# Lecture 7 Frequency Response.

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Lecture 7 Frequency Response

Review of CS, CG and CD Amplifier

Voltage Gain of a CS Amplifier
Interpretation: The resistance at the drain Divided by the resistance in the source path

Voltage Gain of a CD Amplifier

Voltage Gain of a CG Amplifier
If RS=0 and channel length modulation is ignored, Av is

Resistance into the Drain Terminal

Resistance into the Source Terminal

Miller Effect

Miller’s Theorem

Typical Application of Miller’s Theorem
Miller’s theorem is useful when Z appears in parallel with the main signal (i.e. the amplifier)

Limitation of Miller’s Theorem
Limitations: Interaction of poles through R3 and C3.

Association of Poles with Nodes
Each pole is determined by the product of Total capacitance seen from each node to ground Total resistance seen at the node to ground “Each node in the circuit contributes one pole to the transfer function”

Common-Gate Example

CS Stage Output Impedance Input Impedance “Nodal Method”
Miller Approximation “Zx” method Equivalent Circuit Analysis KCL Dominant pole

High Frequency Model of CS Stage

CS Trade-Off L(um) W(um) GDS (uS) CDB (fF) CGD(fF) CGS(fF) 2 5.78
3.613 5.19 1.84 98.16 800n 2.56 3.79 0.915 0.803 17.3 180n 0.86 5.72 0.056 0.273 1.20 120n 0.64 9.55 0.029 0.201 0.55 For Same IOUT, L↓→W↓→GDS↑(Ro↓) →CDS ↓ Specs: AV=10 Vo,cm=0.6V I(M1)=10 uA gm=AV/RD Gmoverid_1=16.67 Trade-offs in GDS and parasitic capacitance.

CS Trade-Off AV I (uA) L(um) W(um) GDS (uS) CDB (fF) CGD(fF) CGS(fF)
10 2 5.78 3.613 5.19 1.84 98.16 15 32.5 5.33 27.5 10.4 517.8 20 668.2 6.66 319.6 239.8 6,041.1 For Same IOUT, L↓→W↓→GDS↑(Ro↓) →CDS ↓ Specs: Vo,cm=0.6V gm=AV/RD Difficult to achieve high gain and high speed at the same time!

Output Impedance Only Valid if Rs is large!

Input Impedance High frequency approximation Exclude CGS
(First order model)

Input Impedance (KCL) High frequency approximation Exclude CGS
(In parallel with CGS)

“Nodal Method”(Miller Approximation)
It is important to identify the high impedance node! Numerical example: RS=50 Ohms L=2.0 um AV=15 fin=4.65 GHz fout=69.9 MHz 16(10.40fF) 517.8 fF CDB=27.51 fF, RD=60 KOhm

Transfer Function

“Nodal Method”(Refined Miller Approximation)
(If RS is large!) (Capacitive) (Resistive)

Equivalent Circuit Analysis

Comparison to Miller Approximation

Dominant Pole Approximation

Transmission Zero Finding a transmission zero in effective Gm.

Source Follower (Strong interaction between XY, making it difficult to associate each pole with each node)

Source Follower

Transmission Zero 𝜔𝑧=−𝑔𝑚/(𝐶𝑔𝑠+𝐶𝑔𝑑)

Input Impedance

Analysis of Input Impedance
Miller Approximation: Av: (Negative Resistance) Can be used to oscillators.

Output Impedance

Equivalent Output Impedance

Issues

Common Gate

Cascode (Gain from A to X)

DC Input Resistance Will a large Rin increase the miller effect of CS dramatically?

Input Resistance of Common Gate
Note that ZL is not infinity if RD is replaced by a current source because ZL is in parallel with CD.

Differential Pair (Differential Mode)
(Differential Half Circuit)

Differential Pair (Common-Mode)
W3 is made as large as possible to minimize VDSAT.

Consequence of Limited CMRR

Differential Pair with High Impedance Load
AC Ground (Dominant Pole)

Differential Pair Example
GM= uS GDS= uS RD=90 Kohm

AC analysis

Use the Waveform Calculator
Add voltages to the calculator Press Eval before you plot

Plot in Magnitude/dB

Transfer Function 3dB Bandwidth: MHz

Differential Pair with Current Mirror

Small Signal Equivalent Model
(Transmission Zero)

Differential Pair with Current Mirror
(Slow Path) (Fast Path)