𝑒

= 10 log 1 + 𝑇𝑇𝑒𝑒 𝐹𝐹dB 10 For a noiseless network giving = 1, or 0 dB. 𝑇𝑇0 Obviously we can write 𝐹𝐹

Noise figure is defined for a matched input source, and for a noise source equivalent to a matched load at = 290 K. Noise figure and equivalent noise temperatures are interchangeable characterizations of a component. 𝑇𝑇0 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 18 NOISE FIGURE: MATCHED LOSSY COMPONENT

Consider a passive, lossy component (e.g., an attenuator, a transmission line), at temperature , connected to a matched source (resistor ) that is also at temperature . 𝑇𝑇 𝑅𝑅 The power gain 𝑇𝑇is less than unity. We can define the 𝐺𝐺loss factor = 1/ .

Because𝐿𝐿 the𝐺𝐺entire system is in thermal equilibrium at the temperature , and has a driving point impedance of R, the output noise power must be 𝑇𝑇

However, we can also think of this power as coming from the source resistor (attenuated by the lossy line), and from the noise generated by the line itself. Thus we also have that

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 19 NOISE FIGURE: MATCHED LOSSY COMPONENT

By comparison of the two expressions

it results

from which

= 𝑁𝑁added 𝑇𝑇𝑒𝑒 In conclusion, the noise figure𝑘𝑘𝑘𝑘 of a lossy component is

If the line is at temperature , then = . For instance, a 6 dB attenuator at room temperature has a noise figure of F = 6 dB. 𝑇𝑇0 𝐹𝐹 𝐿𝐿 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 20 NOISE FIGURE: CASCADED SYSTEMS

When several components are cascaded, each of them may degrade the signal-to-noise ratio. If the noise figures , , , … (or noise temperatures , , , … ) of the individual stages are1 known,2 3 the noise figure of the cascade (or noise temperature ) can𝐹𝐹 𝐹𝐹be𝐹𝐹found. 𝑇𝑇𝑒𝑒1 𝑇𝑇𝑒𝑒2 𝑇𝑇𝑒𝑒3 𝐹𝐹cas Consider the cascade of two components, having gains , , noise figures 𝑇𝑇cas , , and equivalent noise temperatures , . 𝐺𝐺1 𝐺𝐺2 𝐹𝐹1 𝐹𝐹2 𝑇𝑇𝑒𝑒1 𝑇𝑇𝑒𝑒2

The output of the first stage is

and after the second stage

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 21 NOISE FIGURE: CASCADED SYSTEMS

Considering an equivalent component with gain , noise figure and noise temperature , the output noise is 𝐺𝐺1𝐺𝐺2 𝐹𝐹cas 𝑇𝑇cas

By equating to

we obtain

By iterating the procedure, in case of many components it results

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 22 NOISE FIGURE: CASCADED SYSTEMS

Since

it appears that the noise characteristics of a cascaded system are dominated by the characteristics of the first stage since the effect of the second stage is reduced by the gain of the first (assuming > 1). Thus, for the best overall system noise performance, the first stage should 𝐺𝐺1 have a low noise figure and at least moderate gain, later stages have a diminished impact on the overall noise performance. For this reason, a low noise amplifier (LNA) is the first and most critical component in a receiving chain.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 23 NOISE FIGURE: PASSIVE TWO-PORT NETWORK

Consider a passive networks (no active devices, only thermal noise) possibly mismatched. The network is at temperature , the input noise power = is applied at port 1, and is the available power gain from port 1 to port 2. The available output noise power at𝑇𝑇port 2 is 𝑁𝑁1 𝑘𝑘𝑘𝑘𝑘𝑘 𝐺𝐺21

where is the noise power generated internally by the network. 𝑁𝑁added The available power gain is given in terms of the S-parameters and the port mismatches as

where

(see Pozar’s book sec. 12.1).

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 24 NOISE FIGURE: PASSIVE TWO-PORT NETWORK

Since the input noise power is = , the network is passive, and at

temperature , the network is in thermodynamic1 equilibrium, the available output noise power must be =𝑁𝑁 .𝑘𝑘𝑘𝑘𝑘𝑘 𝑇𝑇 From 𝑁𝑁2 𝑘𝑘𝑘𝑘𝑘𝑘 =

we derive 𝑘𝑘𝑘𝑘𝑘𝑘

from which

The results are similar to those for the lossy line, but here we are using the available gain of the network , which accounts for impedance mismatches between the network and the external circuit. 𝐺𝐺21

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 25 NOISE FIGURE: MISMATCHED LOSSY LINE

