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■◆✸✶✼✵✴✹✶✼✵✱ ❙♣r✐♥❣ ✷✵✷✵

P❤✐❧✐♣♣ ❍ä✢✐❣❡r ❤❛❢❧✐❣❡r❅✐❢✐✳✉✐♦✳♥♦ ❊①❝❡r♣t ♦❢ ❙❡❞r❛✴❙♠✐t❤ ❈❤❛♣t❡r ✾✿ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❇❛s✐❝ ❈▼❖❙ ❆♠♣❧✐✜❡rs

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

■♥t❡rr✉♣t✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t

❚r❛♥s❢❡r ❋✉♥❝t✐♦♥

❚❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ H(s) ♦❢ ❛ ❧✐♥❡❛r ✜❧t❡r ✐s ◮ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ ✐ts ✐♠♣✉❧s❡ r❡♣♦♥s❡ h(t)✳ ◮ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ■✴❖ r❡❛❧t✐♦♥s❤✐♣ t❤❛t ✐s t❤❡♥ s♦❧✈❡❞ ❢♦r Vout(s)

Vin(s)

◮ ✭t❤✐s ❧❡❝t✉r❡✦✦✦✮ t❤❡ ❝✐r❝✉✐t ❞✐❛❣r❛♠ s♦❧✈❡❞ q✉✐t❡ ♥♦r♠❛❧❧② ❢♦r

Vout(s) Vin(s) ❜② ♣✉tt✐♥❣ ✐♥ ✐♠♣❡❞❛♥❝❡s Z(s) ❢♦r ❛❧❧ ❧✐♥❡❛r ❡❧❡♠❡♥ts

❛❝❝♦r❞✐♥❣ t♦ s♦♠❡ s✐♠♣❧❡ r✉❧❡s✳

■♠♣❡❞❛♥❝❡s ♦❢ ▲✐♥❡❛r ❈✐r❝✉✐t ❊❧❡♠❡♥ts

r❡s✐st♦r✿ R ❝❛♣❛❝✐t♦r✿

✶ sC

✐♥❞✉❝t♦r✿ sL ■❞❡❛❧ s♦✉r❝❡s ✭❡✳❣✳ t❤❡ id = gmvgs s♦✉r❝❡s ✐♥ s♠❛❧❧ s✐❣♥❛❧ ♠♦❞❡❧s ♦❢ ❋❊❚s✮ ❛r❡ ❧❡❢t ❛s t❤❡② ❛r❡✳

❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ✐♥ ❘♦♦t ❋♦r♠

❚r❛♥s❢❡r ❢✉♥❝t✐♦♥s H(s) ❢♦r ❧✐♥❡❛r ❡❧❡❝tr♦♥✐❝ ❝✐r❝✉✐ts ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❞✐✈✐❞✐♥❣ t✇♦ ♣♦❧②♥♦♠✐❛❧s ♦❢ s✳ H(s) = a✵ + a✶s + ... + amsm ✶ + b✶s + ... + bnsn H(s) ✐s ♦❢t❡♥ ✇r✐tt❡♥ ❛s ♣r♦❞✉❝ts ♦❢ ✜rst ♦r❞❡r t❡r♠s ✐♥ ❜♦t❤ ♥♦♠✐♥❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r♦♦t ❢♦r♠✱ ✇❤✐❝❤ ✐s ❝♦♥✈❡♥✐❡♥t❧② s❤♦✇✐♥❣ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❇♦❞❡✲♣❧♦ts✳ ▼♦r❡ ♦❢ t❤❛t ❧❛t❡r✳ H(s) = a✵ (✶ + s

z✶ )(✶ + s z✷ )...(✶ + s zm )

(✶ + s

ω✶ )(✶ + s ω✷ )...(✶ + s ωn )

❇♦❞❡ P❧♦ts

P❧♦ts ♦❢ ♠❛❣♥✐t✉❞❡ ✭❡✳❣✳ |H(s)|✮ ✐♥ ❞❇ ❛♥❞ ♣❤❛s❡ ❡✳❣✳ ∠H(s) ♦r φ✮ ✈s✳ log(ω)✳ ■♥ ❣❡♥❡r❛❧ ❢♦r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❧② r❡❛❧ ♣♦❧❡s ❛♥❞ ♥♦ ③❡r♦s ✭♣✉r❡ ❧♦✇✲♣❛ss✮✿ ❛✮ ω → ✵+ |H(s)| ✐s ❝♦♥st❛♥t ❛t t❤❡ ❧♦✇ ❢r❡q✉❡♥❝② ❣❛✐♥ ❛♥❞ ∠H(s) = ✵o ❜✮ ❢♦r ❡❛❝❤ ♣♦❧❡ ❛s ω ✐♥❝r❡❛s❡s t❤❡ s❧♦♣❡ ♦❢ |H(s)| ✐♥❝r❡s❡s ❜② − ✷✵dB

