Understanding P1dB and TOI: Formulas and Applications

Understanding the 1dB compression point (P1B) and the third-order intercept point (TOI) is essential in RF and microwave circuit design. These parameters define the linearity and distortion behavior of amplifiers and mixers, directly impacting system performance. This guide explains the formulas, derivations, and practical significance of P1B and TOI with real-world examples.

TOI: Third Order Intercept Point

As mentioned, TOI refers to the Third Order Intercept Point. As we know from our discussion on intermodulation distortion, a saturated or overdriven amplifier produces unwanted products of input signals. This is known as IMD (Intermodulation Distortion).

When two or more signals are fed to a nonlinear device (whether passive or active), many combinations of input signals are produced at the output. These contain first-order, second-order, and third-order products. Out of these intermodulation distortion (IMD) products, only third-order products are vulnerable to interference due to their higher amplitude and close proximity to the desired signals. For input signals $f_1$ and $f_2$, output third-order products, namely $2f_1 - f_2$ as well as $2f_2 - f_1$, lead to IMD interference.

Each 1dB increase in P1dB results in a 3dB increase in 3rd order IM (Intermodulation) power. The TOI point is a figure of merit, which characterizes an RF receiver’s tolerance subjected to multiple RF signals outside the passband of interest. The TOI points on the X-axis and Y-axis are designated as IIP3 and OIP3, respectively. Refer to IIP3 versus OIP3, which describes the difference between IIP3 and OIP3.

The figure depicts an IP3 measurement setup.

OIP3 = $P_{OUT}$ + $\Delta P$/2

Where:

$\Delta P$ = $P_{OUT}$ - $P_{OUT\_3}$

Here, $P_{OUT}$ refers to the power of fundamental signals when they are equal in magnitude, and $P_{OUT\_3}$ refers to the power of third-order intermodulation products when they are equal in magnitude.

Refer to the IIP3 calculator, which calculates IIP3 and OIP3 when the powers of both fundamental signals are equal and when the powers of IMD products are also equal.

Refer to the IM3 calculator, which calculates third-order intercept points at the input (i.e., IIP3) and at the output (i.e., OIP3) using the IM3 formula mentioned above.

P1dB: 1dB Compression Point

P1dB or the 1dB compression point is the part of the curve where a 2dB change in the input power fed to the amplifier (RF circuit) results in a 1 dB change in the output power. The input and output powers are called $P1B_{in}$ and $P1dB_{out}$, respectively.

In reality, the Intercept point is usually about 10 dB higher in value compared to P1dB (1dB compression point). Both IIP3 and OIP3 are results of third-order intermodulation distortion products.

The RF circuit does not function correctly or amplify the input signal properly at the IIP3 input level. IIP3 mentioned above is the 3rd order intercept point at the input of the amplifier for OIP3 (Output third Order Intercept point).

The following equation mentions the relation between IIP3 and OIP3 with respect to gain:

Gain + IIP3 = OIP3

The formula is useful to characterize the linearity of an RF receiver for signals below the saturation level.

Conclusion

P1B and TOI are fundamental concepts in assessing the linearity and dynamic range of RF devices. By mastering these formulas and their practical applications, engineers can design and optimize circuits for superior performance in communication systems.