[HW1 Solution]

1.

Rin = 𝑅𝑅1 ∥ 𝑅𝑅2 ≥ 20𝑘𝑘Ω

IDS =12�

200µAV2 � �

200.18

� (𝑉𝑉𝐺𝐺𝐺𝐺 − 𝑉𝑉𝑡𝑡)2 = 1𝑚𝑚𝑚𝑚

VGS = 0.7𝑉𝑉,𝑉𝑉𝐺𝐺𝐺𝐺 =𝑅𝑅2

𝑅𝑅1 + 𝑅𝑅2

𝐑𝐑𝟏𝟏 ≥ 𝟓𝟓𝟏𝟏.𝟒𝟒𝟒𝟒𝛀𝛀,𝑹𝑹𝟐𝟐 ≥ 𝟑𝟑𝟐𝟐.𝟕𝟕𝟕𝟕𝛀𝛀

2.

(a) Rout

RS

vx, ixrOvgs gmvgs

vgs = −𝑖𝑖𝑥𝑥𝑅𝑅𝐺𝐺 = −𝑉𝑉𝑠𝑠,𝑣𝑣𝑥𝑥 − 𝑉𝑉𝑠𝑠𝑟𝑟𝑜𝑜

+ gm𝑣𝑣𝑔𝑔𝑠𝑠 = 𝑖𝑖𝑥𝑥

𝑣𝑣𝑥𝑥 − 𝑖𝑖𝑥𝑥𝑅𝑅𝐺𝐺𝑟𝑟𝑜𝑜

− gm𝑖𝑖𝑥𝑥𝑅𝑅𝐺𝐺 = 𝑖𝑖𝑥𝑥

𝐑𝐑𝐨𝐨𝐨𝐨𝐨𝐨 =𝒗𝒗𝒙𝒙𝒊𝒊𝒙𝒙

= {𝒓𝒓𝒐𝒐 + (𝟏𝟏 + 𝒈𝒈𝒎𝒎𝒓𝒓𝒐𝒐)𝑹𝑹𝑺𝑺}

VDD=1.8 V

500Ω

M1

R1

R2

(b) Rin

rOvgs gmvgs

vx, ix

RL

vx = −𝑣𝑣𝑔𝑔𝑠𝑠, �ix + 𝑔𝑔𝑚𝑚𝑣𝑣𝑔𝑔𝑠𝑠�ro + ix𝑅𝑅𝐿𝐿 = 𝑣𝑣𝑥𝑥

ix(𝑟𝑟𝑜𝑜 + 𝑅𝑅𝐿𝐿) = 𝑣𝑣𝑥𝑥(1 + 𝑔𝑔𝑚𝑚𝑟𝑟𝑜𝑜)

𝐑𝐑𝐢𝐢𝐢𝐢 =𝒗𝒗𝒙𝒙𝒊𝒊𝒙𝒙

=𝒓𝒓𝒐𝒐 + 𝑹𝑹𝑳𝑳𝟏𝟏 + 𝒈𝒈𝒎𝒎𝒓𝒓𝒐𝒐

3.

rOvin=vgs gmvgs

RD

RS voutix

ix = 𝑔𝑔𝑚𝑚𝑣𝑣𝑔𝑔𝑠𝑠 +𝑣𝑣𝑜𝑜𝑜𝑜𝑡𝑡𝑅𝑅𝐷𝐷 ∥ 𝑟𝑟𝑜𝑜

=𝑣𝑣𝑖𝑖𝑖𝑖 − 𝑣𝑣𝑜𝑜𝑜𝑜𝑡𝑡

𝑅𝑅𝐺𝐺

�1

RS− 𝑔𝑔𝑚𝑚� 𝑣𝑣𝑖𝑖𝑖𝑖 = �

1𝑅𝑅𝐺𝐺

+1

𝑅𝑅𝐷𝐷 ∥ 𝑟𝑟𝑜𝑜� vout

𝐀𝐀𝐕𝐕 =𝒗𝒗𝒐𝒐𝒐𝒐𝒐𝒐𝒗𝒗𝒊𝒊𝒊𝒊

=𝟏𝟏 − 𝒈𝒈𝒎𝒎𝑹𝑹𝑺𝑺

𝟏𝟏 + 𝑹𝑹𝑺𝑺/(𝑹𝑹𝑫𝑫 ∥ 𝒓𝒓𝒐𝒐)

4.