❞❡❝❛❞❡ ❝✮ ❡❛❝❤ ♣♦❧❡ ❝♦♥tr✐❜✉t❡s ✲✾✵o t♦ t❤❡

♣❤❛s❡✱ ❜✉t ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐st✐♦♥ s♦ t❤❛t ❛t ❛ ❢r❡q✉❡♥❝② ❡①❛❝t❧② ❛t t❤❡ ♣♦❧❡ ✐t ✐s ❡①❛❝t❧② ✲✹✺o

• ❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡

♣❧♦ts ✭✶✴✸✮

■♥ ❣❡♥❡r❛❧ ❢♦r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❧② r❡❛❧ ♣♦❧❡s ❛♥❞ ♥♦ ③❡r♦s ✭♣✉r❡ ❧♦✇✲♣❛ss✮✿ ❛✮ ω → ✵+ |H(s)| ✐s ❝♦♥st❛♥t ❛t t❤❡ ❧♦✇ ❢r❡q✉❡♥❝② ❣❛✐♥ ❛♥❞ ∠H(s) = ✵o ❜✮ ❢♦r ❡❛❝❤ ♣♦❧❡ ❛s ω ✐♥❝r❡❛s❡s t❤❡ s❧♦♣❡ ♦❢ |H(s)| ✐♥❝r❡s❡s ❜② − ✷✵dB

❞❡❝❛❞❡ ❝✮ ❡❛❝❤ ♣♦❧❡ ❝♦♥tr✐❜✉t❡s ✲✾✵o t♦ t❤❡

♣❤❛s❡✱ ❜✉t ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐st✐♦♥ s♦ t❤❛t ❛t ❛ ❢r❡q✉❡♥❝② ❡①❛❝t❧② ❛t t❤❡ ♣♦❧❡ ✐t ✐s ❡①❛❝t❧② ✲✹✺o

• ❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡

♣❧♦ts ✭✶✴✸✮

❛✮ ✜♥❞ ❛ ❢r❡q✉❡♥❝② ωmid ✇✐t❤ ❡q✉❛❧ ♥✉♠❜❡r k ♦❢ ③❡r♦s ❛♥❞ ♣♦❧❡s ✇❤❡r❡ z✶, ..., zk, ω✶, ..., ωk < ωmid ⇒ |H(s)| ≈ K |z(k+✶)...|zm| |ω(k+✶)|...|ωn| ∠H(s) ≈ ✵o ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❜♦t❤ |H(s)| ❛♥❞ ∠H(s) ✐s ③❡r♦

• ❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡

♣❧♦ts ✭✷✴✸✮

❜✮ ♠♦✈✐♥❣ ❢r♦♠ ωmid ✐♥ t❤❡ ♠❛❣♥✐t✉❞❡ ♣❧♦t ❛t ❡❛❝❤ |ωi| ❛❞❞ ✲✷✵❞❇✴❞❡❝❛❞❡ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ❣r❛❞✐❡♥t ❛♥❞ ❢♦r ❡❛❝❤ |zi| ❛❞❞ ✰✷✵❞❇✴❞❡❝❛❞❡ ❝✮ ♠♦✈✐♥❣ ❢r♦♠ ωmid ✐♥ t❤❡ ♣❤❛s❡ ♣❧♦t t♦ ❤✐❣❤❡r ❢r❡q✉❡♥❝✐❡s ❛t ❡❛❝❤ ωi ❛❞❞ ✲✾✵o t♦ t❤❡ ♣❤❛s❡ ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐t✐♦♥ ✭r❡s♣❡❝t✐✈❡❧② −✹✺o r✐❣❤t ❛t t❤❡ ♣♦❧❡s✮ ❛♥❞ ✈✐❝❡ ✈❡rs❛ t♦✇❛r❞s ❧♦✇❡r ❢r❡q✉❡♥❝✐❡s✳

• ❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡

♣❧♦ts ✭✸✴✸✮

❞✮ ❋♦r t❤❡ ③❡r♦s t♦✇❛r❞s ❤✐❣❤❡r ❢r❡q✉❡♥❝✐❡s ✐❢ t❤❡ ♥♦♠✐♥❛t❡r ✐s ♦❢ t❤❡ ❢♦r♠ (✶ + s

zi ) ❛❞❞ ✰✾✵o

❛♥❞ ✐❢ ✐ts ♦❢ t❤❡ ❢♦r♠ (✶ − s

zi ) ✭r❡❢❡r❡❞ t♦ ❛s

r✐❣❤t ❤❛❧❢ ♣❧❛✐♥ ③❡r♦ ❛s t❤❡ s♦❧✉t✐♦♥ ❢♦r s ♦❢ ✵ = (✶ − s

zi ) ✐s ♣♦s✐t✐✈❡✮ ❛❞❞ ✲✾✵o t♦ t❤❡ ♣❤❛s❡

✐♥ ❛ s♠♦♦t❤ tr❛♥s✐t✐♦♥ ✭✐✳❡✳ r❡s♣❡❝t✐✈❡❧② ±✹✺o r✐❣❤t ❛t t❤❡ ③❡r♦s✮ ❛♥❞ ✈✐❝❡ ✈❡rs❛ t♦✇❛r❞s ❧♦✇❡r ❢r❡q✉❡♥❝✐❡s✳

