The circuit can be characterized by the Equations (14) and (15).

(14)

(15)

By eliminating v_{s}_{1} in (14) and (15), gY_{out21} is expressed as

(16)

Then gY_{out2} =1/gZ_{out2} is expressed as

. (17)

By definition, A_{1} corresponds to the voltage gain v_{s}_{2o}/v_{g}_{1}, where v_{s}_{2o} is the v_{s}_{2} node voltage when open, and can be derived using the equivalent circuit shown in Figure 7. The circuit can be characterized by the Equations (18) and (19).

Figure 6. AC equivalent circuit to find gZ_{out2}.

Figure 7. AC equivalent circuit to find A_{1}.

(18)

(19)

By eliminating v_{s}_{1} in (18) and (19), we get

(20)

where and

.

gZ_{out1} corresponds to the impedance seen to the left of the v_{o} node with v_{s}_{2} = 0 in Figure 2. Since g_{m}_{2}v_{s}_{2} and (C_{gs}_{2} + C_{js}_{2}) do not function when v_{s}_{2} = 0, gY_{out1} = 1/gZ_{out1} is simply expressed as

. (21)

A_{2} corresponds to the voltage gain v_{oo}/v_{s}_{2}, where v_{oo} is the v_{o} node voltage when open, and A_{2} derivation can be done in the similar fashion to the one for A_{1} derivation. The resulting A_{2} is expressed as

. (22)

In Figure 2, the available input power P_{i}, which is supplied to the LNA when impedance matched, is defined as

. (23)

The maximum output power P_{o}, which is supplied to the resistive load R_{so} when impedance matched, is expressed as

, (24)

where v_{o} and v_{out} are defined in Figure 2, and R_{p} is the transformed parallel resistance of R_{so}, which is already defined relating (1).

Then the available power gain G is expressed as

(25)

where A_{v}_{1}, A_{v}_{2}, and A_{v}_{3} can be easily derived from Figure 5 as follows.

(26)

(27)

(28)

4. Automatic Sizing Algorithm

Figure 8 shows the automatic sizing algorithm developed in this work. The inputs to the algorithm include design and process specifications, and the outputs include synthesized design variable values are for R_{DB}, W, nfb, L_{s}, L_{g}, C_{i}, R_{1}, L_{1}, C_{o}. Here, we explain the procedures from top to bottom in accordance with each step, which is explicitly indicated in Figure 8.

4.1. 1^{st} Step: Entering Design and Process Specifications

The 1^{st} step in the automatic sizing is to enter the design

Figure 8. Automatic sizing algorithm.

and process specifications. The design specifications include the operating frequency f, the input output terminations R_{si} and R_{so}, the supply current I_{DD}, the desired power gain Gain_design. Instead of I_{DD}, the power consumption PWR and the supply voltage V_{DD} can be entered to calculate I_{DD} by PWR/V_{DD}. The process specifications include the transistor channel length L, the transistor channel width per finger WF, and the maximum finger number nf_max defined for one unit of transistors.

4.2. 2^{nd} Step: Calculation of Optimum Transistor Width

The next step is to calculate the transistor channel width W for optimum noise performance. The width for optimum noise performance is usually too large for practical use, and therefore the power-constrained noise optimization (PCNO) device width W_{optP} [11] is adopted as W in this work. W_{optP} is calculated according to the last rough equation in (29).

(29)

As shown in (29), W_{optP} increases continuously as the frequency decreases. Therefore it may be necessary to define a maximum value for W considering lower frequency design. We suggest to limit W below 1000 μm.

If W_{F} and nf_max are defined, the finger number nf is first calculated as W/W_{F}, and the number of the maximum-fingered units m is calculated as the integer value of nf/nf_max, and the residual finger number nf_residue is determined as the residue to give an information for the transistor layout. Then the final W is determined by W = W_{F} × (m × nf_max + nf_residue). We note that W_{F} and nf_max are usually defined in most of recent processes.

4.3. 3^{rd} Step: Calculation of Bias Circuit Design Variables and Getting DC Operating Point Information

The next step is to determine the bias circuit variable values and to get the dc operating point information.

The finger number for the bias transistor nfb and the drain bias resistance R_{DB} in Figure 1 should be determined. By limiting the bias circuit current around 100 μA, for example, we can determine nfb by nfb = (100 μA/I_{DD}) × nf. For the decoupling resistor R_{B}, we can simply use 5 kΩ, which is a reasonable value.

The next procedure is to determine R_{DB}, which, however, is very difficult to determine by calculation. Since I_{DD} is sensitive to the value of R_{DB}, it should be manually determined to give the specified I_{DD} value by dc circuit simulations. This procedure is one obstacle against full design automation in this work. However, it is an essential procedure since it provides the accurate operating point information to proceed with the remaining part of the design automation. The needed operating point information include the values of g_{m}, g_{ds}, C_{gs}, C_{sg}, C_{gd}, C_{dg}, C_{ds}, C_{sd}, C_{js}, and C_{jd} of M_{1} and M_{2} in Figure 1, which should be imported into the automatic sizing algorithm.