Consider a lossy line at temperature , with a power loss factor = 1/ , mismatched to its input circuit ( = ( )/( + ) 0). 𝑇𝑇 𝐿𝐿 𝐺𝐺 The scattering Γ𝑠𝑠 𝑍𝑍𝑔𝑔 − 𝑍𝑍0 𝑍𝑍𝑔𝑔 𝑍𝑍0 ≠ matrix can be written as

Using the previous formulas

and

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 26 NOISE FIGURE: MISMATCHED LOSSY LINE

Finally, the equivalent noise temperature is

Using the previous formulas

For a lossy and mismatched line ( > 1, | | > 0), the noise temperature is greater than the one of the matched lossy line ( = ( 1) ). 𝐿𝐿 Γ𝑠𝑠 The reason is that the lossy line actually delivers noise power to both its 𝑇𝑇𝑒𝑒 𝐿𝐿 − 𝑇𝑇 ports, but when the input port is mismatched some of this noise power is reflected back into port 1 and transmitted to port 2. When the generator is matched to port 1, none of the available power from port 1 is reflected back into the line, so the noise power at port 2 is a minimum. Therefore, is important in minimizing noise temperature and noise figure.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 27 NOISE FIGURE: MISMATCHED AMPLIFIER

Consider an amplifier matched at the output port, but mismatched at the input port. Previous results for a passive component cannot be used. Looking for the noise figure, the input noise power is = , and the output noise is 𝑁𝑁1 𝑘𝑘𝑇𝑇0𝐵𝐵 𝑇𝑇𝑒𝑒 The output signal is given by

and the noise figure of the amplifier results

increases as the mismatch increases. This result demonstrates that good noise figure requires good impedance matching. 𝐹𝐹𝑚𝑚 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 28 NONLINEAR DISTORTION

In addition to nonlinearity due to noise effects at very low signal levels, practical components become nonlinear at high signal levels. In the case of active devices (e.g., diodes, transistors) this may be due to effects such as gain compression or the generation of spurious frequency components due to device nonlinearities, but all devices ultimately fail at very high power levels. We will now address the distortion effects that makes the component nonlinear at high power. dynamic range Nonlinearity due to these effects can be useful for some functions, such as amplification, detection, frequency conversion (mixing), rectification.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 29 NONLINEAR DISTORTION

Beside the useful functions based on nonlinearity, however, these effects may lead to detrimental effects, such as increased losses, signal distortion, and possible interference with other radio channels or services. Some of the many possible effects of nonlinearity in RF and microwave circuits are: • Harmonic generation (multiples of a fundamental signal) • Saturation (gain reduction in an amplifier) • Intermodulation distortion (products of a two-tone input signal) • Cross-modulation (modulation transfer from one signal to another) • AM-PM conversion (amplitude variation causes phase shift) • Spectral regrowth (intermodulation with many closely spaced signals)

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 30 NONLINEAR DISTORTION

Consider a general nonlinear network, having an input voltage and an output voltage . Nonlinear𝑣𝑣𝑖𝑖 In the most general sense, the output response 𝑣𝑣𝑜𝑜 device or of a nonlinear circuit can be modeled as a Taylor 𝑣𝑣𝑖𝑖 network 𝑣𝑣𝑜𝑜 series in terms of the input signal voltage:

where

…

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 31 NONLINEAR DISTORTION

Different functions can be obtained from the nonlinear network depending on the dominance of particular terms in the expansion:

• the constant term, with coefficient , leads to rectification, converting an AC input signal to DC. The 𝑎𝑎0 • the linear term, with coefficient , models a linear attenuator ( < 1) or amplifier ( > 1). 𝑎𝑎1 𝑎𝑎1 • the second-order term, with coefficient , can be used for mixing and 𝑎𝑎1 other frequency conversion functions. 𝑎𝑎2 Practical nonlinear devices usually have a series expansion containing many nonzero terms, and a combination of several of the above effects will occur. We will consider some important special cases in the following.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 32 GAIN COMPRESSION

Consider the case of a single-frequency sinusoid input a nonlinear network:

The output voltage is

Retaining only terms up to the third order, this result leads to the voltage gain of the signal component at frequency

𝜔𝜔0

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 33 GAIN COMPRESSION

The result

shows that the voltage gain is equal to , the coefficient of the linear term,

as expected, but with an additional term1 proportional to the square of the input voltage amplitude. 𝑎𝑎 In most practical amplifiers typically has the opposite sign of , so that

the output of the amplifier tends3 to be reduced from the expected1 linear dependence for large values𝑎𝑎of . 𝑎𝑎 This effect is called gain compression or saturation. 𝑉𝑉0 Physically, this is usually due to the fact that the instantaneous output voltage of an amplifier is limited by the power supply voltage used to bias the active device.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 34 GAIN COMPRESSION: 1 DB COMPRESSION POINT

For an amplifier, the plot of the output power versus input power shows a deviation from the ideal response.