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

▼❖❙❋❊❚ ■❈ tr❛♥s❢❡r ❢✉♥❝t✐♦♥

❇❲ ❛♥❞ ●❇✭P✮

• ❇❂❇❲ ∗ AM
• ❇✿ ❣❛✐♥ ❜❛♥❞✇✐❞t❤ ♣r♦❞✉❝t✱ ❇❲✿ ❜❛♥❞✇✐❞t❤✱ AM✿ ♠✐❞✲❜❛♥❞ ❣❛✐♥✳

❚❤❡r❡ ✐s ✉s✉❛❧❧② ❛ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ ❇❲ ❛♥❞ AM✳ ■❢ t❤✐s tr❛❞❡ ♦✛ ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧✱ t❤❡ ●❇ ✐s ❝♦♥st❛♥t✱ ❡✳❣✳ ✐♥ ♦♣❛♠♣ ❢❡❡❞❜❛❝❦ ❝♦♥✜❣✉r❛t✐♦♥s✳

❇❲ ❛♥❞ ●❇✭P✮ ❢♦r ❈▼❖❙ ✐♥t❡❣r❛t❡❞ ❈✐r❝✉✐ts

❋♦r ✐♥t❡❣r❛t❡❞ ❝✐r❝✉✐ts ✇❤✐❝❤ ♥♦r♠❛❧❧② ❤❛✈❡ ❛ ♣✉r❡ ❧♦✇✲♣❛ss ❝❤❛r❛❝t❡r✐st✐❝s ✭✐✳❡✳ ♥♦ ❡①♣❧✐❝✐t ❆❈✲❝♦✉♣❧✐♥❣ ❛t t❤❡ ✐♥♣✉t✮ ②♦✉ ❝❛♥ s✉❜st✐t✉t❡ AM ✇✐t❤ ADC✱ ✐✳❡✳ t❤❡ ❣❛✐♥ ❛t ❉❈✳ ❆♥❞✿ ❇❲ = fH = f−✸❞❇ ❲❤❡r❡ fH ✐s t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝② ❝✉t♦✛ ❛♥❞ ✐s t❤❡ ❢r❡q✉❡♥❝② ❛t ✇❤✐❝❤ ♣♦✐♥t AM ✐ r❡❞✉❝❡❞ ❜② ✲✸❞❇✱ ✐✳❡✳ t❤❡ s✐❣♥❛❧ ♣♦✇❡r ✐s r❡❞✉❝❡❞ ❜② ✶

✷✱

❛s ✶✵ log✶✵

✶ ✷ = ✸.✵

▼❖❙❋❊❚ ✬P❛r❛s✐t✐❝✬ ❈❛♣❛❝✐t❛♥❝❡s ■❧❧✉str❛t✐♦♥

▼❖❙❋❊❚ ✬P❛r❛s✐t✐❝✬ ❈❛♣❛❝✐t❛♥❝❡s ❊q✉❛t✐♦♥s

Cgs = CoxW (✷ ✸L + Lov) (✾.✷✷) Cgd = CoxWLov (✾.✷✸) Csb/db = Csb✵/db✵

• ✶ +

VSB/DB V✵

(✾.✷✹/✾.✷✺)

❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ✭✶✴✷✮

❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ✭✷✴✷✮

❲✐t❤ ♦♥❧② t❤❡ t✇♦ ♠♦st r❡❧❡✈❛♥t ♣❛r❛s✐t✐❝ ❝❛♣❛❝✐t♦rs✳

❯♥✐t② ●❛✐♥ ❋r❡q✉❡♥❝② fT

❙❤♦rt ❝✐r❝✉✐t ❝✉rr❡♥t ❣❛✐♥✳ ❆ ♠❡❛s✉r❡ ❢♦r t❤❡ ❜❡st ❝❛s❡ tr❛♥s✐st♦r s♣❡❡❞✳ ◆❡❣❧❡❝t✐♥❣ t❤❡ ❝✉rr❡♥t t❤r♦✉❣❤ Cgd✿ io ii = gm s(Cgs + Cgd) (✾.✷✽) fT = gm ✷π(Cgs + Cgd) (✾.✷✾)

❚r❛❞❡✲♦✛ fT ✈s Ao ✭✐✳❡✳ ●❇✮

fT = gm ✷π(Cgs + Cgd) (✾.✷✾) ≈ ✸µnVov ✹πL✷ A✵ = gmro (✼.✹✵) ≈ ✷ λVov = ✷L λL

• ❝♦♥st

Vov

❙✉♠♠❛r② ❈▼❖❙ ❍❋ ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❈❙ ❆♠♣❧✐✜❡r ❍❋ s♠❛❧❧ s✐❣♥❛❧ ♠♦❞❡❧

