Proposed Input Matching Method

The input impedance of the LNA is designed to match with the antenna, in order to pre-vent the incoming signal from reflecting back and forth between the LNA and the antenna.

Generally, the antenna has 50-Ω load to the LNA. Unfortunately, the input impedance of a MOSFET device is inherently capacitive, so the matching with 50-Ω resistive input impedance is not an easy task. Thus, a capacitive feedback matching technique is proposed to overcome this problem [75].

The common-source amplifier as the input stage is shown in Fig. 3.1, where Z_s is the impedance seen from the right node of the input matching inductor L_g, Z_in is the input impedance of M1, C_gd is the gate-drain capacitance, C_gs is the gate-source capacitance, C_out is the output loading capacitance which is the input capacitance of the next stage, R_s is the signal-source resistance, and V_s is the input signal voltage source. By using Millers theorem on C_gd, the input impedance Z_in can be derived as

Z_in = R_f

g_m is the transconductace of M1, and ω_o is the operating angular frequency. As it can be seen from the above equations, both C_gd and C_out with g_m together providing a real term R_f which contributes to the real input impedance in Z_in. They are called the capacitive feedback matching network.

3.2 The Noise Analysis of the Proposed Input Match-ing Method

In the LNA design, input power match is essential but not sufficient. It is also vital for a LNA to satisfy the noise performance requirement, so that the circuit itself does not

degrade the output signal-to-noise ratio (SNR) to an unacceptable level. Thus, a careful noise analysis on the capacitive feedback matching technique is developed to establish the principle of operation clearly and find the limits on noise performance. A brief review of the standard CMOS noise sources will facilitate the analysis.

3.2.1 Noise Sources

Fig. 3.2 shows the small-signal model of the equivalent circuit for the noise analysis.

Three noise sources have been considered in Fig. 3.2. They are the thermal noise of the source resistance i_n,Rs, the thermal noise of the channel current i_n,d, and the gate induced current noise i_n,g. They can be expressed as [48]

i²_n,Rs = 4kT 1

R_sΔf (3.2.1)

i²_n,d = 4kT γg_d0Δf (3.2.2)

i²_n,g = 4kT δω²C_gs²

5g_d0 Δf, (3.2.3)

where k is the Boltzmann’s constant, T is the absolute temperature, R_s is the source resistance, g_d0 is the zero-bias drain conductance, γ is the thermal noise coefficient, Δf is the noise bandwidth in hertz, and δ is the gate induced current noise factor.

According to [48], there is a correlation between the gate induced current noise i_n,g and the thermal noise of the channel current i_n,d. This correlation can be treated by separating i_n,g into two parts. i_n,gc is the part that fully correlated with thermal noise of the channel current i_n,d, whereas i_n,gu is the uncorrelated part. Hence, the gate induced current noise can be written as

where the correlation coefficient c is defined as [48]

jc = i_n,gi^∗_n,d 

i²_n,g i²_n,d

. (3.2.5)

Because of the correlation, special attention must be paid to the reference polarity of the correlated component. The value of c is positive for the polarity shown in Fig. 3.2.

3.2.2 Capacitive Feedback Matching Network Noise Analysis

Noise performance is usually evaluated with noise figure (N F ) which indicates the noise suppression ability of the circuit. Noise figure is defined as

N F = 10log(F ), (3.2.6)

where F is the noise factor which is defined as the total output noise power divided by the noise power at the output due to the input source. F can be expressed as

F = i²_n,o,tot.

i²_n,o,Rs = i²_n,o,Rs+ i²_n,o,g+d

i²_n,o,Rs , (3.2.7)

where i²_n,o,tot. is the mean-squared of the total output noise current, i²_n,o,Rs is the mean-squared output noise current due to i_n,Rs, and i²_n,o,g+d is the mean-squared output noise current due to i_n,d and i_n,g. Considering correlation, i²_n,o,g+d can be re-expressed as

i²_n,o,g+d = (i^∗_n,o,d+ i^∗_n,o,g)(i_n,o,d + i_n,o,g)

= i²_n,o,d+ 2Re(i_n,o,di^∗_n,o,g) + i²_n,o,gu. (3.2.8) From (3.2.5), (3.2.7), and (3.2.8). Noise factor can be expressed as

F =

A²i²_n,d+ 2Re(AB^∗c 

i²_n,g i²_n,d) + B²i²_n,gu

D²i²_n,Rs , (3.2.9)

where Finally, the noise factor for a capacitive feedback matching network is obtained from (3.2.9)-(3.2.10) as to R_s and L_g and let the derivatives equal to zero, optimum source noise impedance, Z_n,opt = R_sn,opt+ jωL_gn,opt, corresponding to minimum noise figure can be written as

Z_n,opt =

Using (5.1.1), (3.2.14) can be re-expressed in Z_n,opt^∗ as

Z_n,opt^∗ = Re[Z_n,opt] + ηImag[Z_in]. (3.2.15)

η =

From (3.2.15), the imaginary part of Z_n,opt^∗ is nearly the same as the imaginary part of Z_in and expressed as ηImag[Z_in].

For the circuit in Fig. 3.1, the condition for maximum input power transfer (thus power-gain) is Z_in = Z_s^∗ and that for the minimum noise figure is Z_s = Z_n,opt. Ideally, we have the condition for maximum power-gain and minimum noise figure is Z_in = Z_n,opt^∗ . From (5.1.1), (3.2.14), and (3.2.15), this condition can be satisfied if η is close to 1 and

1 g_m(Q²_f + 1)

C_out

C_gd = R_sn,opt. (3.2.17)

In the proposed technique of capacitive feedback matching network, the value of η is more close to 1 when compared to the inductive source-degeneration technique. Moreover, (3.2.17) can be satisfied by designing suitable device parameters g_m, C_gd, and C_gs of the device M1 as shown in Fig. 3.3, which are related to gate-source voltage V_gs and transistor size W/L. In other words, the proposed design technique and input stage in Fig. 3.1 help to maximize the power-gain and to minimize the noise figure simultaneously by bringing Z_in close to the optimum source noise conjugate impedance Z_n,opt^∗ .

It can be seen from (5.1.1) that without the feedback gate-drain capacitance C_gd, the input impedance of the common-source amplifier would have no real part. However, the optimum source noise impedance Z_n,opt in (3.2.14) has a real part. Then it is impossible to satisfy the condition Z_in = Z_n,opt^∗ . In the proposed technique, C_gd and the capacitive feedback matching network are used to satisfy the condition. Thus the technique is called the technique of capacitive feedback matching network.