Stay Simple – Square-Law Equation Related

The 1st post in 2013. Happy middle new year! 😉

Nowadays most of us in academy are designing analog circuits in deep-submicron nodes using short-channel transistors. Putting them together with the digital, building up a system, and selling it with extraordinary performance in top conferences or journals—that’s a dream (sometimes triggers a real nightmare) of a PhD student…

Then do you miss the old time when you are designing circuits using long-channel transistors?

I miss! The square-law equation is so neat that the relation among the drain current, the effective voltage, and the transconductance can simply be expressed by

I_D = \frac{V_{GS}-V_T}{2} g_m

With certain effective gate-source voltage, the transconductance linearly follows the biasing current. It is quite straightforward to tell how much power you will pay for your target gain.

Soon, the short-channel transistors together with low voltage come ;-). Life becomes not that straightforward. In [1] and [2], the authors introduce a parameter Veff, defined by

V_{eff}=\frac{I_D}{g_m}

They also compare their parameter with the square-law equation using the following figure.

Veff versus VGS-VT [1][2]

Fig. 1 Veff versus VGS-VT [1][2].

Yes! For modern short-channel transistors, they are often biased in the transition region between sub-threshold and saturation regions. They are making classical equations useless! The proposing of Veff is so decent that it provides a simple way for us to analyze the power bounds of analog circuits in modern CMOS [1][2].

If you ponder on the above simulated curves, do you remember something which also has the beauty of simplicity and continuity?

Yes. The EKV model!  In this model, the Veff, using inversion coefficient, can be derived by [3]

V_{eff}= \frac{I_D}{g_m} = n U_T (\sqrt{IC + 0.25}+0.5)

And the effective gate-source voltage is expressed as [3]

V_{GS}-V_T= 2 n U_T \textnormal{ln}(e^{\sqrt{IC}}-1)

Out of curiosity, I plotted Veff versus VGS-VT based on the EKV model, together with the simulated curves, which I obtained using a similar method as used in [1][2] but for different processes. The figure is shown as bellow.

Veff versus VGS-VT with EKV model included

Fig. 2 Veff versus VGS-VT with EKV model included (Different inversion regions are also indicated).

Though the analog world can’t be as simple as 0 or 1, still the simpler the better!

Reference

[1] T. Sundström, B. Murmann, and C. Svensson, “Power dissipation bounds for high-speed Nyquist analog-to-digital converters”, TCASI, 2008.

[2] C. Svensson and J. J. Wikner, “Power consumption of analog circuits: a tutorial”, Analog Springer, 2010.

[3] D. M. Binkley, B. J. Blalock, and J. M. Rochelle, “Optimizing drain current, inversion level, and channel length in analog CMOS design”, Analog Springer, 2006.

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One Response to Stay Simple – Square-Law Equation Related

  1. Pingback: Gm/Id-Design Methodology | EveryNano Counts

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