❯s✐♥❣ t❤❡ ▼✐❧❧❡r ❊✛❡❝t

◆♦t❡ t❤❡ s✐♠♣❧②❢②✐♥❣ ❛ss✉♠♣t✐♦♥ t❤❛t vo = gmrovgs✱ ✐✳❡✳ ♥❡❣❧❡❝t✐♥❣ ❢❡❡❞ ❢♦r✇❛r❞ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ igd ✇❤✐❝❤ ✇✐❧❧ st✐❧❧ ❜❡ ✈❡r② s♠❛❧❧ ❛r♦✉♥❞ fH ❛♥❞ ♠❛❦❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ q✉✐t❡ ❡①❛❝t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❞♦♠✐♥❛♥t ♣♦❧❡✬s ❢r❡q✉❡♥❝② fP ≈ fH

❈❙ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❞♦♠✐♥❛♥t Rsig✭✶✴✹✮

vgs(s(Cgs + Ceq) + ✶ R′

sig

) = vsig ✶ R′

sig

❈❙ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❞♦♠✐♥❛♥t Rsig✭✷✴✹✮

vgs vsig =

✶ R′

sig

s(Cgs + Ceq) +

✶ R′

sig

vgs vsig = ✶ ✶ + sR′

sig(Cgs + Ceq)

❈❙ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❞♦♠✐♥❛♥t Rsig✭✸✴✹✮

vo vsig = AM ✶ + s

ω✵

(✾.✺✶) ωH = ✶ R′

sig(Cgs + Ceq) (✾.✺✸)

❈❙ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❞♦♠✐♥❛♥t Rsig✭✹✴✹✮

❈❙ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡✱ ❉♦♠✐♥❛♥t CL ✭✶✴✷✮

vo(s(Cgd + CL) + GL) + gmvgs = vgssCgd vo vgs = sCgd − gm s(Cgd + CL) + GL = −gmRL ✶ − s Cgd

gm

s(Cgd + CL)RL + ✶ (✾.✻✺)

❈❙ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡✱ ❉♦♠✐♥❛♥t CL ✭✷✴✷✮

ωz = gm Cgd (✾.✻✻) ωp = ✶ (Cgd + CL)RL (✾.✻✼) ωz ωp = gmRL

• ✶ + CL

Cgd

• (✾.✻✽)

ωt = gm CL + Cgd (✾.✻✾)

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

◆♦t❛t✐♦♥ ✐♥ t❤✐s ❇♦♦❦

A(s) = AMFHs FH(s) = (✶ +

s ωz✶ )...(✶ + s ωzn )

(✶ +

s ωp✶ )...(✶ + s ωpm )

❉♦♠✐♥❛♥t P♦❧❡ ❆♣♣r♦①✐♠❛t✐♦♥

■❢ ωp✶ < ✹ωp✷ ❛♥❞ ωp✶ < ✹ωz✶ t❤❡♥ A(S) ≈ ✶ ✶ +

s ωp✶

ωH ≈ ωp✶

❆♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❲✐t❤♦✉t ❛ ❉♦♠✐♥❛♥t P♦❧❡

✷nd ♦r❞❡r ❡①❛♠♣❧❡ |FH(ωH)|✷ = ✶ ✷ = (✶ + ω✷

H

ω✷

z✶ )(✶ + ω✷ H

ω✷

z✷ )

(✶ + ω✷

H

ω✷

p✶ )(✶ + ω✷ H

ω✷

p✷ )

= ✶ + ω✷

H

• ✶

ω✷

z✶ +

✶ ω✷

z✷

• + ω✹

H

• ✶

ω✷

z✶ω✷ z✷

• ✶ + ω✷

H

• ✶

ω✷

p✶ +

✶ ω✷

p✷

• + ω✹

H

• ✶

ω✷

p✶ω✷ p✷

• ⇒ ωH

≈ ✶ ✶

ω✷

p✶ +

✶ ω✷

p✷ −

✷ ω✷

z✶ −

✷ ω✷

z✷

(✾.✼✻)

❆♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❲✐t❤♦✉t ❛ ❉♦♠✐♥❛♥t P♦❧❡

❣❡♥❡r❛❧✿ ωH ≈ ✶ ✶

ω✷

p✶ +

✶ ω✷

p✷ ... +

✶ ω✷

pm −

✷ ω✷

z✶ −

✷ ω✷

z✷ ... −

✷ ω✷

zn

(✾.✼✼) ■❢ ωp✶ ✐s ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ ❛❧❧ ♦t❤❡r ♣♦❧❡✲ ❛♥❞ ③❡r♦✲❢r❡q✉❡♥❝✐❡s t❤✐s r❡❞✉❝❡s t♦ t❤❡ ❞♦♠✐♥❛♥t ♣♦❧❡ ❛♣♣r♦①✐♠❛t✐♦♥✳

