Frequency Response of Amplifier

Jack OuSonoma State University

RC Low Pass (Review)

𝑉 𝑜𝑢𝑡 (𝑆 )𝑉 𝑖𝑛 (𝑆 )

=1/(1+𝑠𝑅𝐶)

A pole: a root of the denomintor1+sRC=0→S=-RC

Laplace Transform/Fourier Transform

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

(𝑠)=1/(1+𝑠𝑅𝐶)

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

( 𝑗ω)=1 /(1+ 𝑗ω 𝑅𝐶)

𝑠= 𝑗ω

)|=1/| +p|

Phase=-tan-1(/p)

p=1/(RC)

(Fourier Transform)

(Laplace Transform)

𝑗ω

-p

𝑗ω| +p|

Location of the zero in the left complexplane

𝜎

Complex s plane

Rules of thumb: (applicable to a pole)Magnitude:1. 20 dB drop after the cut-off frequency2. 3dB drop at the cut-off frequencyPhase:3. -45 deg at the cut-off frequency4. 0 degree at one decade prior to the cut-frequency5. 90 degrees one decade after the cut-off frequency

RC High Pass Filter (Review)

𝑉 𝑜𝑢𝑡 (𝑆 )𝑉 𝑖𝑛 (𝑆 )

=𝑠𝑅𝐶 /(1+𝑠𝑅𝐶)

A zero at DC.A pole from the denominator.1+sRC=0→S=-RC

Laplace Transform/Fourier Transform

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

(𝑠)=𝑠𝑅𝐶 / (1+𝑠𝑅𝐶)

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

( 𝑗ω)= 𝑗ω𝑅𝐶 /(1+ 𝑗ω𝑅𝐶)

𝑠= 𝑗ω

)|=| |/| +p|

Phase=90-tan-1(/p)

p=1/(RC)Zero at DC.

(Fourier Transform)

(Laplace Transform)

𝑗ω

-p

𝑗ω| +p|

Location of the zero in the left complexplane

𝜎

Complex s plane

Zero at the origin.Thus phase(f=0)=90 degrees.The high pass filter has a cut-off frequency of 100.

RC High Pass Filter (Review)

𝑉 𝑜𝑢𝑡 (𝑆 )𝑉 𝑖𝑛 (𝑆 )

= 𝑅1𝑅1+𝑅2

1+𝑠𝑅1𝐶1+𝑠𝑅12𝐶

R12=(R1R2)/(R1+R2)A pole and a zero in the left complex plane.

Laplace Transform/Fourier Transform (Low Frequency)

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

(𝑠)=𝑅1

𝑅1+𝑅21+𝑠𝑅1𝐶1+𝑠𝑅12𝐶

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

( 𝑗ω)=𝑅1

𝑅1+𝑅21+ 𝑗ω𝑅1𝐶1+ 𝑗ω𝑅12𝐶

𝑠= 𝑗ω

z=1/(RC)p=1/(R12C)

(Fourier Transform)

(Laplace Transform)

𝑗ω

-p

𝑗ω| +p|

Location of the zero in the left complexplane

𝜎

Complex s plane

| +z|

-z

At low frequencies, | +p|>| +p|.

Laplace Transform/Fourier Transform (High Frequency)

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

(𝑠)=𝑅1

𝑅1+𝑅21+𝑠𝑅1𝐶1+𝑠𝑅12𝐶

𝑉 𝑜𝑢𝑡𝑉 𝑖𝑛

( 𝑗ω)=𝑅1

𝑅1+𝑅21+ 𝑗ω𝑅1𝐶1+ 𝑗ω𝑅12𝐶

𝑠= 𝑗ω

z=1/(RC)p=1/(R12C)

(Fourier Transform)

(Laplace Transform)

𝑗ω

-p

𝑗ω| +p|

Location of the zero in the left complexplane

𝜎

Complex s plane| +z|

-z

At high frequency, | +p|is almost equal to | +p|.

Design

• ωz=1/R1C

• ωp=1/(R12)C

• Note that R12<R1

• If R2<<R1, ωp/ ωp=R1/R2

• Design for ωp/ ωp=1000

High Frequency

Examples

Source Follower

Device Setup

Gmoverid:

Gm=17.24 mSRS=1000 OhmsGMBS=2.8 mSCGS=62.79 fF

Small Signal Parameters

Design Constraints:1. 1/(gm+gmbs)=50 Ohms2. Large R1 to minimize Q

R2=58 OhmsR1=1102 OhmsL=4.013 nH

Simulation Results

Current Mirror Example

Gm1=201.3uSGM3=201uSCGS3=CGS4=306.9fFGDS4=3.348uSGDS2=5.119uSRload=118 KohmsCload=1 pF

Fp1=1.347 MHzFp2=52.11 MHzFz=104.2 MHz

Magnitude

AvDC,matlab=27.52AvDC,sim=27.45

Fp1matlab=1.34MHzFp1sim=1.23 MHz

Phase

Transit Frequency

Transit Frequency Calculation

Understanding Transit Frequency

Since fT depends on VGS-VT, fT depndes on gm/ID.fT depends on L.

Overdrive Voltage as a function of gm/ID

gm/ID=2/(VOV)

Transit Frequency as function of gm/ID

gm/gds as a function of gm/ID

Trade-off of gm/gds and fT

15-20

fT

gm/gds

gm/ID

Numerical Example

L=120n gm/gds fT(Hz)

gm/ID=5 12.05 84.32G

gm/ID=10 15.71 64.05G

gm/ID=15 17.19 43.94G

gm/ID=20 17.54 22.76G

gm/ID=25 17.05 0.42 G

VDS=0.6 V

Numerical Example

gm/ID=20 gm/gds fT(Hz)

L=0.12um

17.54 22.7 G

L=0.18 um

29.88 12.6 G

L=0.25 um

37.35 7.96 G

L=1 um 46.00 714.4 M

L=2 um 47.26 190.3 MVDS=0.6 V

gm/ID Principle

Use to gm/ID principle to find capacitance

• gm/ID→(fT,I/W,gm/gds)

• fT=gm/cgg, cgg=cgs+cgb+cgd

• cgs/cgg is also gm/ID dependent.

Example

• Assume gm/ID=20, L=120 nm, VDS=0.6V, I=100uA.

• fT=22.76 GHz• cgg=gm/fT=13.98 fF

• cgd/cgg=0.29→cgd=4.1 fF

• cgs/cgg=0.75 →Cgs=10.5 fF

Noise

Noise is not deterministic

The value of noise cannot be predicted at any time even if the past values are known.

Average Power of a Random Signal

Observe the noise for a long time.

Periodic voltage to a loadresistance.

Unit: V2 rather than W.

It is customary to eliminate RL from PAV.

Power Spectral Density

PSD shows how much Power the signal carriesat each frequency.

Sx(f1) has unit of V2/Hz.

PSD of the Output Noise

PSD of the Output Noise

Output Noise

PSD of the Input Noise

Input Noise

Noise Shaping

Correlated and Uncorrelated Sources

(How similar two signals are.)

Pav=Pav1+Pav2

Superposition holds for only for uncorrelated sources.

Uncorrelated/Correlated Sources

(Multiple conversations in progress)

(clapping)

Resistor Thermal Noise

Example

Vnr1sqr=2.3288 x 10-19

Vnr3sqr=7.7625 x 10-20

Vnoutsqr=3.1050 x10-19

Analytical Versus Simulation

as a function of length

Corner Frequency (fco)

fco as a function of length