4.4. 4^{th} Step: Iterations to Determine Design Variable Values

There are three main iteration loops in the automatic sizing algorithm as shown in Figure 8. The 1^{st} loop finds G_{max}, which corresponds to the case with the upper limit of R_{1}, which is chosen arbitrarily large enough as 10 kΩ in this work. To find G_{max}, we need to find all the design variable values for the G_{max} case simultaneously. Iteration is needed since the input and output matching designs affect each other. The 2^{nd} loop finds G_{min}, which corresponds to the case with the lower limit of R_{1}, which is arbitrarily chosen small as 40 Ω in this work to allow a larger allowable gain range. This iteration is also needed for the same reason explained for the G_{max} case. The 3^{rd} loop finds the proper R_{1} value for the desired gain Gain_ design by the bisection method, which lies within the lower and upper boundaries G_{min} and G_{max}, and its inner loop finds the corresponding design variable values for the present gain value during iteration similarly as in the 1^{st} and 2^{nd} iteration loops.

4.4.1. Iterations to Solve for the G_{max} Case

As explained above, Z_{in1} is affected by output matching design, and Z_{out} is affected by input matching design. Therefore we need some iteration to determine L_{s}. Since Z_{in2} is affected by Z_{o}, which is unknown yet, we need an initial guess for Z_{o} to find the 1^{st} L_{s} value. As shown in Figure 8, an initial guess for Z_{oL} = Z_{o}//(1/sC_{L}) is given as 50/g·m^{2}, which is shown to be large enough for all possible situations in the procedure, to solve for Z_{in2} by (5).

The impedance seen at the gate of M_{1} is equal to Z_{in1}, which is derived in (9). By setting the real part of Z_{in1} Re(Z_{in1}) equal to R_{si} for input impedance matching, we can find L_{s}. However this equation Re(Z_{in1}) = R_{si} is too complicated to get the solution directly with the other present design variables values given, and therefore L_{s} is solicited numerically within the lower and upper boundaries of 0.1 nH and 5 nH. We use the bisection method for this purpose.

The next procedure is to calculate L_{g} and C_{i}, which nullify the imaginary part of Z_{in1} Im(Z_{in1}) in Figure 2. Z_{in1} is usually capacitive to give a negative value for Im(Z_{in1}), and therefore L_{g} can be calculated using the equation Im(Z_{in1}) – 1/(ωC_{i}) + ωL_{g} = 0, where C_{i} is simply a large dc blocking capacitor. We first calculate L_{g}_{1}, which nullifies Im(Z_{in1}) using Im(Z_{in1}) + ωL_{g}_{1} = 0. Although C_{i} is larger the better, considering the layout size, 1/(ωC_{i}) = ωL_{g}_{1}/10 is used to determine C_{i}. L_{g} is then recalculated using Im(Z_{in1}) – 1/(ωC_{i}) + ωL_{g} = 0.

Depending on to the operating frequency and the desired gain, Z_{in1} may happen to be inductive, or this situation can happen in the middle of the iterations. For this case, a nominal single bond wire inductance of 1 nH is assumed for L_{g} and Im(Z_{in1}) – 1/ωC_{i} + ωL_{g} = 0 is used to calculate the required C_{i} value.

In the next procedure, the design variables L_{1} and C_{o} are determined using the equations Re(Z_{out}) = R_{so} and Im(Z_{out}) = 0 for output impedance matching to R_{so}, where Re(Z_{out}) is the real part of Z_{out} expressed in (13).

If we let Z_{out1} in (12) equal to A + jB, the real and imaginary parts of Z_{out1}//jωL_{1} in (13) are expressed as

(30)

Then by letting Re(Z_{out}) = Re(Z_{out1}//jωL_{1}) = R_{so}, L_{1} is expressed as

(31)

By letting Im(Z_{out}) = Im(Z_{out1}//jωL_{1}) – 1/(ωC_{o}) = 0, C_{o} is expressed as

. (32)

Using (31) and (32), L_{1} and C_{o} can be simply calculated.

Now the 1^{st} set of the design variable values are ready to update Z_{oL} and the remaining iterations are performed to find the final design variable values for the G_{max} case. It was found that the iteration number for this loop should be larger than 10.

Right after the iteration loop, A_{1}, gZ_{out2}, A_{2}, and gZ_{out1} are calculated using (20), (17), (22), and (21), respectively, and G_{max} is calculated using (25).

If the G_{max} value is smaller than the desired gain, the routine gives a warning and stops.

4.4.2. Iterations to Solve for the G_{min} Case

The 2^{nd} loop finds the design variable values for the G_{min} case. The same iteration as above with the last Z_{oL} value as an initial guess is performed to find G_{min} using (25) again.

4.4.3. Iterations to Solve for the Gain_Design Case

The 3^{rd} loop finds the proper R_{1} value for the desired gain Gain_design using the bisection method while the inner loop finds the corresponding design variable values for the present gain value. This inner iteration loop is exactly same as the 1^{st} and 2^{nd} loops. After all the design variables are determined for the present gain value, the gain is calculated using (25) again. If the calculated gain is equal to Gain_design within the allowed tolerance, the calculation stops to output the final set of the design variable values, which include W, nf, m, nf_residue, nfb, L_{s}, L_{g}, C_{i}, R_{1}, L_{1}, and C_{o}.