(a)

with ID = 0.5mA, VG = 2𝑉𝑉,𝑉𝑉𝐺𝐺 = 1𝑉𝑉.𝑉𝑉𝐺𝐺𝐺𝐺 = 1𝑉𝑉.𝑉𝑉𝑂𝑂𝑂𝑂 = 0.3𝑉𝑉

0.5mA =12

𝑘𝑘𝑖𝑖𝑉𝑉𝑂𝑂𝑂𝑂2 ,𝟕𝟕𝐢𝐢 = 𝟏𝟏𝟏𝟏.𝟏𝟏𝒎𝒎𝟏𝟏/𝑽𝑽𝟐𝟐

𝐕𝐕𝐃𝐃 = 𝟓𝟓 − (𝟓𝟓𝟒𝟒)(𝟎𝟎.𝟓𝟓𝒎𝒎𝟏𝟏) = 𝟐𝟐.𝟓𝟓𝑽𝑽

𝐕𝐕𝐆𝐆𝐃𝐃 = −𝟎𝟎.𝟓𝟓𝑽𝑽 < 𝑽𝑽𝒐𝒐 ∶ 𝑴𝑴𝟑𝟑 𝒊𝒊𝒊𝒊 𝒊𝒊𝒊𝒊 𝑺𝑺𝑺𝑺𝒐𝒐𝒐𝒐𝒓𝒓𝑺𝑺𝒐𝒐𝒊𝒊𝒐𝒐𝒊𝒊 𝑹𝑹𝑹𝑹𝒈𝒈𝒊𝒊𝒐𝒐𝒊𝒊

(b)

𝐑𝐑𝐢𝐢𝐢𝐢 = 𝟐𝟐𝟎𝟎𝟎𝟎𝟒𝟒 ∥ 𝟑𝟑𝟎𝟎𝟎𝟎𝟒𝟒 = 𝟏𝟏𝟐𝟐𝟎𝟎𝟒𝟒𝛀𝛀

AV =𝑣𝑣𝑜𝑜𝑣𝑣𝑠𝑠𝑖𝑖𝑔𝑔

= −𝑅𝑅𝑖𝑖𝑖𝑖

120𝑘𝑘 + 𝑅𝑅𝑖𝑖𝑖𝑖𝑔𝑔𝑚𝑚(5𝑘𝑘 ∥ 𝑟𝑟𝑜𝑜 ∥ 5𝑘𝑘)

gm =2𝐼𝐼𝐷𝐷𝑉𝑉𝑂𝑂𝑂𝑂

= 3.33𝑚𝑚𝑚𝑚, 𝑟𝑟𝑜𝑜 =𝑉𝑉𝐴𝐴𝐼𝐼𝐷𝐷

= 100𝑘𝑘Ω

𝐀𝐀𝐕𝐕 = −𝟒𝟒.𝟎𝟎𝟎𝟎

(c)

𝑉𝑉sig = 𝑉𝑉𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 sin𝑤𝑤𝑤𝑤 ,𝑉𝑉𝑜𝑜 = 𝑉𝑉𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 sin𝑤𝑤𝑤𝑤

gm(5𝑘𝑘 ∥ 5𝑘𝑘 ∥ 100𝑘𝑘) = 8.12

Vgd + 𝑉𝑉𝐺𝐺𝐷𝐷 = 𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 +𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

8.12− 0.5 ≤ 𝑉𝑉𝑡𝑡 = 0.7 𝑉𝑉

𝐕𝐕𝒐𝒐,𝒑𝒑𝑹𝑹𝑺𝑺𝟒𝟒 < 𝟏𝟏.𝟎𝟎𝟕𝟕𝑽𝑽

5V

300kΩ 5kΩ

5kΩ

2kΩ

Vo

M3Vsig

Rin

200kΩ

120kΩ CC1

CS

CC2

𝐕𝐕𝒑𝒑𝑹𝑹𝑺𝑺𝟒𝟒 <𝑽𝑽𝒐𝒐,𝒑𝒑𝑹𝑹𝑺𝑺𝟒𝟒

𝟏𝟏𝑽𝑽= 𝟐𝟐𝟎𝟎𝟏𝟏𝟐𝟐𝐕𝐕

(d)

Vgs = 𝑂𝑂𝑔𝑔1+𝑔𝑔𝑚𝑚𝑅𝑅𝑆𝑆

= 𝑣𝑣𝑔𝑔2 when 𝐑𝐑𝐒𝐒 = 𝟏𝟏

𝒈𝒈𝒎𝒎= 𝟑𝟑𝟎𝟎𝟎𝟎𝛀𝛀

(Also correct if you’ve considered 2kΩ resistor.

