The Miller OTA

Analysis

Christian Enz

Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland

# Initialization¶

# Introduction¶

Schematic of the Miller OTA.

This notebook presents the design of the basic two-stage or Miller OTA shown in the above schematic. The gain of the simple OTA can be enhanced by adding a 2nd gain stage implemented by M2. Because of the higher gain, the OTA needs to be frequency compensated to ensure that it will remain stable in all feedback configurations. The compensation is achieved by adding capacitor $C_c$ which takes advantage of the Miller effect hence its name of Miller compensation or Miller OTA. The amplifier small-signal transfer function is analyzed next.

In this Notebook we will focus on minimizing the power consumption at the cost of a larger area compared to the minimum area Notebook.

# Analysis¶

## Small-signal analysis¶

Small-signal schematic of the Miller OTA.

We start with the small-signal analysis. The small-signal schematic corresponding to the basic two-stage OTA is shown in the above figure. Assuming perfect matching and differential operation $\Delta V_{Gb1} = -\Delta V_{Ga1} = V_{id}/2$, the voltage at node 4 remains unchanged and hence $\Delta V_4=0$. The source transconductances can then be omitted leading to the simplified small-signal circuit in differential mode shown below.

Small-signal schematic of the Miller OTA in differential mode.

If the capacitance $C_3$ at the current mirror node 3 is neglected, then $\Delta V_3 = -G_{m1}/G_{m3} \cdot V_{id}/2$. The two transconductances connected to node 1 can then be replaced by a single transconductance resulting in the simplified schematic shown below.

Simplified small-signal schematic of the two-stage OPAMP in differential mode.

The small-signal differential gain of the simplified circuit shown above is then given by \begin{equation*} A_{dm} \triangleq \frac{\Delta V_{out}}{\Delta V_{id}} = A_{dc} \cdot \frac{1-s/z_1}{(1-s/p_1)(1-s/p_2)} = A_{dc} \cdot \frac{1+n_1\,s}{1+d_1\,s+d_2\,s^2} = A_{dc} \cdot \frac{1-\frac{s}{z_1}}{1-\frac{s}{p_1+p_2}+\frac{s^2}{p_1\,p_2}} \end{equation*} with \begin{align*} A_{dc} &= \frac{G_{m1}}{G_1} \cdot \frac{G_{m2}}{G_2},\\ n_1 &= -\frac{1}{z_1} = -\frac{C_c}{G_{m2}},\\ d_1 &= -\left(\frac{1}{p_1}+\frac{1}{p_2}\right) = \frac{C_1}{G_1}+\frac{C_2}{G_2}+\frac{C_c}{G_1}\,\left(1+\frac{G_1}{G_2}+\frac{G_{m2}}{G_2}\right),\\ d_2 &= \frac{1}{p_1\,p_2} = \frac{C_c\,C_2+C_c\,C_1+C_1\,C_2}{G_1\,G_2}. \end{align*}

Note that $G_{m1}/G_1$ and $G_{m2}/G_2$ are the voltage gains of the first and second stage, respectively. $G_1$ and $G_2$ are the total conductances at nodes 1 and 2 \begin{align*} G_1 &= G_{ds1b} + G_{ds4b},\\ G_2 &= G_{ds2} + G_{ds5b}, \end{align*} whereas $C_1$ and $C_2$ are the total capacitances at nodes 1 and 2. Note that $C_2$ is usually dominated by $C_L$.

If we ignore the compensation capacitor ($C_c=0$), the zero disappears and the two poles are simply given by \begin{align*} p_1' &= -\frac{G_1}{C_1},\\ p_2' &= -\frac{G_2}{C_2}. \end{align*} We see that the poles are actually associated to the nodes 1 and 2 (output).

The compensation capacitor introduces a right half-plane (RHP) zero $z_1=G_{m2}/C_c$ and has two real poles $p_1$ and $p_2$. Assuming that $C_1 \ll C_2$ and that $G_1$ and $G_2$ are of the same order of magnitude, the poles can be considered as widely separated $|p_1| \ll |p_2|$ then \begin{equation*} d_1 \cong -\frac{1}{p_1} = \frac{C_1}{G_1}+\frac{C_2}{G_2}+\frac{C_c}{G_1} \cdot \left(1+\frac{G_1}{G_2}+\frac{G_{m2}}{G_2}\right). \end{equation*} We can further assume that $G_{m2}/G_2 \gg 1$ and the dominant pole $p_1$ is approximately given by \begin{equation*} p_1 \cong -\frac{G_1\,G_2}{G_{m2}\,C_c}. \end{equation*} The gain-bandwidth product $GBW$ is then approximately given by \begin{equation*} GBW = |p_1| \cdot A_{dc} \cong \frac{G_{m1}}{C_c}. \end{equation*} Note that $|p_2|$ must be at least equal to $GBW$ for the above approximation to hold \begin{equation*} GBW < \frac{G_{m2}}{C_2}. \end{equation*}

The mechanism of pole splitting.

The compensation process.

