The EKV Normalized Functions
For the Design of Analog Circuits (Version 1)
1 Introduction
This notebook presents all the normlaized functions that are used for the design of analog circuits using the \(G_m/I_D\) approach with the inversion coefficient [1] [2] [3] [4].
2 Large-signal functions
2.1 Normalized current versus charge
2.1.1 Long-channel
The normalized drain current or inversion coefficient \(IC\) is defined as the drain current in saturation normalized to the specific current \(I_{spec}\) \[\begin{equation} IC \triangleq \frac{\left.I_{D}\right|_{\mathrm{saturation}}}{I_{spec}}, \end{equation}\] where the specific current is given by \[\begin{equation} I_{spec} = I_{spec\Box} \cdot \frac{W}{L} \end{equation}\] with \[\begin{equation} I_{spec\Box} \triangleq 2n \cdot \mu \cdot C_{ox} \cdot U_T^2. \end{equation}\] The inversion coefficient \(IC\) gives the level of inversion of the transistor according to \[\begin{align} IC < 0.1 & \textsf{weak inversion (WI)},\\ 0.1 \leq IC < 10 & \textsf{moderate inversion (MI)},\\ 10 \leq IC & \textsf{strong inversion (SI)}. \end{align}\]
The inversion coefficient for a long-channel transistor is a function of the normalized source charge according to \[\begin{equation}\label{eqn:ic_qs} IC = q_s^2 + q_s \end{equation}\] where \(q_s\) is the inversion charge \(Q_i\) evaluated at the source and normalized to \(Q_{spec} \triangleq -2 n C_{ox} U_T\) \[\begin{equation} q_s \triangleq \frac{Q_i(x=0)}{Q_{spec}}. \end{equation}\] The expression of \(IC\) versus \(q_s\) can be inverted to express the normalized charge as a function of the inversion coefficient according to \[\begin{equation} q_s = \frac{\sqrt{4IC+1}-1}{2} \end{equation}\]
2.1.2 Short-channel
The normalized drain current in the simplified EKV (sEKV) model is given by \[\begin{equation}\label{eqn:ic_qs_short} IC \triangleq \frac{\left.I_D\right|_{saturation}}{I_{spec}} = \frac{2(q_s^2+q_s)}{1 + \sqrt{1 + \lambda_c^2 (q_s^2+q_s)}}, \end{equation}\] where parameter \(\lambda_c\) is the velocity saturation parameter which scales as \[\begin{equation} \lambda_c = \frac{L_{sat}}{L} \end{equation}\] where \(L_{sat} = 2 \mu U_T/v_{sat}\) is the length over which the carriers velocity is saturating to \(v_{sat}\). \(\lambda_c\) is therefore the fraction of the channel over which the carriers are in full velocity saturation. The long-channel case is obtained by setting \(\lambda_c = 0\) for which the normalized drain current \(\eqref{eqn:ic_qs_short}\) reduces to \(\eqref{eqn:ic_qs}\).
The short-channel asymptotes are given by \[\begin{equation} IC \cong \begin{cases} q_s & \textsf{in weak inversion ($\lambda_c \cdot q_s \ll 1$)},\\ \frac{2 q_s}{\lambda_c} & \textsf{in strong inversion ($\lambda_c \cdot q_s \gg 1$)}. \end{cases} \end{equation}\]
The inversion charge \(IC\) is plotted versus \(q_s\) in Figure 1 for the long- and short-channel cases.
2.2 Normalized charge versus current
Equation \(\eqref{eqn:ic_qs_short}\) can be inverted to express the normalized source charge in function of the inversion coefficient \[\begin{equation}\label{eqn:qs_ic_short} q_s = \frac{\sqrt{4\,IC+1+(\lambda_c\,IC)^2}-1}{2} \end{equation}\] which for long-channel (\(\lambda_c=0\)) reduces to \[\begin{equation}\label{eqn:qs_ic} q_s =\frac{\sqrt{4\,IC+1}-1}{2}. \end{equation}\]
The normalized source charge \(q_s\) is plotted versus the inversion coefficient \(IC\) in Figure 2 for both the long- and short-channel cases.
2.3 Normalized saturation voltage versus charge
The charge is related to the voltage according to \[\begin{equation}\label{eqn:vps_qs} v_p-v_s = 2 q_s + \ln(q_s) \end{equation}\] which is plotted which is plotted in Figure 4.
2.4 Normalized saturation voltage versus inversion coefficient
The source charge can be expressed in terms of the inversion coefficient as \(\eqref{eqn:qs_ic}\) for a long-channel transistor or \(\eqref{eqn:qs_ic_short}\). Replacing in \(\eqref{eqn:vps_qs}\) allows to express the saturation voltage in terms of the inversion coefficient. For a long-channel transistor we get \[\begin{equation} v_p-v_s = \ln(\sqrt{4 IC+1}-1) + \sqrt{4 IC+1} - 1 -\ln(2) \end{equation}\] The saturation voltage \(v_p-v_s\) is plotted versus the inversion coefficient \(IC\) in Figure 4 for both the long- and short-channel case.