To quantify the linear operating range of the amplifier, the 1 dB com- pression point is defined as the power level for which the output power has decreased by 1 dB from the ideal linear characteristic.

This power level is usually denoted by , and can be stated in terms of either input power ( ) or output power ( ). 𝑃𝑃1dB The 1 dB compression point is typically given as the larger of these two 𝐼𝐼𝑃𝑃1dB 𝑂𝑂𝑃𝑃1dB options (usually for amplifiers, for mixers).

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 35 𝑂𝑂𝑂𝑂1dB 𝐼𝐼𝐼𝐼1dB HARMONIC AND INTERMODULATION DISTORTION

Consider again a single-frequency sinusoid input a nonlinear network:

The output signal is

A portion of the single input frequency signal at is converted to harmonics at frequencies , for = 0, 1, 2, . . . . 𝜔𝜔0 In amplifiers, the presence of other frequency components will lead to 𝑛𝑛𝜔𝜔0 𝑛𝑛 signal distortion if those components are in the passband. However, these harmonics are at least one octave from , and, often, lie outside the passband of the amplifier. 𝜔𝜔0 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 36 HARMONIC AND INTERMODULATION DISTORTION

The situation is different, however, when the input signal consists of a two- tone input voltage with closely spaced frequencies and : = (cos + cos ) 𝜔𝜔1 𝜔𝜔2 The output is 𝑣𝑣𝑖𝑖 𝑉𝑉0 𝜔𝜔1𝑡𝑡 𝜔𝜔2𝑡𝑡

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 37 HARMONIC AND INTERMODULATION DISTORTION

The output spectrum consists of harmonics of the form + ( , = 0, ±1, ±2, ±3, . . .)

These combinations of the two𝑚𝑚input𝜔𝜔1 𝑛𝑛frequencies𝜔𝜔2 𝑚𝑚 are𝑛𝑛 called intermodulation products, and the order of a given product is defined as | | + | |. For example, the squared term gives rise to the following four 𝑚𝑚 𝑛𝑛 intermodulation products of second order: 2 (second harmonic of ) = 2 = 0 order = 2 2 (second harmonic of ) = 0 = 2 order = 2 𝜔𝜔1 𝜔𝜔1 𝑚𝑚 𝑛𝑛 (difference frequency) = 1 = 1 order = 2 𝜔𝜔2 𝜔𝜔2 𝑚𝑚 𝑛𝑛 + (sum frequency) = 1 = 1 order = 2 𝜔𝜔1 − 𝜔𝜔2 𝑚𝑚 𝑛𝑛 − Second-order products are undesired in an amplifier, but in a mixer the sum 𝜔𝜔1 𝜔𝜔2 𝑚𝑚 𝑛𝑛 or difference frequencies is the desired outputs. If and are close, all of the second-order products will be far from or and can easily be filtered. 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1 2 Microwaves𝜔𝜔, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 38 HARMONIC AND INTERMODULATION DISTORTION

A typical spectrum of the second- and third-order two-tone intermodulation products is shown in this figure: Amplifier second-order and third-order passband

0 2 2 3 3 2 2 + 2 + 2 + 𝜔𝜔2 − 𝜔𝜔1 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1 𝜔𝜔2 𝜔𝜔 𝜔𝜔1 − 𝜔𝜔2 𝜔𝜔2 − 𝜔𝜔1 𝜔𝜔1 𝜔𝜔2 𝜔𝜔1 𝜔𝜔2 𝜔𝜔2 𝜔𝜔1 The third-order intermodulation products 2 , 2 are near the

original input signals at and , and so cannot1 2be easily2 filtered1 from the passband of an amplifier. 𝜔𝜔 − 𝜔𝜔 𝜔𝜔 − 𝜔𝜔 𝜔𝜔1 𝜔𝜔2 For an arbitrary input signal consisting of many frequencies of varying amplitude and phase, the resulting in-band intermodulation products will cause distortion of the output signal. This effect is called third-order intermodulation distortion.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 39 THIRD-ORDER INTERCEPT POINT

Consider again the two-tone excitation = (cos + cos ) :

𝑣𝑣𝑖𝑖 𝑉𝑉0 𝜔𝜔1𝑡𝑡 𝜔𝜔2𝑡𝑡 𝑣𝑣𝑜𝑜

The third-order products falling in the bandpass increases as . Power is proportional to the square of voltage, so the output power of3third-order 𝑉𝑉0 products must increase as the cube of the input power. Although the , the third-order intermodulation products will increase quickly as input power increases. 𝑎𝑎3 ≪ 𝑎𝑎1 Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 40 THIRD-ORDER INTERCEPT POINT