❖♣❡♥✲❈✐r❝✉✐t ❚✐♠❡ ❈♦♥st❛♥ts ▼❡t❤♦❞

ωH ≈ ✶

• i CiRi

❲❤❡r❡ Ci ❛r❡ ❛❧❧ ❝❛♣❛❝✐t♦rs ✐♥ t❤❡ ❝✐r❝✉✐t ❛♥❞ Ri ✐s t❤❡ r❡s✐st❛♥❝❡ s❡❡♥ ❜② Ci ✇❤❡♥ t❤❡ ✐♥♣✉t s✐❣♥❛❧ s♦✉r❝❡ ✐s ③❡r♦❡❞ ❛♥❞ ❛❧❧ ♦t❤❡r ❝❛♣❛❝✐t♦rs ❛r❡ ♦♣❡♥ ❝✐r❝✉✐t❡❞✳

❖♣❡♥✲❈✐r❝✉✐t ❚✐♠❡ ❈♦♥st❛♥ts ▼❡t❤♦❞ ❊①❛♠♣❧❡ ❈❙ ❆♠♣

❚❤❡ ❉✐✣❝✉❧t ❖♥❡ ✐s Rgd

ix = − vgs Rsig = vgs + vx RL + vgsgm = vx RL − ixRsig ✶ RL + gm

• Rgd = vx

ix = [RL + Rsig (✶ + gmRL)]

❖♣❡♥ ❈✐r❝✉✐t ❚✐♠❡ ❈♦♥st❛♥t

τH = RsigCgs + RLCL + [RL + Rsig (✶ + gmRL)] Cgd = Rsig [Cgs + (✶ + gmRL) Cgd] + RL [Cgd + CL] (✾.✽✽) Pr❡✈✐♦✉s❧②✿ ωH = ✶ R′

sig(Cgs + (✶ + gmRL)Cgd) (✾.✺✸)

ωH = ✶ (Cgd + CL)RL (✾.✻✼)

❈♦♠♣❛r✐♥❣ ❆♣♣r♦①✐♠❛t✐♦♥s

■❢ ②♦✉ ❝♦♠❜✐♥❡ t❤❡ ♣r✈✐♦✉s❧② tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ❢♦r vgs

vsig ❞❡r✐✈❡❞ ❢r♦♠

✭✾✳✹✻✮ ❛♥❞ vo

vgs ❢r♦♠ ✭✾✳✻✺✮ ❛s A(s) = vgs vsig vo vgs ②♦✉ ❣❡t ❜♦t❤ ♦❢ t❤❡s❡

♣r❡✈✐♦✉s ωH ❛s ♣♦❧❡s ❛♥❞ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ❝♦♠❜✐♥❡❞ ωH ❛❝❝♦r❞✐♥❣ t♦ ✭✾✳✼✼✮✿ τH = ✶ ωH ≈

• [R′

sig(Cgs + Ceq)]✷ + [(Cgd + CL)RL]✷ (✾.✼✼)

❙♦ t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ r❛t❤❡r t❤❛♥ t❤❡ s✉♠ ✳✳✳

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❈● ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡

◆❖❚❊✿ ♥♦ ▼✐❧❧❡r ❡✛❡❝t✦

❈● ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡ ❚✲♠♦❞❡❧

❈● ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡ ✇✐t❤♦✉t ro

τp✶ = Cgs

• Rsig|| ✶

gm

• τp✷ = (Cgd + CL)RL

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgs

❑✐r❝❤♦✛ Rin ❢♦r ♥♦❞❡ vs✿ vs(gm + ✶ ro ) = vo ✶ ro + is vo( ✶ ro + ✶ RL ) = vs(gm + ✶ ro ) vo = vs gm + ✶

ro ✶ ro + ✶ RL

❝♦♠❜✐♥❡✿ vs(gm + ✶ ro ) = vs gm + ✶

ro ✶ ro + ✶ RL

✶ ro + is vs(gm + ✶ ro − gm + ✶

ro

✶ + ro

RL

) = is vs((gm + ✶ ro )(✶ − ✶ ✶ + ro

RL

)) = is

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgs

❑✐r❝❤♦✛ Rin ❢♦r ♥♦❞❡ vs ✭❝♦♥t✐♥✉❡❞✮✿ vs((gm + ✶ ro )(

✶ RL ✶ ro + ✶ RL

)) = is vs is = ✶ gm + ✶

ro ✶ ro + ✶ RL ✶ RL

Rs = RL gm + ✶

ro

( ✶ ro + ✶ RL ) Rs = RLro gmro + ✶ RL + ro RLro Rs = RL + ro gmro + ✶

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgs

τgs = Cgs

• Rsig||ro + RL

gmro

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgd + CL

❑✐r❝❤♦✛ ❢♦r Ro ❛♥❞ vo✿ vo ✶ ro = io + vs ✶ ro + gmvs vs(gm + ✶ Rsig + ✶ ro ) = vo ✶ ro vs = vo

✶ ro

gm +

✶ Rsig + ✶ ro

❝♦♠❜✐♥❡✿ vo ✶ ro = io + vo

✶ ro

gm +

✶ Rsig + ✶ ro

( ✶ ro + gm)

