As a increases, the linear term increases as a¹. The second-order IMD term

increases as a² and the third-order term increases as a³. When these coefficients are plotted as a function of a, on a log-log plane, the exponent simply becomes the slope of each of the three lines, as shown in Fig. 3. Note that the nth-order term increases at a rate n times that of the linear term.

In reality, an amplifier's output doesn't increase without bound as Fig. 3 suggests. Rather, the amplifier's output can be limited by, among other factors, its power supply rails. The finite power supply can force the transfer function to limit or compress as the input amplitude increases (Fig. 4). This compression, modeled by the higher-order terms in the power series, arises as a result of higher-order terms becoming a significant portion of the output.

It is common to identify the point where the actual output deviates from the ideal output by 1 dB. This point is known as the 1-dB compression point, represented as P_1dB.

For convenience, a line 1 dB below the ideal output is shown in Fig. 4.

Parameter P_1dB can be estimated when the third order term dominates. Such is the case in differential amplifiers where the transfer function exhibits odd-symmetry. It is simple to find the input amplitude where the distorted output is 1 dB less than the ideal, undistorted output:

The intersections of the second-order and third-order lines with the line produced by the linear term are known as the intercept points (IP). The second-order intercept point is called IP2 and the third-order intercept point is called IP3. The intercept points are further defined with respect to the input amplitude or the output amplitude where the intersections occur. For example, the input IP3 is represented as IIP3, whereas the output IP3 is represented as OIP3.

The intercept points are defined as if the linear term and higher-order terms increased without bound. They provide a means of quantifying the linearity of a two-port, and as a result, facilitate comparison with other two-ports. The concept of the IP is a powerful and versatile tool and is used very frequently. The coefficients {c₂, c₃, …, c_n} have the effect of shifting their respective lines up or down (in a log-log plane). As c₂ and c₃ approach zero, their lines shift down, intuitively causing their respective intercept points to increase.

As was mentioned above, the intercept points occur where the coefficients of the IMD terms are at parity with the coefficient of the linear term. Thus, if the linear coefficient is equated with the second order IMD term, and solving for a, IIP2 can be found with Eq. 14:

Similarly, IIP3 can be found with Eq. 15:

In practice, c₂ and c₃ are unknown. Moreover, they are difficult to measure directly. Fortunately, another method can be used to measure IP2 and IP3.

As was seen earlier, the slopes of the linear, second-order, and third-order lines are m₁ = 1, m₂ = 2 and m₃ = 3, respectively. Referring to Fig. 5, equations that describe the two lines can be written as:

The lines intersect according to the conditions of Eq. 18:

Solving the system of equations for P_out and noting that m_n = n, results in Eq. 19:

It is possible to use a spectrum analyzer, in conjunction with Eq. 19, to measure OIP2 and OIP3. For a given P_in1, the output amplitude of the fundamental (Pout1) and the IMD term (P_outn) are measured. Simple substitution of the measured values into Eq. 19 yields OIPn. This measurement is routinely performed and should be made such that the amplifier's fundamental output is not strongly influenced by the higher-order terms. That is, it should be made while the input amplitude is well below P_1dB. Otherwise, P_out will contain an error component.

To point out one final interesting relationship, refer to Eqs. 13 and 15. It can be seen that IIP3 and P_1dB are related to one another as:

or in decibels as:

This is true, of course, as long as the second-order term is insignificant. In practice, this relationship is commonly observed for a wide variety of amplifier topologies.

In summary, the concepts of linearity and intercept points were reviewed, and it was shown that a spectrum analyzer could be used to measure intercept points. Problems caused by linearity can arise in all parts of a system. By measuring, analyzing, understanding, and optimizing a system's linearity, it is possible to dramatically improve its performance. By understanding the types of problems that nonlinearities can create, and how to overcome them, the designer is better equipped to achieve success.

For further reading
Behzad Razavi, Design of Analog CMOS Integrated Circuits, McGraw Hill, 2001.

David M. Pozar, Microwave Engineering, Addison Wesley, 1990.

Jarek Lucek and Robbin Damen, "LNA Design for CDMA Front End," RF Design, February, 1999.

Ken Kundert, "Accurate and Rapid Measurement of IP2 and IP3," www.designersguide.org.