If we assume the linear response (first order) has a slope of unity, the third- order products has a slope of 3. Second-order products (slope of 2), generally not in the passband, is not not plotted. Linear and third-order re- sponses exhibit compres- sion at high input powers. Extrapolating (dashed lines) their hypothetical intersect (typically above (dBm)

compression), is called the out 𝑃𝑃 third-order intercept point, denoted as . It may be specified as an 𝐼𝐼𝐼𝐼3 input power level , or

an output power level 3 (dBm) ( = ). 𝐼𝐼𝐼𝐼𝑃𝑃 𝑂𝑂𝐼𝐼𝐼𝐼3 𝑃𝑃in Microwaves𝑂𝑂𝑂𝑂𝑂𝑂, a.a3 . 2019/20𝐺𝐺 � 𝐼𝐼𝐼𝐼𝐼𝐼3 Prof. Luca Perregrini Noise and non-linear distortion, pag. 41 THIRD-ORDER INTERCEPT POINT

generally occurs at a higher power level than the 1 dB compression

point3 . Many practical components follow the approximate rule that is𝐼𝐼𝐼𝐼10–15 dB greater than . 𝑃𝑃1dB 𝐼𝐼𝐼𝐼3 From the expression of 𝑃𝑃1,dBthe output power of the desired frequency and of the harmonic 2 are 𝑣𝑣𝑜𝑜 𝜔𝜔1 𝜔𝜔1 − 𝜔𝜔2

By definition, these two powers are equal at the third-order intercept point, therefore, the input signal voltage at the intercept point is

𝑉𝑉𝐼𝐼𝐼𝐼

is equal to the linear response of at the intercept point:

𝑂𝑂𝐼𝐼𝐼𝐼3 𝑃𝑃𝜔𝜔1

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 42 INTERCEPT POINT OF A CASCADED SYSTEM

A cascade connection of components usually degrade (lower) the third- order intercept point. Unlike noise powers, however, intermodulation products are deterministic and may be in phase coherence.

coherent (in-phase) case noncoherent case

intermodulation products not very close to the fundamental signals (relatively random phases from each stage)

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 43 PASSIVE INTERMODULATION

Also in passive component passive intermodulation products (PIM) can be generated. It occurs when signals at two or more closely spaced frequencies mix to produce spurious products in metal-to-metal contact. It may be due to poor mechanical contact, oxidation of junctions between ferrous-based metals, contamination of conducting surfaces at RF junctions, or the use of nonlinear materials such as carbon fiber composites or ferromagnetic materials. In addition, when high powers are involved, thermal effects may contribute to the overall nonlinearity of a junction. It is very difficult to predict PIM levels from first principles, so measurement techniques must usually be used. Because of the third-power dependence of the third-order intermodulation products with input power, passive intermodulation is usually only significant when input signal powers are relatively large. This is frequently the case in cellular telephone base station (powers of 30–40 dBm, many closely spaced RF channels).

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 44 PASSIVE INTERMODULATION

It is often desired to maintain the PIM level below −125 dBm, with two 40 dBm transmit signals. This is a very wide dynamic range, and requires careful selection of components used in the high-power portions of the , including cables, connectors, and antenna components. Because these components are often exposed to the weather, deterioration due to oxidation, vibration, and sunlight must be offset by a careful maintenance program. Communications satellites often face similar problems with passive intermodulation. Passive intermodulation is generally not a problem in receiver systems due to the much lower power levels.

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 45 LINEAR AND SPURIOUS FREE DYNAMIC RANGE

We can define dynamic range in a general sense as the operating range for which a component or system has desirable characteristics. For a power amplifier this may be the power range that is limited at the low end by noise and at the high end by the compression point. This is essentially the linear operating range for the amplifier, and is called the linear dynamic range (LDR). For low-noise amplifiers or mixers, operation may be limited by noise at the low end and the maximum power level for which intermodulation distortion becomes unacceptable. This is effectively the operating range for which spurious responses are minimal, and it is called the spurious-free dynamic range (SFDR).

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 46 LINEAR DYNAMIC RANGE

We can find the linear dynamic range LDR as the ratio of to the noise level (in dBm) of the component: 𝑃𝑃1dB 𝑁𝑁0

Someone define it in terms of a minimum detectable power, which is more appro- priate for a receiver system rather than an LDR individual component, as it depends on external factors (SNR, modulation, error- correcting coding).

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 47 SPURIOUS FREE DYNAMIC RANGE

Maximum output signal power for which the power of the third-order intermodulation product is equal to the noise level of the component, divided by the output noise level. SFDR

It can be demonstrated that:

Microwaves, a.a. 2019/20 Prof. Luca Perregrini Noise and non-linear distortion, pag. 48