MOS Transistors Yannis Tsividis

Small-Signal Modeling y-Parameter Model

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 1

These slides are based on Y. Tsividis and C. McAndrew, “Operation and Modeling of the MOS Transistor”, Copyright © Oxford University Press, 2011. They are meant to be part of a lecture, and may be incomplete or may not even make sense without the accompanying narration.

2

𝑣𝑔 𝑡 = 𝑀 cos 𝜔𝑡 + 𝜙

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑣𝑔(𝑡) 𝑖𝑔(𝑡)

𝑖𝑠(𝑡) 𝑖𝑑(𝑡)

𝑖𝑏(𝑡) 𝑣𝑠(𝑡)

𝑣𝑏(𝑡)

𝑣𝑑(𝑡)

𝑉𝑔 𝐼𝑔

𝐼𝑠 𝐼𝑑

𝐼𝑏 𝑉𝑠

𝑉𝑏 𝑉𝑑

𝑉𝑔 = 𝑀𝑒𝑗𝜙

3 Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑦𝑑𝑑 ≡𝐼𝑑𝑉𝑑

𝑉𝑔,𝑉𝑏,𝑉𝑠=0

𝐼𝑑

𝑉𝑑

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑦𝑑𝑔 ≡𝐼𝑑𝑉𝑔

𝑉𝑑,𝑉𝑏,𝑉𝑠=0

𝑉𝑔

𝐼𝑑

4

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑦𝑑𝑏 ≡𝐼𝑑𝑉𝑏

𝑉𝑔,𝑉𝑑,𝑉𝑠=0

𝐼𝑑

𝑉𝑏

5

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑦𝑑𝑠 ≡𝐼𝑑𝑉𝑠

𝑉𝑑,𝑉𝑔,𝑉𝑏=0

𝐼𝑑

𝑉𝑠

6

7

𝐼𝑑 = 𝐼𝑑 𝑉𝑔,𝑉𝑏,𝑉𝑠=0

+ 𝐼𝑑 𝑉𝑑,𝑉𝑏,𝑉𝑠=0

+ 𝐼𝑑 𝑉𝑑,𝑉𝑔,𝑉𝑠=0

+ 𝐼𝑑 𝑉𝑑,𝑉𝑔,𝑉𝑏=0

𝐼𝑑 = 𝑦𝑑𝑑 𝑉𝑑 + 𝑦𝑑𝑔𝑉𝑔 + 𝑦𝑑𝑏𝑉𝑏 + 𝑦𝑑𝑠𝑉𝑠

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

When all voltages are applied simultaneously:

𝑉𝑔

𝐼𝑑

𝑉𝑠

𝑉𝑏 𝑉𝑑

8

𝐼𝑑 = 𝑦𝑑𝑑𝑉𝑑 + 𝑦𝑑𝑔𝑉𝑔 + 𝑦𝑑𝑏𝑉𝑏 + 𝑦𝑑𝑠𝑉𝑠

𝐼𝑔 = 𝑦𝑔𝑑𝑉𝑑 + 𝑦𝑔𝑔𝑉𝑔 + 𝑦𝑔𝑏𝑉𝑏 + 𝑦𝑔𝑠𝑉𝑠

𝐼𝑏 = 𝑦𝑏𝑑𝑉𝑑 + 𝑦𝑏𝑔𝑉𝑔 + 𝑦𝑏𝑏𝑉𝑏 + 𝑦𝑏𝑠𝑉𝑠

𝐼𝑠 = 𝑦𝑠𝑑𝑉𝑑 + 𝑦𝑠𝑔𝑉𝑔 + 𝑦𝑠𝑏𝑉𝑏 + 𝑦𝑠𝑠𝑉𝑠

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

𝑦𝑘𝑙 ≡𝐼𝑘𝑉𝑙

𝑉𝑛=0, 𝑛≠𝑙

𝑉𝑔 𝐼𝑔

𝐼𝑠 𝐼𝑑

𝐼𝑏 𝑉𝑠

𝑉𝑏 𝑉𝑑

Similarly for the other currents:

9

𝑦𝑑𝑑 + 𝑦𝑑𝑔 + 𝑦𝑑𝑏 + 𝑦𝑑𝑠 = 𝑦𝑑𝑑 + 𝑦𝑔𝑑 + 𝑦𝑏𝑑 + 𝑦𝑠𝑑 = 0

𝑦𝑔𝑔 + 𝑦𝑔𝑑 + 𝑦𝑔𝑏 + 𝑦𝑔𝑠 = 𝑦𝑔𝑔 + 𝑦𝑑𝑔 + 𝑦𝑏𝑔 + 𝑦𝑠𝑔 = 0

𝑦𝑏𝑏 + 𝑦𝑏𝑑 + 𝑦𝑏𝑔 + 𝑦𝑏𝑠 = 𝑦𝑏𝑏 + 𝑦𝑏𝑑 + 𝑦𝑔𝑏 + 𝑦𝑠𝑏 = 0

𝑦𝑠𝑠 + 𝑦𝑠𝑑 + 𝑦𝑠𝑔 + 𝑦𝑠𝑏 = 𝑦𝑠𝑠 + 𝑦𝑑𝑠 + 𝑦𝑔𝑠 + 𝑦𝑏𝑠 = 0

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

Working as in the case of capacitance parameters, we obtain:

10

𝐼𝑑 = 𝑦𝑑𝑑𝑉𝑑𝑠 + 𝑦𝑑𝑔𝑉𝑔𝑠 + 𝑦𝑑𝑏𝑉𝑏𝑠

𝐼𝑔 = 𝑦𝑔𝑑𝑉𝑑𝑠 + 𝑦𝑔𝑔𝑉𝑔𝑠 + 𝑦𝑔𝑏𝑉𝑏𝑠

𝐼𝑏 = 𝑦𝑏𝑑𝑉𝑑𝑠 + 𝑦𝑏𝑔𝑉𝑔𝑠 + 𝑦𝑏𝑏𝑉𝑏𝑠

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

and, keeping three independent equations:

11

𝑦𝑚 = 𝑦𝑑𝑔 − 𝑦𝑔𝑑

𝑦𝑚𝑏 = 𝑦𝑑𝑏 − 𝑦𝑏𝑑

𝑦𝑚𝑥 = 𝑦𝑏𝑔 − 𝑦𝑔𝑏

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011

Completely general!

𝐼𝑔

−𝑦𝑔𝑠 −𝑦𝑔𝑑 𝑦𝑚𝑉𝑔𝑠

−𝑦𝑠𝑑

𝑦𝑚𝑏𝑉𝑏𝑠

𝐼𝑑

−𝑦𝑏𝑠 𝑦𝑚𝑥𝑉𝑔𝑏 −𝑦𝑔𝑏 −𝑦𝑏𝑑

𝐼𝑏

𝐼𝑑 = −𝑦𝑔𝑑𝑉𝑑𝑔 − 𝑦𝑠𝑑𝑉𝑑𝑠 − 𝑦𝑏𝑑𝑉𝑏𝑑 + 𝑦𝑚𝑉𝑔𝑠 + 𝑦𝑚𝑏𝑉𝑏𝑠

𝐼𝑔 = −𝑦𝑔𝑑𝑉𝑔𝑑 − 𝑦𝑔𝑏𝑉𝑔𝑏 − 𝑦𝑔𝑠𝑉𝑔𝑠

𝐼𝑏 = −𝑦𝑏𝑑𝑉𝑏𝑑 − 𝑦𝑔𝑏𝑉𝑏𝑔 + 𝑦𝑚𝑥𝑉𝑔𝑏 + 𝑦𝑏𝑠𝑉𝑏𝑠

Transformation of the above equations results in:

Based on Tsividis/McAndrew; Copyright © Oxford University Press, 2011 12

Compare to our complete quasi-static model:

𝑖𝑔

𝐶𝑔𝑠 𝐶𝑚

𝑑𝑣𝑔𝑠

𝑑𝑡

𝑔𝑚𝑣𝑔𝑠

𝑔𝑠𝑑

𝐶𝑔𝑑

𝑖𝑠 𝑖𝑑 𝐶𝑔𝑏 𝐶𝑠𝑑

𝑔𝑚𝑏𝑣𝑏𝑠

𝐶𝑚𝑏

𝑑𝑣𝑏𝑠

𝑑𝑡

𝐶𝑚𝑥

𝑑𝑣𝑔𝑏

𝑑𝑡 𝐶𝑏𝑠 𝐶𝑏𝑑

𝑖𝑏

𝐼𝑔

−𝑦𝑔𝑠 −𝑦𝑔𝑑 𝑦𝑚𝑉𝑔𝑠

−𝑦𝑠𝑑

𝑦𝑚𝑏𝑉𝑏𝑠

𝐼𝑑

−𝑦𝑏𝑠 𝑦𝑚𝑥𝑉𝑔𝑏 −𝑦𝑔𝑏 −𝑦𝑏𝑑

𝐼𝑏