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgd + CL

❑✐r❝❤♦✛ ❢♦r Ro ❛♥❞ vo ✭❝♦♥t✐♥✉❡❞✮✿ vo ✶ ro (✶ −

✶ ro + gm

gm +

✶ Rsig + ✶ ro

) = io vo ✶ ro (

✶ Rsig

gm +

✶ Rsig + ✶ ro

) = io vo( ✶ gmroRsig + ro + Rsig ) = io Ro = gmroRsig + ro + Rsig

❈● ❆♠♣❧✐✜❡r ♦♣❡♥ ❝✐r❝✉✐t t✐♠❡✲❝♦♥st❛♥t ✇✐t❤ ro ❢♦r Cgd + CL

τgd = (Cgd + CL) (Rsig||(ro + Rsig + gmroRsig))

❈● ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡ ❈♦♥❝❧✉s✐♦♥

◆♦ ▼✐❧❧❡r ❡✛❡❝t t❤❛t ✇♦✉❧❞ ❝❛✉s❡ ❧♦✇ ✐♠♣❡❞❛♥❝❡ ❛t ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s✱ ❜✉t ❞✉❡ t♦ ❧♦✇ ✐♥♣✉t r❡s✐st❛♥❝❡ t❤❡ ✐♠♣❡❞❛♥❝❡ ✐s ❛❧r❡❛❞② ❧♦✇ ❛t ❉❈ ⇒ ❧♦✇ AM ❢♦r Rsig > ✵

❈❛s❝♦❞❡ ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡

τgs✶ = Cgs✶Rsig τgd✶ = Cgd✶ [(✶ + gm✶Rd✶)Rsig + Rd✶] ✇❤❡r❡ Rd✶ = ro✶||ro✷ + RL gm✷ro✷ τgs✷ = (Cgs✷ + Cdb✶)Rd✶ τgd✷ = (CL + Cgd✷)(RL||(ro✷ + ro✶ + gm✷ro✷ro✶)) τh ≈ τgs✶ + τgd✶ + τgs✷ + τgd✷

❈❛s❝♦❞❡ ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡

❘❡❛rr❛♥❣✐♥❣ τh ❣r♦✉♣✐♥❣ ❜② t❤❡ t❤r❡❡ ♥♦❞❡s✬ r❡s✐st♦rs✿ τh ≈ Rsig [Cgs✶ + Cgd✶(✶ + gm✶Rd✶)] +Rd✶(Cgd✶ + Cgs✷ + Cdb✶) +(RL||Ro)(CL + Cgd✷) ❚❤✉s✱ ✐❢ Rsig > ✵ ❛♥❞ t❡r♠s ✇✐t❤ Rsig ❛r❡ ❞♦♠✐♥❛♥t ♦♥❡ ❝❛♥ ❡✐t❤❡r ❣❡t ❧❛r❣❡r ❜❛♥❞✇✐❞t❤ ❛t t❤❡ s❛♠❡ ❉❈ ❣❛✐♥ t❤❛♥ ❛ ❈❙ ❛♠♣❧✐✜❡r ✇❤❡♥ RL ≈ ro ♦r ❣❡t ♠♦r❡ ❉❈ ❣❛✐♥ ❛t t❤❡ s❛♠❡ ❜❛♥❞✇✐t❤ t❤❛♥ ❛ ❈❙ ❛♠♣❧✐✜❡r ✇❤❡♥ RL ≈ gmr✷

• ♦r ✐♥❝r❡❛s❡ ❜♦t❤

❜❛♥❞✇✐t❤ ❛♥❞ ❉❈ ❣❛✐♥ t♦ ❧❡ss t❤❛♥ t❤❡✐r ♠❛①✐♠✉♠ ❜② t✉♥✐♥❣ RL s♦♠❡✇❤❡r❡ ✐♥❜❡t✇❡❡♥✳

❈❛s❝♦❞❡ ❆♠♣❧✐✜❡r ❍❋ ❘❡s♣♦♥s❡

❘❡❛rr❛♥❣✐♥❣ τh ❣r♦✉♣✐♥❣ ❜② t❤❡ t❤r❡❡ ♥♦❞❡s✬ r❡s✐st♦rs✿ τh ≈ Rsig [Cgs✶ + Cgd✶(✶ + gm✶Rd✶)] +Rd✶(Cgd✶ + Cgs✷ + Cdb✶) +(RL||Ro)(CL + Cgd✷) ❲✐t❤ Rsig ≈ ✵ ♦♥❡ ❝❛♥ tr❛❞❡ ❤✐❣❤❡r ❇❲ ❢♦r r❡❞✉❝❡❞ ADC ♦r ❤✐❣❤❡r ADC ❢♦r r❡❞✉❝❡❞ ❇❲ ❝♦♠♣❛r❡❞ t♦ ❛ ❈❙ ❛♠♣✱ ❦❡❡♣✐♥❣ t❤❡ ✉♥✐t② ❣❛✐♥ ❢r❡q✉❡♥❝② ✭✐✳❡✳ t❤❡ ●❇✮ ❝♦♥st❛♥t✳

