"Determination of input stage PMOS $g_m$ and $c_{iss}$"

Determination of input stage PMOS $g_m$ and $c_{iss}$

Noise model for determination of $g_m$ and $c_{iss}$

Determination of the output noise

The budget for the DIN A weighted RMS noise equals $10.5\,\mu V$.

Determination of the lowest possible unweighted RMS output noise

The output noise spectral density in V$^2$/Hz is obtained as:

\begin{equation} S_{vo}=\frac{1.638 \cdot 10^{-7}}{\pi^{2} f^{2}} + \frac{8.48 \cdot 10^{-56} \cdot \left(1.344 \cdot 10^{66} \pi^{2} f^{2} + 1.786 \cdot 10^{73}\right)}{\pi^{2} f^{2} \cdot \left(9.216 \cdot 10^{17} \pi^{2} f^{2} + 1.135 \cdot 10^{27}\right)} + \frac{1.306 \cdot 10^{19}}{\pi^{2} f^{2} \cdot \left(9.216 \cdot 10^{17} \pi^{2} f^{2} + 1.135 \cdot 10^{27}\right)} + \frac{1.6 \cdot 10^{-53} \cdot \left(7.845 \cdot 10^{68} \pi^{4} c_{iss}^{2} f^{4} + 3.649 \cdot 10^{77} \pi^{2} c_{iss}^{2} f^{2} - 6.047 \cdot 10^{68} \pi^{2} c_{iss} f^{2} + 6.37 \cdot 10^{59} \pi^{2} f^{2} + 7.844 \cdot 10^{68}\right)}{\pi^{2} f^{2} g_{m} \left(9.216 \cdot 10^{17} \pi^{2} f^{2} + 1.135 \cdot 10^{27}\right)} + \frac{7.741 \cdot 10^{-16}}{\pi^{3} c_{iss} f^{3}} \end{equation}

Notice that this spectrum decreases with increasing $g_m$. Hence. increase of the inversion coefficient at a given $c_{iss}$ results in a lower noise.

The unweighted squared RMS output noise $V_{onoise}^2$ over the frequency range of interest is found as:

\begin{equation} V_{onoise}^{2}=\frac{33.51 c_{iss}^{2}}{g_{m}} - \frac{4.438 \cdot 10^{-8} c_{iss}}{g_{m}} + 5.716 \cdot 10^{-11} + \frac{3.548 \cdot 10^{-15}}{g_{m}} + \frac{1.384 \cdot 10^{-22}}{c_{iss}} \end{equation}

If we substitute $g_m=2\pi f_T c_{iss}$, and take the maximum $f_T=f_{T_{peak}}\approx 10\cdot 10^9$ Hz for CMOS18 PMOS, we can calculate the minimum noise that can be obtained in this technology and for this application.

To this end we calculate the optimum value $c_{iss_{opt}}$ for $c_{iss}$ from $\frac{d}{d\,c_{iss}}V_{onoise}^2 = 0$.

Substitution of $c_{iss_{opt}}$ in the expression for $V_{onoise}^2$ yields the minimum unweighted noise$V_{onoise_{min}}$; we find: $V_{onoise_{min}}=7.56 \cdot 10^{-6}$. This is much lower than required.

Determination $g_m$ and $c_{iss}$ for the best performance-to-cost ratio

The budget for the DIN A weighted squared noise equals $1.1 \cdot 10^{-10}\, V^2$.

As a result of the DIN A weighting this budget for the unweighted squeared noise be factor $1.64$ more, which equals: $2.87 \cdot 10^{-10}\, V^2$.

This factor has been determined after a few iterations of the procedure below, of which the results of the last iteration are shown.

The unweighted squared RMS output noise over the frequency range of interest was found as $V_{onoise}^2$.

We obtain valid combinations of $g_m$ and $c_{iss}$ by solving $g_m$ from:

\begin{equation} V_{onoise}^{2}=2.869 \cdot 10^{-10} \end{equation}

This yields:

\begin{equation} g_{m}=\frac{1.047 \cdot 10^{20} c_{iss} \left(5.0 \cdot 10^{29} c_{iss}^{2} - 6.622 \cdot 10^{20} c_{iss} + 5.294 \cdot 10^{13}\right)}{3.589 \cdot 10^{38} c_{iss} - 2.162 \cdot 10^{26}} \end{equation}

A positive value for $g_m$ is obtained if both the numerator and the denominator of this expression are positive.

This occurs if $c_{iss} > 6.023 \cdot 10^{-13}$.

A minimum transconductance is required if:

\begin{equation} \frac{dg_{m}}{dc_{iss}}=0 \end{equation}

This minimum $g_{mmin}$ is found at $c_{iss}=7.23 \cdot 10^{-10}$:

\begin{equation} g_{m min}=1.54 \cdot 10^{-5} \end{equation}

The product $g_m c_{iss}$ can be taken as a measure for the $costs$. The input capacitance for minimum $costs$ is obtained from:

\begin{equation} \frac{dcosts}{dc_{iss}}=0 \end{equation}

This yields: $c_{iss}=1.2$ pF and $g_m > 30.9\,\mu $S.

Design verification

The noise spectrum at the output of the receive coil amplifier is shown below.

The DIN A weighted RMS output noise over the frequency range of interest amounts $13.1\,\mu$V.

Go to noiseCissGm_p_index

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Last project update: 2024-01-07 12:49:39