Detailed explanation of third-order intermodulation distortion and testing

btty038 · Published on 2022-10-21 14:37

Detailed explanation of third-order intermodulation distortion and testing [Copy link]

This post was last edited by btty038 on 2022-3-31 22:31

A very important common parameter is the third-order intermodulation distortion (3rd ^- order IMD), which will be the focus of this article.

Any semiconductor device has a certain degree of nonlinearity, especially in the case of large signal input, the nonlinearity will be more obvious. Since the amplifier has a certain gain, it means that the amplifier has more obvious nonlinearity than other semiconductor devices, which is why the nonlinearity of the amplifier is particularly concerned in practice. The following will take the amplifier as an example to discuss intermodulation distortion and its test methods.

1. What are the effects of intermodulation distortion?

In wireless communication systems, intermodulation distortion not only affects the performance of the transmitting link, but also affects the performance of the receiving link.

For the transmission link, the most nonlinear component is the power amplifier. When the signal is a broadband modulated signal, whether within or outside the signal bandwidth, a relatively rich intermodulation product will be generated. Intermodulation products outside the band will cause interference to other channels, which is usually characterized by ACLR or ACPR in communication, known as adjacent channel leakage ratio. Intermodulation products within the band will interfere with the signal itself, deteriorating the signal-to-noise ratio/signal-to-interference ratio, which is usually characterized by the noise power ratio NPR, which is a parameter of concern in satellite communications.

For the receiving link, the main consideration is the intermodulation distortion of the front-end low noise amplifier. When there is strong dual-tone or multi-tone interference near the signal, the intermodulation distortion product will fall into the signal band, thereby deteriorating the sensitivity of the receiver. One of the very targeted test items is the "dual-tone sensitivity" of the mobile phone, that is, when there is dual-tone interference at the position of the adjacent channel, the sensitivity is tested at this time. The frequency and amplitude of dual-tone interference are defined in the specification, and the sensitivity must meet certain requirements. This requires the RF front-end LNA to have relatively excellent linearity!

Figure 1. Intermodulation distortion falls in-band and causes direct interference.

In summary, intermodulation distortion has a very important impact on the performance of the entire transceiver system of wireless communication. In the design and debugging of RF amplifiers, nonlinear performance is a consideration that cannot be ignored.

2. Overview of the intermodulation distortion generation mechanism

When a single-tone signal (i.e., a single-frequency signal) is input to the amplifier, the amplifier will output the fundamental frequency and its harmonic components. When a dual-tone or multi-tone signal is input, the nonlinearity of the amplifier will cause different frequencies to combine and produce different frequency components, which are called intermodulation distortion products.

How does intermodulation distortion occur?

The output signal of the nonlinear circuit is expanded in Taylor series as follows:

For simplicity, let us consider the input signal to be a two-tone signal with equal amplitude. Let the input excitation signal be

Substituting into the above formula, we can get

After expanding the above equation, we find that v _out (t) contains the following three types of frequency components:

(1) Fundamental wave and harmonics of $\omega$ ₁ and $\omega$ _{2 ;}

(2) The combination frequency of $\omega$ ₁ and $\omega$ ₂ : m $\omega$ ₁ ± n $\omega$ ₂ (m, n are positive integers);

(3) DC component.

The second frequency component mentioned above is the intermodulation distortion product. The sum of m and n determines the order of the intermodulation product. For example, intermodulation distortion within the 4th order includes:

3 $\omega$ ₁ ± $\omega$ ₂ : m=3, n=1, fourth-order intermodulation product;
3 $\omega$ ₂ ± $\omega$ ₁ : m=1, n=3, fourth-order intermodulation product;
2 $\omega$ ₂ ± 2 $\omega$ ₁ : m=2, n=2, fourth-order intermodulation product;
2 $\omega$ ₁ ± $\omega$ ₂ : m=2, n=1, third-order intermodulation product;
2 $\omega$ ₂ ± $\omega$ ₁ : m=1, n=2, third-order intermodulation product;
$\omega$ ₂ ± $\omega$ ₁ : m=1, n=1, second-order intermodulation product;

Among the many nonlinear distortion items, the difference frequency third-order intermodulation products are closest to the baseband signal in terms of frequency spectrum: (2 $\omega$ ₁ - $\omega$ ₂ ) and (2 $\omega$ ₂ - $\omega$ ₁ ). In broadband communication systems, they are most likely to interfere with the signal itself and adjacent channels. Moreover, among the intermodulation products, the amplitude of third-order intermodulation is relatively strong. Therefore, third-order intermodulation is the most concerned distortion item. The intermodulation distortion parameters of active devices usually given basically refer to third-order intermodulation distortion.

So does third-order intermodulation distortion only come from the third-order terms in the Taylor series expansion? In fact, in addition to the third-order terms, fifth-order, seventh-order and other odd-numbered higher-order terms can also be generated, but the higher the order, the less contribution.

To facilitate quantitative analysis, the following table gives the coefficients of the fundamental frequency and third-order intermodulation distortion within the fifth-order terms after Taylor series expansion.