The non-dominant pole $p_2$ is then approximately given by \begin{equation*} p_2 \cong \frac{1}{p_1\,d_2} \cong -\frac{G_{m2} C_c}{C_c C_2 + C_c C_1 + C_1 C_2}. \end{equation*} We see that the dominant pole magnitude $|p_1|$ decreases as $C_c$ increases, whereas $|p_2|$ increases as $C_c$ increases. Thus, increasing $C_c$ causes the poles to split apart as illustrated in Fig. \ref{fig:pole_splitting}. If $C_2$ and $C_c$ can be considered much larger than $C_1$, the non-dominant pole is approximately set by the output capacitance \begin{equation*} p_2 \cong -\frac{G_{m2}}{C_2}. \end{equation*}

It can be shown that if the zero is ten times higher than the $GBW$, then in order to achieve a phase margin better than $60^{\circ}$, the second pole must be placed at least 2.2 times higher than the $GBW$. This translates into the following constraints \begin{align*} |z_1|> 10\,GBW\;\Rightarrow\;\frac{G_{m2}}{C_c}>10\,\frac{G_{m1}}{C_c}\;\Rightarrow\;G_{m2}>10\,G_{m1},\\ % |p_2|>2.2\,GBW\;\Rightarrow\;\frca{G_{m2}}{C_o}>2.2\,\frac{G_{m1}}{C_c}. \end{align*} which results in \begin{equation*} C_c > 2.2\,\frac{G_{m1}}{G_{m2}}\,C_2 > 0.22\,C_2. \end{equation*}

The dominant-pole is often called a Miller pole because it takes advantage of the Miller effect. The dominant-pole can actually be found by using the Miller approximation. Using the result obtained earlier without the compensation capacitor and replacing $C_1$ by the Miller capacitance \begin{equation*} C_M \cong \frac{G_{m2}}{G_2}\cdot C_2 \end{equation*} resulting in \begin{equation*} p_1 \cong -\frac{G_1 G_2}{G_{m2}\,C_c} \end{equation*} which is identical to the result found above. However, the Miller approximation does account for the RHP zero. The later introduces very undesirable effects with regards to stability: it increases the phase shift and at the same time increases the magnitude. The effects of the RHP zero can be mitigated by different means.

## Noise analysis¶

Small-signal schematic for noise analysis.

The small-signal schematic for the noise analysis is shown in the above figure. We can reuse the noise analysis performed for the Miller OTA. If we neglect the capacitances at the 1st-stage current mirror node and assume a perfect matching, the noise coming from the first stage can be modeled by the noisy current source $I_{n1}$ due to transistors M1a-M1b and M3a-M3b, whereas $I_{n2}$ models the noise coming from transistors M2 and M5b. The input-referred equivalent noise voltage is then given by \begin{equation*} V_{neq} = \frac{I_{n1}}{G_{m1}} - \frac{G_1}{G_{m1}\,G_{m2}} \cdot \frac{1+s\,(C_1+C_c)/G_1}{1-s\,C_c/G_{m2}} \cdot I_{n2}. \end{equation*} For $\omega \ll G_1/(C_1+C_c)<G_{m2}/C_c$, $V_{neq}$ can be approximated by \begin{equation*} V_{neq} \cong \frac{I_{n1}}{G_{m1}} - \cdot \frac{G_1}{G_{m1}\,G_{m2}} \cdot I_{n2} \end{equation*}

The input-referred PSD is then given by \begin{equation*} S_{V_{neq}} \cong \frac{S_{I_{n1}}}{G_{m1}^2} + \left(\frac{G_1}{G_{m1}\,G_{m2}}\right)^2 \cdot S_{I_{n2}}. \end{equation*}

### Input-referred thermal noise¶

For thermal noise we have \begin{align*} S_{I_{n1}} &= 4 kT \cdot 2 \cdot (\gamma_{n1} \cdot G_{m1}+\gamma_{n4} \cdot G_{m4}),\\ S_{I_{n2}} &= 4 kT \cdot (\gamma_{n2} \cdot G_{m2}+2\gamma_{n5} \cdot G_{m5}). \end{align*}

The input-referred thermal noise resistance $R_{nin,th}$ is then given by \begin{equation*} R_{nin,th} = \frac{2 \gamma_{n1}}{G_{m1}} \cdot (1 + \eta_{th}) \end{equation*} where \begin{equation*} \eta_{th} = \frac{\gamma_{n4}}{\gamma_{n1}}\,\frac{G_{m4}}{G_{m1}} + \frac{G_1^2}{2 G_{m1}\,G_{m2}} \cdot \left(\frac{\gamma_{n2}}{\gamma_{n1}} + \frac{2\gamma_{n5}}{\gamma_{n1}} \cdot \frac{G_{m5}}{G_{m2}}\right) \end{equation*} is the contribution of the current mirror M4a-M4b, 2nd-stage M2 and current mirror M5a-M5b relative to that of the differential pair.

