Second Order Intercept Point (IP2)

Second order intercept point(IP2) is a used to quantify weak non-linearity in an amplifier or RF system.
It is an interpolated point where the fundamental component power curve meets the second order intermod(IMD2) component power curve.

Consider a nonlinear amplifier whose input-output characteristic is represented by a 3rd order polynomial given by

(1)   \begin{equation*} v_o(t) = \alpha_o + \alpha_{\tiny 1} v_{i}(t) + \alpha_{\tiny 2} v_{i}^{\tiny 2}(t) + \alpha_3 v_{i}^3(t) \end{equation*}

where, v_o(t) is the output voltage of the amplifier
and v_i(t) is the input voltage applied to the amplifier

Generally a two tone test is used to compute IP3 for an amplifier.

(2)   \begin{equation*} v_{i} = A_{i,1} \cos(\varphi_{1}) + A_{i,2} \cos(\varphi_{2}) \end{equation*}

\varphi_{\tiny 1}=\omega_{\tiny 1}t+\phi_{\tiny 1} and
\varphi_{\tiny 2}=\omega_{\tiny 2}t+\phi_{\tiny 2}.

For two tone input, the output of the amplifier is given by substituting Eq.(2) in Eq.(1)

(3)   \begin{eqnarray*} v_o &=& \underbrace{\left(\alpha_{1}+ \alpha_{3}\left(\frac{3}{4}A_{i,1}^{2} + \frac{3}{2}A_{i,2}^{\tiny 2} \right) \right)A_{i,1}\cos{\varphi_{\tiny 1}} + \left(\alpha_{\tiny 1}+ \alpha_{\tiny 3}\left(\frac{3}{4}A_{i,2}^{\tiny 2} + \frac{3}{2}A_{i,1}^{\tiny 2}\right) \right) A_{i,2} \cos{\varphi_{\tiny 2}}}_{fundamental} \\ & &+\underbrace{\frac{1}{2}\alpha_{\tiny 2} \left(A_{i,1}^{\tiny 2}\cos{2\varphi_{\tiny 1}}+ A_{i,2}^{\tiny 2} \cos{2\varphi_{\tiny 2}} \right)}_{2^{nd}~~harmonic }\\ & &+\underbrace{\frac{1}{4}\alpha_{\tiny 3} \left(A_{i,1}^{\tiny 3} \cos(3\varphi_{\tiny 1}) + A_{i,2}^{\tiny 3} \cos(3\varphi_{\tiny 2})\right)}_{3^{rd}harmonic}\\ & &+\underbrace{\frac{1}{2}\alpha_{\tiny 2}A_{i,1}A_{i,2}\left(\cos(\varphi_{\tiny 1}-\varphi_{\tiny 2}) + \cos(\varphi_{\tiny 1}+\varphi_{\tiny 2}) \right)}_{2^{nd}~~order~~intermods}\\ & &+\underbrace{\frac{3}{4}\alpha_{\tiny 3}\left(A_{i,1}A_{i,2}^{\tiny 2}\cos(\varphi_{\tiny 1}-2\varphi_{\tiny 2}) + A_{i,1}^{\tiny 2}A_{i,2}\cos(2\varphi_{\tiny 1}-\varphi_{\tiny 2})+ A_{i,1}A_{i,2}^{\tiny 2}\cos(\varphi_{\tiny 1}+2\varphi_{\tiny 2}) + A_{i,1}^{\tiny 2}A_{i,2}\cos(2\varphi_{\tiny 1}+\varphi_{\tiny 2})\right)}_{3^{rd}~~order~~intermods} \end{eqnarray*}

From Eq.(3), the output contain fundamental, harmonics, second order intermod and third order intermod components. The second order intermod(IMD2) components are generated at frequencies \omega_1-\omega_2 and \omega_1+\omega_2 . If \omega_1 and \omega_2 are close-by, \omega_1-\omega_2 component falls in the signal band and \omega_1 +\omega_2 falls out of signal band.

At second order intercept point, fundamental component is equal to second order intermod component.
From Eq.(3),

(4)   \begin{equation*} \left(\alpha_1+\alpha_{3}\left(\frac{3}{4}A_{i,1}^{2}\right)\right) A_{i,1} = {1 \over 2}\alpha_{2}A_{i,1}A_{i,2} \end{equation*}

At low power levels, \alpha_1 \gg \alpha_{3}\left(\frac{3}{4}A_{i,1}^{2}\right). So only \alpha_1 is extrapolated to IP2 point.

(5)   \begin{equation*} \alpha_1 A_{i,1} \approx {1 \over 2}\alpha_{2}A_{i,1}A_{i,2} \quad\quad  \rightarrow \quad\quad  \boxed{ A_{IP2} = 2 {\alpha_1 \over \alpha_2} } \end{equation*}

In the above equation A_{IP2} is the amplitude of input tone corresponding to IIP2 point. We can write this in power to get IIP2 point. Here we computed IIP2 point from system parameters. Rather we can also compute IIP2 by measuring output power level of intermod and signal as follows:

From Eq.(3), the ratio of fundamental component to second order intermod component in output voltage is

(6)   \begin{equation*} { v_{o,1} \over v_{o,IMD2} } = { \alpha_1 A_{i,1}^{2}   \over {1 \over 2}\alpha_2 A_{i,1}A_{i,2} } = {  A_{IP2} \over A_{i,2} } \end{equation*}

Taking log on both sides, and rearranging

(7)   \begin{eqnarray*} 20 \log(v_{o,1}) - 20 \log(v_{o,IMD2}) &=& 20 \log\left( {A_{IP2} \over A_{i,2}} \right)  \\ 20 \log( {A_{IP2}) &=& 20 \log(A_{i,2}) +  \left[ 20 \log(v_{o,1}) - 20 \log(v_{o,IMD2}) \right] \end{eqnarray*}

Writing the equation in terms of power,

(8)   \begin{eqnarray*} IIP2 &=& P_{i,2} + \left[ P_{o,1} - P_{o,IMD2} \right] \\ &=& P_{i,2} + \Delta P_{IMD2} \\ \end{eqnarray*}

where \Delta P_{IMD2} is the IMD2 power relative to fundamental component power. In an two tone test P_{i,1} and P_{i,2} are of equal power. So P_{i,1} = P_{i,2} = P_{i}

(9)   \begin{equation*} \boxed{IIP2 &=& P_{i} + \Delta P_{IMD2} } \end{equation*}

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