# Fundamentals of Microelectronics

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6.4 (a) Q ( x ) = WC ox ( V GS - V ( x ) - V TH ) = WC ox ( V GS - V TH ) - WC ox V ( x ) Increasing V DS L WC ox ( V GS - V TH ) x Q ( x ) The curve that intersects the axis at x = L (i.e., the curve for which the channel begins to pinch oF) corresponds to V DS = V GS - V TH . (b) R Local ( x ) 1 μQ ( x ) L x R Local Increasing V DS
Note that R Local diverges at x = L when V DS = V GS - V TH .

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6.15 V GS I D V TH Increasing V DS Initially, when V GS is small, the transistor is in cutof and no current Fows. Once V GS increases beyond V TH , the curves start ±ollowing the square-law characteristic as the transistor enters saturation. However, once V GS increases past V DS + V TH (i.e., when V DS < V GS - V TH ), the transistor goes into triode and the curves become linear. As we increase V DS , the transistor stays in saturation up to larger values o± V GS , as expected.

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6.17 I D = 1 2 μ n C ox W L ( V GS - V TH ) α , α < 2 g m d ∂I D ∂V GS = α 2 μ n C ox W L ( V GS - V TH ) α - 1 = αI D V GS - V TH

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6.21 Since they’re being used as current sources, assume M 1 and M 2 are in saturation for this problem. To Fnd the maximum allowable value of λ , we should evaluate λ when 0 . 99 I D 2 = I D 1 and 1 . 01 I D 2 = I D 1 , i.e., at the limits of the allowable values for the currents. However, note that for any valid λ (remember, λ should be non-negative), we know that
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ch06sol - 6.4(a Q(x = W Cox(VGS V(x VT H = W Cox(VGS VT H W...

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