Noise Factor Optimization under Power Matching Constraints

Chapter 3 A 24-GHz CMOS Front-End 39

3.2 Common-Gate With Resistive Feedthrough LNA

3.2.6 Noise Factor Optimization under Power Matching Constraints

As we pointed out, because perfect noise matching and power matching can not be achieved simultaneously in either CS or CGRF LNA design, the final choice of the amplifier’s input impedance is a trade-off between low noise performance and signal power transfer. In many applications, a good power matching with sufficient margin is mandatory, requiring LNA to be designed by optimizing F under power matching constraints instead of designing forF_min. In this subsection we derive the expression for the noise factor of the CGRF stage with a perfect power match and compare to that of CS topology.

Amplifier’s gain requirement results in g_mR_f >>1, indicating i_n,d>>i_n,f. Hence we ignore i_n,f as well as i_n,RL in our analysis from now on. Instead, for a more

accurate result, we take g_g , g_mb, and the difference between g_m and g_d0 into account. We define

m mb

g

≡ g

χ (3.47)

0 d

m

g

≡ g

α (3.48)

( )

₀ ^{0 2}

5 ⎟⎟⎠

⎜⎜ ⎞

⎝

≈ ⎛

≡

T m

g

g g

ω α ω ω

η (3.49)

A more accurate expression for the input impedance of the small-signal circuits in Figure 3.7 is given by

1 0) ) (

1 (

1 ⁻

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +

+ +

= + _m

L f

f m

in g

R R

R

Z g χ η ω

(3.50) The fact thatR_f /(R_f +R_L)<1 results in a lower-bound on gm which is given by

)) ( 1 (

1 ω0

η

≥ +

s

m R

g (3.51)

A value of gm lower than this will make it impossible to achieve a perfect input match.

Assuming Zin is perfectly matched to R_s, the effective transconductance of CGRF stage is given by

(

1 ( )

)

2 1

, _{m s}η ω0

s

mCGRF g R

G = R − (3.52)

which indicates a large gm can degrade the gain. This is because the increase of gm

results in a larger gg, making more signal loss through the gate. We choose the higher- bound of gm as the value that causes 3dB Gm degradation from its low-frequency value, i.e.,

) ( 2

1

0 3

, ^dB _sη ω

m

m g R

g ≤ ₋ = (3.53)

Input matching criterion and gain consideration set the limits on the design parameter gm. The following discussion on noise figure is based on the assumption that the input is perfectly matched and the noise of the following stage is negligible. Therefore the expressions following are only valid for gm inside the range specified by (3.51) and (3.53). A lower gm will violate the input match assumption and a larger gm can

tremendously degrade gain, making the noise contribution of the following stages significant.

At perfect power matching the following expression for F is yielded, as explained in Appendix 3.1:

S m s

m s

m

CGRF g R g R

R

F g1 ( ) 2 ( ) ( )

1

1 1 ² _{0 0 0}

2

ω δη ω

η ω

χ η α

γ ₊

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

≈ (3.54)

where the second term represents the contribution of channel thermal noise and the third term accounts for the contribution of induced gate noise. By equating the derivative of F_CGRF to zero and solving for g_m, we obtain an optimum gm for the lowest noise figure, i.e.,

2 1

0 2 0 2 ,

, 1 (1 ) ( ) ( ) ⁻

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

= χ η ω η ω

γ δα

s opt CGRF

m R

g (3.55)

and the corresponding optimized F under power matching constraints given by ) 1 (

) 1 1 (

) 2 1 (

1 2 ² ₀

2 2 0

0

, η ω

χ α

ω γ α η γδ ω χ

χη α

γ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

⎟ +

⎠

⎜ ⎞

⎝ +⎛ + +

+ +

opt =

FCGRF (3.56)

The noise factor of the inductively degenerated common-source stage under perfect power matching and its corresponding lowest value is given by [⁷⁹]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟⎟ + + +

⎠

⎞

⎜⎜

⎝

⎛ + +

⎟⎟⎠

⎜⎜ ⎞

⎝ +⎛

= )

1 5 5 ( 2 5

1 5 2 5

1 1

2 0

2 2

2 0

γ δα ω

α ω γ

δα γ

δα γ δα α

γ ω

ω Q c

c Q F

T s

s T

CS (3.57)

where

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

0

1 ω ω_T

S m

s g R

Q (3.58)

By similar procedure, we obtain the lowest value for F_CS when it is optimized under power matching constraints

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +

⎟⎟+

⎠

⎜⎜ ⎞

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ + +

+

=

2 0 2 0

2 2

, 1 5

5 2 5

2 5 5 1

1 4

T T

opt

CS c c

F ω

ω γ δα ω

ω γ δα γ δα γ

γ δ (3.59)

The difference between F_CS,optand F_CGRF,optcan be evaluated by subtracting (3.56).

from (3.57). Assuming α =1 and χ =0, this difference is given by

2 0 0

,

, 5 5

1 2 5 2 5

5 1

4 ⎟⎟⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎛

⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟⎟⎠ −

⎞

⎜⎜⎝

⎛ + +

=

−

T T

opt CGRF opt

CS

c c F

F ω

γδ ω ω

ω γ

δ γ δ

γδ _(3.60)

With (3.60), it is evident that CGRF LNA is advantageous over common-source LNA in terms of minimum achievable noise figure. And most importantly, this improvement increases with frequency.

Power dissipation is another important factor in LNA design. To illustrate the trade-off between power and noise figure, the noise figure is plotted against g_m in Figure 3.9 for both common-source and CGRF stages at different ω0 to ωT ratio. For CGRF curves the gm is constrained in the range specified by (3.51) and (3.53). To first order estimation, assuming the MOSFET is biased at a fixed ωT,, a larger gm is directly related to a larger transistor width and larger power dissipation. Figure 3.10 shows that the optimum g_m of common-source topology is less than that of a CGRF one. Figure 3.10 also illustrates that at high power level CGRF stage achieves a lower noise figure than common-source one, but at low power level it is opposite. Therefore, the choice of topology depends on both the noise specification and the power budget.

At very high frequencies, where low-noise is a principle challenge and can not be met by using common-source LNA due to its theoretical limitation, CGRF LNA provides a way to design towards lower noise figure at the price of more power consumption.

Low-noise requirements lead to our choice of CGRF in this work. However, we also take power consumption into consideration and a trade-off will be shown in the next section.

3.2.7 Stability

Since in CGRF stage Rf acts as a positive feedback, the stability issue needs to be carefully addressed. Considering the input transistor with feedthrough resistor as a two-port network shown in Figure 3.10, it is a sufficient condition to prevent oscillation that the real parts of both impedances seen looking into the source and the drain of the input transistor are positive. It is easy to show that Re[Zin] and Re[Zout] can be expressed as

) 1 ( 1

] ] Re[

Re[ + +χ

= +

f m

d f

in g R

Z

Z R (3.61)

Figure 3.9: Noise figure of CS and CGRF LNA under power matching constraints

(

¹

)

^{Re[ ]}

]

Re[Z_out =R_f + g_mR_f + Z_s (3.62)

where Zs and Zd are the load impedances at source and drain respectively. (3.61) and (3.62) indicate that as long as Re[Zd] and Re[Zs] are positive, which is true for any passive termination, the stability of the CGRF stage is guaranteed.