5. Verifications

The automatic sizing algorithm explained in Section 4 was coded using Matlab (Version 7.9.0.529) assuming usage of a 90 nm commercial CMOS process. The design variable sets for seven different operating frequencies ranging from 0.7 GHz to 5 GHz were synthesized, and verifications were done by one-time Spectre circuit simulations with the corresponding BSIM4.5.0 MOSFET model [12] for the assumed process.

The design specifications include I_{D} = 5 mA, V_{DD} = 1.2 V, Gain_design = 21 dB, and R_{si} = R_{so} = 50 Ω. The process specifications include L = 75 nm, W_{F} = 3 μm, and nf_max = 64, where 75 nm for L is the effective channel length in this process. The maximum transistor width was set as W_{max} = nf_max × m × W_{F} = 64 × 5 × 3 μm = 960 μm, which is below 1000 μm as we suggested.

As examples of the verifications, Figures 9 and 10 show the simulated LNA characteristics without any tuning for the operating frequency of 1 GHz and 5 GHz, respectively, when the corresponding sets of the design variable values obtained using the automatic sizing algorithm are used for the simulations. The synthesized design variable values are as follows;

For 1 GHz design, R_{DB} = 12.7 kΩ, W = 960 μm (m = 5, nf_residue = 0), nfb = 6, L_{s} = 1.382 nH, L_{g} = 19.557 nH, C_{i} = 14.25 pF, R_{1} = 497.1 Ω, L_{1} = 11.904 nH, C_{o} = 1.447 pF.

For 5 GHz design, R_{DB} = 5.96 kΩ, W = 231 μm (m = 1, nf_residue = 13), nfb = 2, L_{s} = 0.5383 nH, L_{g} = 2.690 nH, C_{i} = 4.142 pF, R_{1} = 1.752 kΩ, L_{1} = 2.813 nH, C_{o} = 0.190 pF.

Table 1 summarizes the simulated results of the seven designs, which reside in the frequency range, where the automatic sizing program could provide the design variable set for Gain_design of 21 dB. Notice that, for the operating frequencies below 1 GHz, the synthesized W values are restricted to below 960 μm, which is equal to the value for W_{max}.

In Table 1, we can see that the input and output matchings (S_{11} and S_{22}) are pretty good for all the designs, and the noise figure is pretty close to the noise figure minimum, which demonstrates the adequacy of the designs.

We note that power gain values are about the same with S_{21} values. The S_{21} values in Table 1 are smaller than the desired gain of 21 dB. This seems to be caused by neglecting g_{mb}, C_{gb}, R_{s}, R_{d}, R_{g}, and R_{sub} in the equivalent circuit in Figure 2. However we believe that the result is pretty good for the first-cut quick design.

(a)(b)

Figure 9. Simulated (a) s parameter and (b) noise characteristics for f = 1 GHz: S_{21} = 20.31 dB, NF = 0.660 dB, NF_{min} = 0.585, S_{11} = –23.6 dB, S_{22} = –23.0 dB.

Figure 10. Simulated s parameters for f = 5 GHz: S_{21} = 17.16 dB, S_{11} = –16.9 dB, S_{22} = –34.8 dB.

Table 2 summarizes the synthesized available gain ranges with the corresponding R_{1} values for each design. We can see that a wide range of power gain can be obtained by varying the R_{1} values as expected.

6. Conclusions

The analytical expressions for the principle parameters

Table 1. Simulation summary for the desired gain Gain_ design of 21 dB.

Table 2. Synthesis summary for the available gain ranges with the corresponding R_{1} values.

were derived using the ac equivalent circuit of the singleended narrow-band cascode CMOS LNA adopting the inductive source degeneration. Based on the expressions, the automatic sizing algorithm was developed by adopting the power-constrained noise optimization criteria. The algorithm was coded using Matlab, and could provide a set of design variable values within seconds. One-time Spectre simulations without any tuning assuming usage of a commercial 90 nm CMOS process were performed to confirm that the automatic sizing program can synthesize the aimed first-cut design with a reasonable accuracy for the frequency range reaching up to 5GHz.

This work showed in detail how the accurate automatic sizing can be done in an analytical approach. The approach can be applied to a common source LNA more easily since the derivation of principal parameters will be simpler with a fewer gain stages. It can be also applied to a differential LNA easily since the derivation will be basically same. The approach seems applicable to more complicated designs even though the derivation procedures will contain enhanced complexity.

The automatic sizing program may be utilized efficiently for additional tuning purpose. For example, after examining the first-cut synthesis result with verifying circuit simulations, a smaller value for W_{M}_{2} compared to the synthesized one for W_{M}_{1} can be entered into the automatic sizing program to obtain another design variable set for better linearity.

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NOTES

^{*}This work was supported by 2011 Hongik University Research Fund (sponsors).