❈❛s❝♦❞❡ ✈s ❈❙

❈❙✿ ADC = −gm(ro||RL) τH = Rsig [Cgs + (✶ + gm(ro||RL)) Cgd] + (ro||RL) [Cgd + CL] ❈❛s❝♦❞❡✿ ADC = (−gm✶(RO||RL) ✇❤❡r❡ RO = gm✷ro✷ro✶ τH = Rsig [Cgs✶ + Cgd✶(✶ + gm✶Rd✶)] +Rd✶(Cgd✶ + Cgs✷ + Cdb✶) +(RL||Ro)(CL + Cgd✷) ✇❤❡r❡ Rd✶ = ro✶||ro✷ + RL gm✷ro✷

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❈♦♥❝❡♣t

Af = A ✶ + Aβ ≈ ✶ β ❢♦r Aβ >> ✶ Af ✿ ❝❧♦s❡❞ ❧♦♦♣ ❣❛✐♥ Aβ✿ ❧♦♦♣ ❣❛✐♥ β ♥♦r♠❛❧❧② ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❢r❡❡ ♦❢ ❛♥② ❢r❡q✉❡♥❝② ❞❡♣❡♥❞❡♥❝②✦✮

A ❍❛s ❛ ❙✐♥❣❧❡ P♦❧❡

Af =

A✵ ✶+A✵β

✶ + s

ωp (✶ + A✵β)

ωpf = ωp(✶ + A✵β)

A ❍❛s ❚✇♦ ❘❡❛❧ P♦❧❡s ✭✶✴✷✮

❙♦❧✈✐♥❣ ❢♦r ♣♦❧❡s✿ ✶ + A(s)β = ✵ ⇒ ✵ = s✷ + s(ωp✶ + ωp✷) + (✶ + A✵β)ωp✶ωp✷ (✶✵.✻✼) s = −✶ ✷(ωp✶ + ωp✷) ± ✶ ✷

• (ωp✶ + ωp✷)✷ − ✹(✶ + A✵β)ωp✶ωp✷ (✶✵.✻✽)
• ❡♥❡r❛❧❧② ♠♦r❡ s❡♣❛r❛t✐♦♥ ❜❡t✇❡❡♥ ωp✶ ❛♥❞ ωp✷ ✭✐✳❡✳ ❛ ❞♦♠✐♥❛♥t

♣♦❧❡✮ ❤❡❧♣s t♦ ❦❡❡♣ t❤❡ ♣♦❧❡s ♦❢ Af r❡♠❛✐♥ r❡❛❧ ✈❛❧✉❡s ❛♥❞ t♦ ❦❡❡♣ Af st❛❜❧❡ ❛t ❤✐❣❤❡r ❧♦♦♣ ❣❛✐♥s✳

A ❍❛s ❚✇♦ ❘❡❛❧ P♦❧❡s ✭✷✴✷✮

❘❡✇r✐t✐♥❣ t❤❡ s❛♠❡✿ ✵ = s✷ + s ω✵ Q + ω✷

✵ (✶✵.✻✾)

s = −✶ ✷(ω✵ Q ) ± ✶ ✷

• ω✷

✵

Q✷ − ✹ω✷

✵

Q =

• (✶ + A✵β)ωp✶ωp✷

ωp✶ + ωp✷ (✶✵.✼✵) ω✵ =

• (✶ + A✵β)ωp✶ωp✷

■❢ Q > ✵.✺ t❤❡ ♣♦❧❡s ❜❡❝♦♠❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❛♥❞ ✇❡ s❡❡ r❡s♦♥❛♥❝❡✳

❲❤② t❤❡ ❘❡s♦♥❛♥❝❡❄

◆❡❣❛t✐✈❡ ❢❡❡❞❜❛❝❦ s❤♦✉❧❞ ♥♦t ❧❡❛❞ t♦ ❛♠♣❧✐✜❝❛t✐♦♥✳ ❚❤❡ ❝r✉①✿ ❛ ♣❤❛s❡ s❤✐❢t φ = −✶✽✵o t✉r♥s ♥❡❣❛t✐✈❡ ❢❡❡❞❜❛❝❦ ✐♥t♦ ♣♦s✐t✐✈❡ ❢❡❡❞❜❛❝❦✳ ■❢ t❤❡ ❧♦♦♣ ❣❛✐♥ Aβ > ✶ ❛t s✉❝❤ ❛ ♣♦✐♥t✱ t❤❡ ❝✐r❝✉✐t ❤❛s ✐♥✜♥✐t❡ ❣❛✐♥✱ ✐✳❡✳ ✐s ✉♥st❛❜❧❡✳ ■♥ ❛ ❧♦✇ ♣❛ss ❝✐r❝✉✐t t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ♣❤❛s❡ ❢r♦♠ ✲✶✽✵o✱ ✐✳❡✳ φ + ✶✽✵o ✇❤❡r❡ t❤❡ ❧♦♦♣ ❣❛✐♥ ❜❡❝♦♠❡s ✉♥✐t② ✭Aβ = ✶ ♦r A = ✶