2.5 Normalized charge versus saturation voltage
The voltage versus charge equation \(\eqref{eqn:vps_qs}\) can actually be inverted using the Lambert W-function of order 0. The Lambert function \(W(z)\) is defined as the function satisfying \[\begin{equation} W(z) \cdot e^{W(z)} = z. \end{equation}\] The voltage versus charge equation can be written as \[\begin{equation}\label{eqn:q_v} 2q \cdot e^{2q} = e^v \end{equation}\] where \(q \triangleq q_s\) and \(v \triangleq v_p-v_s\). Eqn. \(\eqref{eqn:q_v}\) can now be solved for \(q\) using the Lambert W-function by setting \(z=2e^v\) which leads to \[\begin{equation} q(v) =\frac{1}{2} W\left(2e^v\right) \end{equation}\] or \[\begin{equation} q_s = \frac{1}{2} W\left(2 e^{v_p-v_s}\right). \end{equation}\] The Lambert W-function is available in the scipy Python package as lambertw. It is also available in Mathematic as the ProductLog[z] function.
The charge versus voltage can also be approximated by the EKV function which is compared to the exact Lambert W-function in Figure 5.
2.6 Inversion coefficient versus saturation voltage
Finally, the inversion charge \(IC\) is plotted versus the saturation voltage \(v_p-v_s\) in Figure 6 for both the long- and short-channel cases. We see that velocity saturation reduces the current in strong inversion but has no effect in weak inversion.
2.7 Drain-to-source saturation voltage versus inversion coefficient
The \(v_p-v_s\) voltage works well in strong inversion to estimate the saturation voltage for a long-channel transistor. However, it becomes negative in weak inversion and cannot be used for estimating the saturation voltage in weak inversion which should be equal to a few \(U_T\), typically \(4\,U_T\). An approximation of the saturation voltage valid from weak to strong inversion can be defined as \[\begin{equation} v_{ds_{sat}} = 2\,\sqrt{IC + v_{ds_{sat,wi}}}, \end{equation}\] where \(v_{ds_{sat,wi}} \cong 4\).
2.8 Inversion coefficient versus drain-to-source saturation voltage
2.9 Slope factor versus inversion coefficient
The slope factor \(n\) in weak inversion is actually depending on the pinch-off voltage according to
\[\begin{equation} n = n_{wi} = 1 + \frac{\Gamma_b}{2\sqrt{\Psi_0 + V_P}} = 1 + \frac{\gamma_b}{2\sqrt{\psi_0 + v_p}} \end{equation}\]
where \(\Gamma_b\) is the substrate factor given by
\[\begin{equation} \Gamma_b = \gamma_b \cdot \sqrt{U_T} = \frac{\sqrt{2qN_b \epsilon_{Si}}}{C_{ox}} \end{equation}\]
and \(\Psi_0 \cong 2\Phi_F + \text{ a few } U_T\). The slope factor normally depends on the pich-off voltage which depends on the gate voltage. For \(v_s=0\), we can express the \(v_p\) as a function of \(IC\) and plot the slope factor versus \(IC\).
3 Small-signal functions
It can be shown that for a long-channel transistor the source transconductance \(G_{ms}\) is actually proportional to the inversion charge taken at the source \(Q_i(x=0)\) or to the normalized source charge \(q_s\). The normalized source transconductance \(g_{ms}\) can therefore be expressed in terms of the inversion coefficient \(IC\) according to \[\begin{equation} g_{ms} \triangleq \frac{G_{ms}}{G_{spec}} = q_s(IC) = \frac{\sqrt{4IC+1}-1}{2}, \end{equation}\] where the specific conductance \(G_{spec}\) is defined as \[\begin{equation} G_{spec} = \frac{I_{spec}}{U_T} = 2n \cdot \mu \cdot C_{ox} \cdot U_T. \end{equation}\]
If velocity saturation is accounted for the normalized source transconductance becomes \[\begin{equation} g_{ms} = \frac{\sqrt{4\,IC+1+(\lambda_c\,IC)^2}-1}{2+\lambda_c^2\,IC}. \end{equation}\]
The normalized source transconductance in weak and in strong inversion reduces to \[\begin{equation} g_{ms} = \begin{cases} IC & \textsf{in weak inversion ($IC \ll 1$)},\\ \frac{1}{\lambda_c} & \textsf{in strong inversion ($IC \gg 1$)}. \end{cases} \end{equation}\] The normalized source transconductance \(g_{ms}\) is plotted versus the inversion coefficient in Figure 10 for both the long- and short-channel cases. We see that the normalized source transconductance saturates to \(1/\lambda_c\) in SI.
3.1 Inversion coefficient versus transconductance
For long-channel transistor, it is easy to invert the normalized source transconductance to express the inversion coefficient \(IC\) required to achieve a given normalized source transconductance \[\begin{equation}\label{eqn:ic_gms} IC = g_{ms} \cdot (g_{ms}+1). \end{equation}\] The formula becomes much more complicated when including velocity saturation. For \(\lambda_c > 0\) we get \[\begin{equation}\label{eqn:ic_gms_short} IC = \frac{2-\lambda_c^2\,g_{ms}\,(1+2\,g_{ms})-\sqrt{4-(\lambda_c\,g_{ms})^2\,(4-\lambda_c^2)}} {\lambda_c^2\,((\lambda_c\,g_{ms})^2 - 1)}, \end{equation}\] which for \(\lambda_c \rightarrow 0\) reduces to \(\eqref{eqn:ic_gms}\).