Table 1. Fundamental and third-order intermodulation distortion coefficients (within the fifth order)

coefficient	cos(2 $\omega$ ₁- $\omega$ ₂)t	cos $\omega$ ₁t	cos $\omega$ ₂t	cos(2 $\omega$ ₂- $\omega$ ₁)t
(cos $\omega$ ₁t+ cos $\omega$ ₂t)¹	0	1	1	0
(cos $\omega$ ₁t+ cos $\omega$ ₂t)²	0	0	0	0
(cos $\omega$ ₁t+ cos $\omega$ ₂t)³	3/4	9/4	9/4	3/4
(cos $\omega$ ₁t+ cos $\omega$ ₂t)⁴	0	0	0	0
(cos $\omega$ ₁t+ cos $\omega$ ₂t)⁵	25/8	25/4	25/4	25/8

The fundamental frequency and third-order intermodulation distortion components can be written as

In the formula, "∑ higher-order terms" refers to the component contributed by higher-order terms above order 5. The higher the order, the smaller the constant coefficient c _i . For ease of analysis, the higher-order terms can be ignored.

The following takes two frequency signals $\omega$ ₁ and (2 $\omega$ ₁ - $\omega$ ₂ ) as examples to discuss the relationship between their output power and input power.

The input baseband signal power is

The output baseband signal power is

The logarithm is expressed as

It can be seen from the formula that when the input signal is small, the output power of the baseband signal and the input power show an approximately linear relationship.

The third-order intermodulation distortion power is

The logarithm is expressed as

In the logarithmic coordinate system, the following conclusions can be drawn from the above formula:

(1) Whether it is the baseband signal or the third-order intermodulation distortion, the power at the amplifier output side does not change linearly with the input power;

(2) When the input signal power is relatively low, c ₃ V ₀² →0, c ₅ V ₀² →0, c ₅ V ₀⁴ →0, at this time, the output power of the baseband signal and the third-order intermodulation distortion presents an approximately linear relationship with the input power. This is very important because the derivation of the third-order intermodulation point (IP3) power later needs to be based on this approximate linear relationship;

(3) In the approximately linear region, as the input power increases, the power of the third-order intermodulation distortion increases faster than the power of the fundamental frequency component. The former increases three times faster than the latter. In the logarithmic coordinate system of input and output power, the slope of the fundamental frequency power curve is 1, while the slope of the third-order intermodulation power curve is 3, as shown in Figure 2.

(4) In the approximate linear region, it can be seen from the mathematical expression that when the input power is low (usually much less than 0 dBm), the power of the third-order intermodulation product is much smaller than the power of the baseband signal;

(5) As the input power increases further, the nonlinearity of the output power curves of the fundamental frequency and third-order intermodulation distortion becomes more and more obvious, gradually presenting a compression state.

Figure 2. Power output characteristics of the fundamental frequency and third-order intermodulation distortion products caused by nonlinearity.

The third-order intermodulation distortion caused by nonlinearity is usually measured using two parameters: "third-order intermodulation distortion (IMD3 ⁾ " and "third-order intercept point (IP3 ⁾ ". The latter is actually the input or output power corresponding to the third-order intercept point.

In the power output characteristic curve shown in Figure 2, when the input power is low, the power curves of the fundamental frequency and the third-order intermodulation distortion both show an approximately linear relationship. Due to the different slopes, the linear extension of the two will inevitably have an intersection, which is the third-order intermodulation point IP3. Of course, it is impossible to achieve the output power corresponding to IP3 in practice. The introduction of IP3 is only to measure the nonlinear characteristics of semiconductor devices in a unified way when dual-tone or multi-tone signals are input.

How is third-order intermodulation distortion IMD3 defined?

The power ratio of the third-order intermodulation product to the fundamental frequency product is defined as IMD3, which is expressed logarithmically as

Further simplifying

IMD3(dB)=2P_in+Res.

In the formula, the remainder Res. is expressed as follows

In the approximate linear region, the above remainder can be considered as a constant, which means: for every 1dB increase in input power, IMD3 will deteriorate by 2dB; conversely, for every 1dB decrease in input power, IMD3 will improve by 2dB. If it exceeds the approximate linear region, this relationship is no longer satisfied!

What is the relationship between IMD3 and IP3 power?

As mentioned above, IP3 refers to the intersection of the two straight line extensions. If you want to determine this point, you need to perform calculations based on the two straight lines. The formulas for the two straight lines can be written as

The intersection of the two means that the output power of the two signals is the same. Assuming that the input and output powers corresponding to IP3 are IIP3 and OIP3 respectively, substituting them into the above formula is

Subtracting the two gives

In the linear region, the third-order intermodulation distortion IMD3 is

Combining the above two formulas, we can get

Where G is the linear gain of the amplifier.

The above formula is an important basis for calculating IP3 power, but there is a big premise: IMD3 must be tested in the approximate linear region, otherwise the above formula for calculating IP3 power is not valid!

