Non-Linearity Effects2017-04-09T08:12:11+00:00

# Problems caused by nonlinear distortions

## Reception

• Spurii (“signals” show up, even if nonexistent at input)
• Reduce dynamic range
• Reduce sensitivity (desensitization)
• Blocking of desired signals

## Transmission

• Harmonics
• EVM and Image Rejection degradation
• Reduce efficiency (by backoff)

## Compression

The output power of an amplifier typically exhibits a linear correspondence to the input power as it changes the gain, i.e. the output power/input power quotient remains constant. If you successively raise the power of the input signal, starting at a certain point the output power no longer corresponds exactly to the input power. There is an increasing deviation, the closer you come to the amplifier’s maximum output power: the amplifier compresses.

Gain versus output power and definition of the 1 dB compression point at the amplifier output (Pout/1dB)

Gain versus output power and definition of the 1 dB compression point at the amplifier output (Pout/1dB)

## 1 dB compression point

The 1 dB compression point specifies the output power of an amplifier at which the output signal lags behind the nominal output signal by 1 dB.

A linear gain, i.e. a gain with a sufficiently low driving signal, would yield the nominal output signal. The difference in the level of the output signal to the nominal output signal can be at least qualitatively explained by the over-proportional in harmonics with a high driving signal.
To prevent the power of the harmonics from corrupting the measurement result, the output power must be selectively measured. The amplifier compression is best measured by using a setup with a signal generator and spectrum analyzer as subsequently described in chapter 2. If you want to use a power meter instead of the spectrum analyzer to measure the power, a suitable lowpass or bandpass must be connected ahead of the power meter to eliminate the effect of the harmonics on the result. Compression measurements can also be performed with network analyzers using the power sweep function.

# Nonlinearities

An ideal amplifier can be viewed as a linear two port and transfers signals from the input to the output without distorting them. The power transfer function of such a twoport is as follows:

where

Pout(t) – power at output of two-port
Pin(t ) –  power at input of two-port
GP  –  power gain of two-port

The connection to the input and output voltage is as follows:

and

where

Rin – input resistance of two-port (for simplification, assumed real)
RL – load resistance of two-port (for simplification, assumed real)
vin(t)  – voltage at input of two-port
vout(t)  – voltage at output of two-port

The voltage transfer function of the linear two-port is as follows:

where

Gv voltage gain of two-port
For the sake of clarity, the voltage gain is examined in the following. In practice, ideal two-ports are only possible using passive components. For example, resistive attenuators are assumed to be ideal within wide limits. Two-ports that contain semiconductor components – such as amplifiers – exhibit nonlinearities. A nonlinear transfer function can be approached mathematically by a power series (Taylor series).
The following formula is used:

where

vout(t) – voltage at output of two-port
vin(t) – voltage at input of two-port
a0 – DC component
a1 – gain Gv
an – coefficient of the nonlinear element of the voltage gain

In most cases, it suffices to take the square and cubic component into account, which means that equation 5 only has to be developed up to n = 3. The effects of the nonlinearities of a two-port on its output spectrum depend on the input signal.

# Single-tone driving – harmonics

If a single sinusoidal signal vin(t) is applied to the input of the two-port

where

and

this is referred to as single-tone driving. By inserting equation 7 into equation 5, it can be demonstrated that harmonics of the input signal having the frequencies fn.H = n · fin,1 are produced by the nonlinearities

Applying the trigonometric conversion:

to the square and cubic component yields:

Note:
The 2nd harmonic ( 1 , 2 in  ) is phase-shifted by 90° with respect to the fundamental, since the following trigonometric relationship applies: cos(x)  sin( / 2 x)

Spectrum before and after a nonlinear two-port

The levels of these harmonics depend on the coefficients an in equation 2. But they also depend on the order n of the particular harmonic and on the input level. The levels of harmonics increase over-proportionally with their order as the input level increases, i.e. changing the input level by Δ dB changes the harmonic level by n · Δ dB. Data sheet specifications of this type of signal distortion are usually limited to the 2nd and 3rd harmonic, for which the level difference akN to the fundamental at the output of the two-port is specified. Such specifications apply only to a particular input level Pin or output level Pout that must also always be specified. A level-independent specification using the 2nd harmonic intercept (SHI) point is more favorable for comparisons.