The input-referred thermal noise resistance $R_{nin,th}$ can also be written as \begin{equation*} R_{nin,th} = \frac{\gamma_{neq}}{G_{m1}} \end{equation*} where $\gamma_{neq}$ is the amplifier equivalent thermal noise excess factor given by \begin{equation*} \gamma_{neq}= 2\,\gamma_{n1}\,(1+\eta_{th}) = 2\,\gamma_{n1}\,\left[1 + \frac{\gamma_{n4}}{\gamma_{n1}}\,\frac{G_{m4}}{G_{m1}} + \frac{G_1^2}{2 G_{m1}\,G_{m2}} \cdot \left(\frac{\gamma_{n2}}{\gamma_{n1}} + \frac{2\gamma_{n5}}{\gamma_{n1}} \cdot \frac{G_{m5}}{G_{m2}}\right)\right]. \end{equation*} We see that the contribution of the current mirror M4a-M4b can be minimized by choosing $G_{m1} \gg G_{m4}$ (same as for the simple OTA). The contribution of M2 and M5a-M5b are small thanks to the term \begin{equation*} \frac{G_1^2}{2 G_{m1}\,G_{m2}} = \frac{(G_{ds1}+G_{ds4})^2}{2 G_{m1}\,G_{m2}} \gg 1 \end{equation*} which is in the order of the DC gain. The contribution of the current mirror M5a-M5b can be made negligible by choosing $G_{m2} \gg G_{m5}$.

### Input-referred flicker noise¶

For flicker noise we have \begin{align*} S_{I_{n1}} &= \frac{4 kT}{f} \cdot 2 \cdot \left(G_{m1}^2\,\frac{\rho_n}{W_1 L_1} + G_{m4}^2\,\frac{\rho_p}{W_4 L_4}\right),\\ S_{I_{n2}} &= \frac{4 kT}{f} \cdot \left(G_{m2}^2\,\frac{\rho_p}{W_2 L_2} + 2 G_{m5}^2\,\frac{\rho_n}{W_5 L_5}\right). \end{align*}

The input-referred flicker noise resistance is then given by \begin{equation*} f \cdot R_{nin,fl} = 2\,\left[\frac{\rho_n}{W_1 L_1} + \left(\frac{G_{m4}}{G_{m1}}\right)^2\,\frac{\rho_p}{W_4 L_4}\right] + \left(\frac{G_1}{G_{m1}}\right)^2\,\left[\frac{\rho_p}{W_2 L_2} + 2\left(\frac{G_{m5}}{G_{m2}}\right)^2\,\frac{\rho_n}{W_5 L_5}\right] \end{equation*} which can be written in terms of the individual contributions relative to that of the differential pair \begin{equation*} f \cdot R_{nin,fl} = \frac{2 \rho_n}{W_1 L_1} \cdot (1+\eta_{fl}), \end{equation*} with \begin{equation*} \eta_{fl} = \left(\frac{G_{m4}}{G_{m1}}\right)^2\,\frac{\rho_p}{\rho_n}\,\frac{W_1 L_1}{W_4 L_4} + \frac{1}{2}\,\left(\frac{G_1}{G_{m1}}\right)^2\,\left[\frac{\rho_p}{\rho_n}\,\frac{W_1 L_1}{W_2 L_2} + 2\left(\frac{G_{m5}}{G_{m1}}\right)^2\,\frac{W_1 L_1}{W_5 L_5}\right] \end{equation*}

We see that the contribution of the current mirror M4a-M4b can be minimized by choosing $G_{m1} \gg G_{m4}$ (same as for the simple OTA). The contribution of M2 and M5a-M5b are small thanks to the first stage gain \begin{equation*} \left(\frac{G_1}{G_{m1}}\right)^2 = \left(\frac{G_{ds1}+G_{ds4}}{G_{m1}}\right)^2 \gg 1. \end{equation*} The contribution of the current source M5a-M5b can be made negligible by choosing $G_{m2} \gg G_{m5}$.

## Input-referred offset voltage¶

Small-signal schematic for mismatch analysis.

The estimation of the offset voltage can be handled similarly to the noise. It is essentially due to the first stage and is therefore similar to what was done for the simple OTA. Using the schematic shown in Fig. \ref{fig:mismatchschematic}, we can derive the input-referred offset voltage variance as \begin{equation*} \sigma{V{os}}^2 = \sigma{VT1}^2 + \left(\frac{G{m4}}{G{m1}}\right)^2 \cdot \sigma_{VT4}^2 + \left(\frac{Ib}{G{m1}}\right)^2 \cdot \left(\sigma{\beta 1}^2 + \sigma{\beta 4}^2\right) \end{equation} where \begin{align} \sigma{VT1}^2 &= \frac{A{VTn}^2}{W_1 L1},\ \sigma{\beta 1}^2 &= \frac{A_{\beta n}^2}{W_1 L1},\ \sigma{VT4}^2 &= \frac{A_{VTp}^2}{W_4 L4},\ \sigma{\beta 4}^2 &= \frac{A_{\beta_p}^2}{W_4 L_4},\ \end{align*}

# Conclusion¶

This notebook presented the detailed analysis of the basic two-stage or Miller OTA. The analysis allowed to derive the design equations to achieve the target specifications. The design equations will be used in the Design Notebook for the design of the Miller OTA for a given set of specifications.