The above solution ignored 2kΩ resistor considering 2kΩ||300Ω ≒ 300Ω)

gm𝑅𝑅𝐿𝐿′1 + 𝑔𝑔𝑚𝑚𝑅𝑅𝐺𝐺

= �𝑉𝑉𝑜𝑜𝑉𝑉𝑔𝑔� = 4.06

Vo,peak +𝑉𝑉𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

4.06− 0.5V ≤ 0.7V

𝐕𝐕𝐨𝐨,𝐩𝐩𝐩𝐩𝐩𝐩𝟕𝟕 = 𝟎𝟎.𝟗𝟗𝟎𝟎𝑽𝑽

5.

(a)

ID = 0.3𝑚𝑚𝑚𝑚, |𝑉𝑉𝑂𝑂𝑂𝑂| = 0.3𝑉𝑉

VSG = 1.0𝑉𝑉,𝑉𝑉𝐺𝐺 = 0𝑉𝑉,𝑉𝑉𝐺𝐺 = 2.5 − 𝐼𝐼𝐷𝐷𝑅𝑅𝐺𝐺 = 1.0𝑉𝑉

𝐑𝐑𝐒𝐒 = 𝟓𝟓.𝟎𝟎𝟒𝟒𝛀𝛀

(b)

2.5V

RS

VoM3Vsig

Rsig

CC

-2.5V

RD

CS

gm =2𝐼𝐼𝐷𝐷𝑉𝑉𝑂𝑂𝑂𝑂

= 2𝑚𝑚𝑚𝑚,𝑚𝑚𝑂𝑂 =𝑣𝑣𝑜𝑜𝑣𝑣𝑠𝑠𝑖𝑖𝑔𝑔

= −𝑔𝑔𝑚𝑚𝑅𝑅𝐷𝐷 = −10

𝐑𝐑𝐃𝐃 = 𝟓𝟓.𝟎𝟎𝟒𝟒𝛀𝛀

(c)

Vgd + 𝑉𝑉𝐺𝐺𝐷𝐷 ≥ 𝑉𝑉𝑡𝑡𝑝𝑝 = −0.7

− �𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 +𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

10�+ 1𝑉𝑉 ≥ −0.7𝑉𝑉

𝐯𝐯𝐨𝐨,𝐩𝐩𝐩𝐩𝐩𝐩𝟕𝟕 ≤ 𝟏𝟏.𝟓𝟓𝟓𝟓𝑽𝑽

𝐯𝐯𝐩𝐩𝐩𝐩𝐩𝐩𝟕𝟕 ≤𝒗𝒗𝒐𝒐,𝒑𝒑𝑹𝑹𝑺𝑺𝟒𝟒

𝟏𝟏𝟎𝟎= 𝟎𝟎.𝟏𝟏𝟓𝟓𝟓𝟓𝑽𝑽

(d)

vpeak = 50𝑚𝑚𝑉𝑉, change RD

− �𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 +𝑣𝑣𝑜𝑜,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

𝑔𝑔𝑚𝑚𝑅𝑅𝐷𝐷� + (2.5 − 𝐼𝐼𝐷𝐷𝑅𝑅𝐷𝐷) ≥ −0.7𝑉𝑉

for gm = 2 𝑚𝑚𝑚𝑚, 𝐼𝐼𝐷𝐷 = 0.3𝑚𝑚𝑚𝑚

− �1 + 𝑔𝑔𝑚𝑚𝑅𝑅𝐷𝐷𝑔𝑔𝑚𝑚𝑅𝑅𝐷𝐷

� 𝑔𝑔𝑚𝑚𝑅𝑅𝐷𝐷𝑣𝑣𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 + 2.5 − 𝐼𝐼𝐷𝐷𝑅𝑅𝐷𝐷 ≥ −0.7

𝐑𝐑𝐃𝐃 ≤ 𝟕𝟕.𝟖𝟖𝟖𝟖𝟒𝟒𝛀𝛀, 𝐀𝐀𝐕𝐕 = −𝒈𝒈𝒎𝒎𝑹𝑹𝑫𝑫 = −𝟏𝟏𝟓𝟓.𝟖𝟖