Unveiling the Power of Non-Linearity in RF Circuits: A Comprehensive Exploration

Introduction  

In the previous blog, we have seen how the polynomial representation predicts the non-linearities generating the undesired frequencies that are harmonics and different intermodulation products IMn. 

Some of them are concerning, especially the terms from the third order since they appear in the useful data band; causing data loss or distortion. In this blog post, we will see how IMn can be quantified via the intercept points IPn.

Intermodulation Product (IM) to Intercept Point (IP)

Now that we understand the origins of IM products, and particularly IM3, we are better prepared to determine their values and measure them with a common method and units. 

Please note: IMn are the intermodulation products, while IPn is the actual measures. 

The discussions on previous blog posts showed that the terms for i > 1 in the transfer function coefficients Ai quantify the device nonlinearity.

The larger they are, the greater the distortion. Thus we focus our attention on those values A2, A3, ... Ai...An in the polynomial: 

y = A₀ + A₁.x¹ + A₂.x² + A₃.x³ +... + A_i.xⁱ +... A_n.xⁿ  - (Eq. 14)  

And let’s try to display y(t) versus x(t) term by term:

Figure 5. The individual behavior of the polynomial terms 

Let’s now draw them in a log-log axis: they will appear all as straight lines with slopes at 0, 1:1, 2:1, 3:1, etc…

Vertical axis y and horizontal axis x will then be scaled as dB values:

Figure 6. The individual behavior of the polynomial terms in the log axes 

By superposing all the above pieces, we get various straight lines at different slopes and cross one to another.  

Figure 7. Combined behaviors  

Intercept Points 

Let’s analyze more in detail the above set of straight lines.

Figure 8. Intercept Points 

From Figure 8, we find that: 

• The term A0 is a constant value (offset) and independent of the value of x. 
• The term A1 x is the linear portion; in a double-log scales graph, y-x is a straight line with an offset defined by A1 and the slope is just 1dB/dB (doubling x, results in doubling y). 
• The term A2 x² is the quadratic term (second-order); it has an offset determined by A2 and a slope that is exactly twice the previous slope (2dB/dB); or, restated, doubling the input x will result in quadrupling y. 
• The term A3 x3 is the third-order part; it is a straight line in the graph y-x with offset determined by A3; the slope is exactly three times sharper than for the linear term (3dB/dB); or, restated, doubling x will result in multiplying x by 8. 
• This logic is applied to all the following terms and the nth-order line will have a slope of ndB/dB. 

Since the higher-order terms have lines with a sharper slope, soon or later there will be a moment (a point actually) where the high-order line will cross the first-order line. The crossing points are called intercept points (IPn). 

One can easily observe that the more a device is linear, the more the first-order line is high in the graph (compared to the other lines). Therefore, a higher value is reached for IP points. Graphically, this is easy to see (Figure 8). The slope is fixed, so when the device is strongly linear, the nth-order terms will be very small. (The An lines start from deeper values and, hence, will cross the first-order line much later (far away in the axes).

Intercept Point of the Third Order or IP3 

The 3rd order line keeps our special attention since IM3 products are the most dangerous ones, causing data loss or damage. 

Let’s examine the crossing point between the linear line (wanted) and the slope3 line (which quantifies the Intermodulation Products of order 3. At that point (input x, Output y), the useful data amplitude equals the 3d order parasitic. Since order n lines progress quicker than the linear line (order 1), there is always, soon or later, a power where the parasitic will become larger. A good linear device is then characterized by those crossing points as high as possible. 

IP3 is just defined as the level of input/output power where the third order crosses the first order line. You can look at that power at the input level: it is then the IIP3 (input Intercept Point of order 3) or at the output power level: it is then the OIP3. 

Figure 9. IPP3 and OIP3  

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Summary 

In this blog post, one has learned how the different intermodulation products are visualized and quantified via their intercept points with linear slope: the IPn. Each IPi, in reverse mode, gives the quantity of IMi present in the non-linear device. As noted before, IP3 is the most looked one by the engineers.  

In the next blog, we will see how IPn can be evaluated and measured in the lab.