β✮

✐s t❤❡ ♣❤❛s❡ ♠❛r❣✐♥ ✭P▼✮✳ ❆ ❤✐❣❤ P▼ ✐♥❞✐❝❛t❡s ♥♦ ♦r ❧✐tt❧❡ r❡s♦♥❛♥❝❡✳ ❆ ♥❡❣❛t✐✈❡ ♣❤❛s❡ ♠❛r❣✐♥ ✐♥❞✐❝❛t❡s ❛♥ ✉♥st❛❜❧❡ ❝✐r❝✉✐t✳

❙♦✉r❝❡ ❋♦❧❧♦✇❡r ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ P♦ss✐❜✐❧✐t✐❡s

ωp✶,p✷ = −

✶ Qω✵ ±

• ✶

ω✷

✵Q✷ − ✹ ✶

ω✷

✵

✷

■♥t✉✐t✐♦♥ ❢♦r ❘❡s♦♥❛♥❝❡✴■♥st❛❜✐❧✐t②

❉❡♣❡♥❞❡♥❝❡ ♦♥ Q✲❢❛❝t♦r ✭✶✴✷✮

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Re(A)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Im(A) One Real Pole ωp1 =1/1000 1/A(j ω) A(j* ω)|j*[10

0,10 5]

A(j* ω)|j* ω=j*10 A(j* ω)|j* ω=j*10

3

j* ω=j*10

5

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Re(A)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Im(A) Two Identical Real Poles: ω0=1/1000 and Q=0.5 1/A(j ω) A(j* ω)|j*[10

0,10 5]

A(j* ω)|j* ω=j*10 A(j* ω)|j* ω=j*10

3

j* ω=j*10

5

❉❡♣❡♥❞❡♥❝❡ ♦♥ Q✲❢❛❝t♦r ✭✷✴✷✮

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Re(A)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Im(A) Two Complex Poles: ω0=1/1000 and Q=0.707 1/A(j ω) A(j* ω)|j*[10

0,10 5]

A(j* ω)|j* ω=j*10 A(j* ω)|j* ω=j*10

3

j* ω=j*10

5

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Re(A)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Im(A) Two Complex Poles: ω0=1/1000 and Q=1.0 1/A(j ω) A(j* ω)|j*[10

0,10 5]

A(j* ω)|j* ω=j*10 A(j* ω)|j* ω=j*10

3

j* ω=j*10

5

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❙♦✉r❝❡ ❋♦❧❧♦✇❡r ❍❋ ❘❡s♣♦♥s❡

A(s) = AM ✶ +

• s

ωz

• ✶ + b✶s + b✷s✷ = AM

✶ +

• s

ωz

• ✶ + ✶

Q s ω✵ + s✷ ω✷

✵

❈♦♥t❡♥t

■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮

• ❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮

❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❍❋ ❆♥❛❧②s✐s ♦❢ ❈✉rr❡♥t✲▼✐rr♦r✲▲♦❛❞❡❞ ❈▼❖❙ ❆♠♣ ✭✶✴✸✮

◆❡❣❧❡❝t✐♥❣ ro ✐♥ ❝✉rr❡♥t ♠✐rr♦r✿ vg✸(sCm + gm✸) + vid ✷ gm = ✵ io = −vg✸gm✸ + vid ✷ gm vg✸ = −

vid ✷ gm

sCm + gm✸ io = vid ✷ gm(✶ + gm✸ sCm + gm✸ ) io = vid ✷ gm(sCm + ✷gm✸ sCm + gm✸ ) io vid = gm s Cm

✷gm✸ + ✶

s Cm

gm✸ + ✶

❍❋ ❆♥❛❧②s✐s ♦❢ ❈✉rr❡♥t✲▼✐rr♦r✲▲♦❛❞❡❞ ❈▼❖❙ ❆♠♣ ✭✷✴✸✮

◆❡❣❧❡❝t✐♥❣ ro ✐♥ ❝✉rr❡♥t ♠✐rr♦r✿ GM = gm ✶ + s Cm

✷gm✸

✶ + s Cm

gm✸

ωp✷ = gm✸ Cm ωz = ✷gm✸ Cm

❍❋ ❆♥❛❧②s✐s ♦❢ ❈✉rr❡♥t✲▼✐rr♦r✲▲♦❛❞❡❞ ❈▼❖❙ ❆♠♣ ✭✸✴✸✮

vo = vidGMZo vo vid = gmRo

• ✶ + s Cm

✷gm✸

✶ + s Cm

gm✸

✶ ✶ +

✶ sCLRo

• (✾.✶✹✹)

ωp✶ = ✶ CLRo (✾.✶✹✺) ❆♥❞ ωp✶ ✐s ✉s✉❛❧❧② ❝❧❡❛r❧② ❞♦♠✐♥❛♥t✳