\(IC\) is plotted versus \(g_{ms}\) in Figure 11 for the log- and short-channel cases.
3.2 Normalized \(G_m/I_D\) versus inversion coefficient
The normalized \(G_m/I_D\) for the long-channel transistor can then be expressed in terms of the inversion coefficient as \[\begin{equation} \frac{G_m \cdot n \, U_T}{I_D} = \frac{G_{ms} \cdot U_T}{I_D} = \frac{g_{ms}(IC)}{IC} = \frac{\sqrt{4IC+1}-1}{2 IC} \end{equation}\]
For short-channel devices it becomes \[\begin{equation} \frac{G_m \cdot n \, U_T}{I_D} = \frac{G_{ms} \cdot U_T}{I_D} = \frac{g_{ms}}{IC} = \frac{\sqrt{4\,IC+1+(\lambda_c\,IC)^2}-1}{IC\,(2+\lambda_c^2\,IC)}. \end{equation}\]
The normalized source transconductance in strong inversion reduces to \[\begin{equation} \frac{g_{ms}}{IC} = \begin{cases} \frac{1}{\sqrt{IC}} & \text{for long-channel ($\lambda_c = 0$)},\\ \frac{1}{\lambda_c \cdot IC} & \text{for short-channel ($\lambda_c > 0$)}. \end{cases} \end{equation}\]
The normalized \(G_m/I_D\) function is plotted below versus \(IC\).
3.3 Inversion coefficient versus normalized \(G_m/I_D\)
For long-channel transistors, the normalized \(G_m/I_D\) can be inverted to express the inversion coefficient as a function of the desired \(G_m/I_D\). This results in \[\begin{equation} IC = \frac{1-gmsid}{gmsid^2}, \end{equation}\] where \(gmsid = G_m n U_T/I_D\).
\(IC\) is plotted versus g_{ms}/i_d$ in Figure 13.
There is unfortunately no closed-form expression for the inversion coefficient \(IC\) as a function of the normalized \(G_m/I_D\) when including velocity saturation. However, it can be solved numerically using fsolve.
3.4 Noise functions
3.4.1 Thermal noise
4 Normalized intrinsic capacitances
The normalized intrinsic capacitances of a long-channel transistor are given by \[\begin{align} c_{gsi} &\triangleq \frac{C_{GSi}}{C_{OX}} = \frac{q_s}{3}\,\frac{2 q_s+4 q_d+3}{(q_s+q_d+1)^2},\\ c_{gdi} &\triangleq \frac{C_{GDi}}{C_{OX}} = \frac{q_d}{3}\,\frac{2 q_d+4 q_s+3}{(q_s+q_d+1)^2},\\ c_{gbi} &\triangleq \frac{C_{GBi}}{C_{OX}} = \frac{n-1}{n}\,(1-c_{gsi}-c_{gdi}),\\ c_{bsi} &\triangleq \frac{C_{BSi}}{C_{OX}} = (n-1)\,c_{gsi},\\ c_{bdi} &\triangleq \frac{C_{BDi}}{C_{OX}} = (n-1)\,c_{gdi},\\ c_{ggi} &\triangleq \frac{C_{GGi}}{C_{OX}} = c_{gsi} + c_{gdi} + c_{gbi}, \end{align}\] where \(C_{OX} \triangleq W_{eff}\,L_{eff}\,C_{ox}\).
The normalized intrinsic capacitances are plotted versus \(v_p\) in Figure 17. Note that the dashed lines represents the intrinsic capacitances \(c_{gbi}\), \(c_{bsi}\), \(c_{bdi}\) and \(c_{ggi}\) calculated with a constant slope factor \(n\).
5 References
[1]
C. C. Enz and E. A. Vittoz, Charge-Based MOS Transistor Modeling - The EKV Model for Low-Power and RF IC Design, 1st ed. John Wiley, 2006.
[2]
M. Bucher, C. Lallement, C. Enz, F. Théodoloz, and F. Krummenacher, “The EPFL-EKV MOSFET Model Equations for Simulation.” https://github.com/chrisenz/EKV/blob/main/EKV2.6/docs/ekv_v26_rev2.pdf, 1998.
[3]
C. Enz, F. Chicco, and A. Pezzotta, “Nanoscale MOSFET Modeling: Part 1: The Simplified EKV Model for the Design of Low-Power Analog Circuits,” IEEE Solid-State Circuits Magazine, vol. 9, no. 3, pp. 26–35, 2017.
[4]
C. Enz, F. Chicco, and A. Pezzotta, “Nanoscale MOSFET Modeling: Part 2: Using the Inversion Coefficient as the Primary Design Parameter,” IEEE Solid-State Circuits Magazine, vol. 9, no. 4, pp. 73–81, 2017.