# IP3 Intermodulation Calculator Description

#### Background

Various classes of RF devices, such as amplifiers, mixer and switches, are inherently nonlinear. Even if operated in the small-signal/linear regime, a signal passing through them gets distorted. In the frequency domain the distortion appears in the form of harmonics and intermodulations. Harmonics occur at multiples of the carrier frequency, and can be easily removed by using a low pass filter. On the other hand, intermodulations products can fall in the passband where they cannot be filtered out, reducing the channel’s dynamic range.

The most prominent passband intermodulation originates from the third order nonlinearity. This nonlinearity is typically specified by a measure called the third-order intercept point (IP3). Graphically, the IP3 is measured by feeding the device with a two-tone signal, than plotting on a log-log scale the fundamental output power and the third order intermodulation distortion products power (IMD3 or IM3), as a function of the input power. These curves, shown in the left figure, are linear for small input power (black lines). Extrapolating the linear part of the curves (dashed lines), the IP3 is the point where the two lines meet. The input power and output power at this point are called input IP3 (IIP3) and output IP3 (OIP3).

#### Equations

The relation between the IP3 and IM3 is derived from the Taylor expansion of the device’s output voltage $$V_{out}$$ as a function of its input voltage $$V_{in}$$. Under a small signal approximation taken to third order, $$V_{out}$$ is expressed by $$V_{out} = C_{0} + C_{1}V_{in} + C_{2}V_{in}^{2} + C_{3}V_{in}^{3} + ...$$ where $$C_{0}$$ is the output DC voltage, $$C_{1}$$ is the small signal voltage gain, $$C_{2}$$ is the second order distortion coefficient, and $$C_{3}$$ is the third order distortion coefficient.

For an input signal consisting of two-tones of unequal power, $$V_{in}$$ is given by $$V_{in} = v_{low}sin(2 \pi f_{low} t) + v_{hi}sin(2 \pi f_{hi} t)$$ where $$v_{low}$$ and $$v_{hi}$$ are the amplitudes of the input tones, and $$f_{low}$$ and $$f_{hi}$$ are their frequencies.

By combining both of the above equations and rearranging them as a sum of sinusoidal terms, we get the small-signal two-tone transfer function $$V_{out} = C_{1}v_{low}sin(2 \pi f_{low} t) + C_{1}v_{hi}sin(2 \pi f_{hi} t)\\ + \frac{3}{4}C_{3}v_{low}^{2}v_{hi}sin[2 \pi (2f_{low} - f_{hi}) t]\\ + \frac{3}{4}C_{3}v_{low}v_{hi}^{2}sin[2 \pi (2f_{hi} - f_{low}) t]\\ + ...$$ the first two terms are the fundamental signal amplified by $$C_{1}$$, while the next two terms are the IM3. All other terms, not explicitly shown in the equation, include higher order IM and harmonics, which are either small or fall out of band.

For equal power two-tones ($$v_{low}=v_{hi}$$), the OIP3 equals to the fundamental and IM3 power at the intercept point $$OIP3 = \frac{(C_{1}v_{low})^{2}}{Z_{L}} = \frac{(\frac{3}{4}C_{3}v_{low}^{3})^{2}}{Z_{L}}$$ where $$Z_{L}$$ is the load impedance. Solving for OIP3 gives $$OIP3 = \frac{4C_{1}^{3}}{3Z_{L}C_{3}}$$

Using the above results, we get the relation between the OIP3 and IM3 for unequal two-tones $$IM3_{low} = \frac{(\frac{3}{4}C_{3}v_{low}^{2}v_{hi})^{2}}{Z_{L}} = \frac{C_{1}^{6}v_{low}^{4}v_{hi}^{2}}{OIP3^{2}Z_{L}^{3}}$$ $$IM3_{hi} = \frac{(\frac{3}{4}C_{3}v_{low}v_{hi}^{2})^{2}}{Z_{L}} = \frac{C_{1}^{6}v_{low}^{2}v_{hi}^{4}}{OIP3^{2}Z_{L}^{3}}$$ and in units of dBm $$IM3_{low}(dBm) = 2P_{low} + P_{hi} - 2OIP3$$ $$IM3_{hi}(dBm) = P_{low} + 2P_{hi} - 2OIP3$$ where $$P_{low}=\frac{(C_{1}v_{low})^{2}}{Z_{L}}$$ and $$P_{hi}=\frac{(C_{1}v_{hi})^{2}}{Z_{L}}$$ are the fundamental tones output power.