### Definition:

The SHIin or SHIout point corresponds to the fictitious input or output level at which the 2nd harmonic of the output signal would exhibit the same level as the fundamental at the output of the two-port. The fundamental is assumed to be linearly transferred

Graphical definition of the 2nd harmonic intercept (SHI) point at the two-port input (SHIin) and at the two-port output (SHIout)

In practice, this point can hardly ever be reached, since the two-port, as shown in the Fig., compresses already at low input levels. The intercept point can be referenced to the input as well as the output level and is referred to as the input or output intercept point, respectively (called SHIin or SHIout here). Assuming the input level Pin and the harmonic ratio ak2 of the 2nd harmonic are known, this point can be calculated as follows:

If SHIout is referenced to the output, the following applies:

where g: power gain of two-port, in dB.
or also:

if the output level is taken as the reference

# Two-tone driving – intermodulation

Two-tone driving applies a signal vin(t) to the input of the two port. This signal consists of two sinusoidal signals of the same amplitude. The following formula is valid for the input signal:

where ,1,2 in V peak values of the two sinusoidal signals
fin,1, fin,2 signal frequencies

Inserting equation 12 into the nonlinear transfer function according to equation 5 yields at the output of the two-port, among other things, the intermodulation products listed in Table 1-1. The angular frequency ω is always specified, where ω1 = 2·π· fin,1 and ω2 = 2·π· fin,2.
The new frequencies produced are a result of the following trigonometric conversions:

2nd order intermodulation products
as well as:

Intermodulation products up to max. 3rd order with two-tone driving

Output spectrum of a nonlinear two-port with two-tone driving for intermodulation products up to max. 3rd order

Besides generating harmonics, two-tone driving also produces intermodulation products (also referred to as difference frequencies). The order of the intermodulation products corresponds to the sum of the order numbers of the components involved. For example, for the product with 2·fin,1 + 1·fin,2 the order is 2 + 1 = 3. Table 1-1 takes into account intermodulation products only up to the 3rd order.

The frequencies of even-numbered intermodulation products (e.g. 2nd order) are far away from the two input signals, namely at the sum frequency and at the difference frequency. They are in general easy to suppress by filtering.
Some of the odd-numbered intermodulation products (the difference products) always occur in the immediate vicinity of the input signals and are therefore difficult to suppress by filtering.
Depending on the application, products of both even- and odd-numbered order can cause interference. In the case of measurements on cable TV (CATV) amplifiers where a frequency range of more than one octave is to be tested, harmonics as well as intermodulation products of an even-numbered order occur in the range of interest.

Graphical definition of 2nd and 3rd order intercept points at the input (IP2in, IP3in) and at the output (IP2out, IP3out) of an amplifier

As with higher-order harmonics, a level change of the two sinusoidal carriers at the input by Δ dB causes the level of the associated intermodulation product to change by n · Δ dB.

Specifications regarding the level differences between intermodulation products and the fundamentals of the sinusoidal carriers thus always require that the output level of the amplifier be specified, for otherwise no statement can be made about its linearity.
Therefore, here too it is advantageous to calculate the nth-order intercept point. The following formula applies to the nth-order intercept point referenced to the input:

where IPnin nth-order input intercept point, in dBm
aIMn level difference between intermodulation product of nth order and fundamental of input signal, in dB
Pin level of one of the two input signals, in dBm

In most cases, the intercept points of the 2nd and 3rd order are specified (see also Fig. 1-6). They are abbreviated as IP2 or SOI (2nd order intercept) and IP3 or TOI (third order intercept), respectively.
Definition:
The 2nd order intercept point IP2in or IP2out corresponds to the fictitious input or output level at which the 2nd order intermodulation product would exhibit the same level as the fundamental at the output of the two-port.
The 3rd order intercept point IP3in or IP3out corresponds to the fictitious input or output level at which the third order intermodulation product would exhibit the same level as the fundamental at the output of the two-port.
In both cases, the fundamental is assumed to be linearly transferred (see Fig. 1-6).
The following formulas apply, respectively, to the 2nd and 3rd order input intercept points:

and

The output intercept points can be calculated from the input intercept points (or, vice versa, the input intercept points from the output intercept points) by adding the gain (g) of the two-port (in dB).

The following formulas apply, respectively, to the 2nd and 3rd order output intercept points:

or

and

or

2nd order intermodulation products with two-tone driving as well as the second harmonic with single-tone driving are the result of the square component of the nonlinear transfer function. Between SHI and IP2 exists a fixed correlation that is derived from the coefficients from Table 1 (factor of 0.5 of the 2nd order harmonics vis-à-vis the 2nd order intermodulation products):

Second harmonic intercept (SHI) point :

Data sheets therefore mostly only specify IP2 or SHI, rarely are both values published simultaneously. Intercept points are almost always specified in dBm.