3. How to test third-order intermodulation distortion and intermodulation point power?

The IMD3 and IP3 tests are not difficult, but there are some points that require attention during the test. If not handled properly, the accuracy of the test results will be affected.

The third-order intermodulation test requires that the DUT be fed with equal-amplitude dual-tone signals. The dual-tone spacing should be set according to the DUT test requirements. Usually, it is necessary to select the appropriate dual-tone points and frequency spacing according to the actual usage scenario. For the IMD3 test, the dual-tone amplitude can be large or small, but if IP3 is to be tested, as described in the previous section, the amplitude cannot be too large, and the DUT must be ensured to work in the approximate linear region.

Figure 3. Connection diagram for testing third-order intermodulation distortion using two signal sources.

During the test, two signal sources can be used to provide dual-tone signals, which is the most commonly used method for third-order intermodulation testing and can provide relatively pure dual-tone signals. Alternatively, a vector source can be used to edit the waveform file on the baseband side so that a dual-tone signal can be output from a single channel. The signal generated by this method will have certain third-order intermodulation distortion, so it is only used as an alternative solution when two signal sources are not available.

Figure 4. Connection diagram for testing third-order intermodulation distortion using a single vector source.

Figures 3 and 4 show the connection diagrams of the third-order intermodulation test when two dual-tone generation modes are used. The entire test is relatively simple. Use a spectrum analyzer to test the spectrum of the amplifier output, set the appropriate reference level, center frequency, span and RBW, etc., display the spectrum of the baseband and third-order intermodulation signal, and use the Marker function to calibrate IMD3, and calculate the power value of IP3. Currently, most mid-to-high-end spectrum analyzers on the market have direct IMD3 and IP3 test functions, which makes testing more convenient.

The following introduces the points that require special attention during testing from the two aspects of spectrum analyzer and signal source.

(1) Pay special attention to the spectrum analyzer side. Do not allow the spectrum analyzer to enter nonlinearity and produce strong third-order intermodulation distortion during testing. The spectrum analyzer will definitely produce intermodulation distortion during testing, but it should not be too strong, otherwise it will disrupt the test.

Judgment method: Increase the front-end attenuator inside the spectrum analyzer. If the third-order intermodulation component does not change much, the impact of the intermodulation distortion generated by the spectrum analyzer can be ignored. If the third-order intermodulation component becomes smaller, it means that the attenuation needs to be further increased until the third-order intermodulation component does not change much. However, using the attenuator will reduce the test dynamic range of IMD3. If necessary, consider using a notch filter to attenuate the baseband signal to prevent the spectrum analyzer from generating strong intermodulation distortion.

If you are testing the third-order intermodulation distortion of a PA, be sure to use an attenuator with the appropriate power capacity before feeding it into the spectrum analyzer to ensure that it does not damage the spectrum analyzer. If you want to achieve a higher test dynamic, you need to use a notch filter to attenuate the baseband signal.

(2) There are two main points to note on the signal source side, one of which is the dual-tone signal amplitude.

If testing IMD3, there is no high requirement for the dual-tone amplitude, but the IP3 test requires that the input signal amplitude cannot be too high. To ensure that the amplifier works in a nearly linear area, it is recommended that the dual-tone signal amplitude be lower than the 1dB gain compression point input power P _{in, 1dB} at least 20dB. Whether testing IMD3 or IP3, be sure to indicate the dual-tone spacing and amplitude when recording the test results!

Judgment method: If the input power increases by 1dB and the IMD3 deteriorates by 2dB, it means that the amplifier is still operating in the approximately linear region and the IP3 can be calculated.

Another point to note is that even with the test setup shown in Figure 3, third-order intermodulation products may already exist on the combiner output side. The specific reason is related to the automatic power control loop of the signal source, which will be introduced in detail later. In short, due to the limited port isolation of the combiner, the signal is reversely connected to the signal source, and then through the ALC loop, the signal source itself outputs dual-tone and intermodulation distortion signals.

It is recommended that you use a spectrum analyzer to test the dual-tone signal before testing to observe whether there is strong third-order intermodulation distortion.

How to reduce the impact of this situation on testing?

Most signal sources support manually turning off the ALC function of the signal source, which can effectively avoid this situation. However, turning off the ALC function will also reduce the stability of the output power.

Either use a high-isolation coupler as a combiner, or connect an attenuator to the output of each signal source to increase the isolation between them.

bmahu001 · Published on 2022-10-21 14:37

Some signal sources support manually turning off the ALC function of the signal source, which can effectively avoid this situation. However, turning off the ALC function will also reduce the stability of the output power.

btty038 · Published on 2022-10-22 20:14

bmahu001 posted on 2022-10-21 14:37 Some signal sources support manually turning off the ALC function of the signal source, which can effectively avoid this situation. However, after turning off the ALC function, it will also reduce...

I tested it last time and it didn't seem to make the above difference. I guess the signal source is too old.

Agilent E4423】

Detailed explanation of third-order intermodulation distortion and testing [Copy link]

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