Substituted with their expressions under the noise-canceling condition,(4)can be calculated as NF=1+ RF +y(2Rs RF)/4+RL (RF/Rs)2Rs Rs Rs (2Rs+1 2 =1+ Rs +RF RE (5) where y is a parameter greater than 1 for submicron MOSFET. B.Linearity Analysis Considering only the first-order deviation of the transcon- ductance from the square law and weakly nonlinear condition, Fig.2.Topology of a LNA exploiting noise-canceling technique. the drain current of Mi and M2 can be given by id1=91,1UA+91,2v7+91,3v月 (6) teristics,gain greater than 12 dB,NF less than 4 dB and IIP3 ia2=92,1vA+92,2w7+92,3v月 greater than 3 dBm within 50-860 MHz frequency range. =n(g1,1vA+g1,2v2+g1,3u) (7) III.PARAMETER ANALYSIS AND CIRCUIT DESIGN where gij means the jthorder distortion of the transcon- ductance of MOSFET Mi (for i 1,2 and j=1,2,3),and A.Noise-canceling Under Input/output Impedance Matching n =g2.j/g1.j =1+RF/Rs.The voltage of node B can be calculated by small-signal analysis as The LNA exploiting a noise-canceling technique [2]is shown in Fig.2.The primary purpose of this circuit is to UB =(1-91.1RF)vA -91.2RFvA-91.3RFUA (8) eliminate the noise contribution of M channel thermal-noise in.,which is a dominating noise source.It flowing through Then the input IP3 of the first stage is Rg and Rs causes two instantaneous noise voltages at nodes B and A with the same phase.Then the voltage of node -91,1RF AIP3.fs- 91,1 (9) A is amplified by a common-source stage M2 and node B 3 91,3RF 91,3 91,3RF is followed by a source-follow stage M3.Consequently,the Here,the first term is the non-linearity contribution of thermal noise of M could be counteracted at the output due the common-source topology M and the latter is that of to the opposite voltage gain sign of M2 and M3. the feedback resistor RF.Normally g1,3<0 when Mi is The input and output impedances can be calculated as R;= in saturation region,therefore the IP3 of the first stage is 1/gmi and Ro=1/gm3.respectively.And the load impedance decreased due to RF. is RL=Rs =501.Then the impedance matching condition The third-order distortion of M3 can be ignored owing to is the over-driven voltage of M3 is quite large (VGs3 >1V). 9m1 =9m3 =1/Rs (1) Therefore,the voltage at the output can be written as Considering the influence of load impedance RL,the noise- vo =-(ng1.1Rs+91.1RF-1)vA+91.2(nRs RF)vA canceling condition is +91,3(nRs+Rr)u]/2 (10) 9m2=(1+RF/Rs)/Rs (2) And the input IP3 of the total circuit is where Rp is the feedback resistor.Thus,under the impedance AIP3,total ng1,1Rs+91,1RF-1 matching and noise-canceling condition,the voltage gain from 3 91.3(nRs +RF) node A to output is 4 91.1 1 (11) Av Uo/VA=-RF/Rs (3) 391,3 91,3(Rs+2RF) According to [4],the noise figure (or noise factor)can be From (3),(5)and (11),it can be seen that Av,NF and IP3 represented as are only depended on the feedback resistor RF.High gain,low noise figure and high linearity can be achieved simultaneously NF=1+ 吃M十吃Rp十吃，M2,Mg十院RL when Rr is large enough.However,a large RF needs a large (4) A·4kTRs gm2 to meet the noise-canceling condition(2),and this would lead much greater power consumption.The relationships of where话，i,哈，，哈.wa,and疏-are the noise contribu-- Ay,NF,IIP3,Idd(total current)versus the feedback resistor tions at the output of Mi,RF,M2-M3 and RL,respectively. RF are shown in Fig.3. 260B I RF M1 A B M2 M3 Vdd RS S v o v RL Ri Ro 1 n M, Ci i Co Fig. 2. Topology of a LNA exploiting noise-canceling technique. teristics, gain greater than 12 dB, NF less than 4 dB and IIP3 greater than 3 dBm within 50-860 MHz frequency range. III. PARAMETER ANALYSIS AND CIRCUIT DESIGN A. Noise-canceling Under Input/output Impedance Matching The LNA exploiting a noise-canceling technique [2] is shown in Fig. 2. The primary purpose of this circuit is to eliminate the noise contribution of M1 channel thermal-noise in,M1, which is a dominating noise source. It flowing through RF and RS causes two instantaneous noise voltages at nodes B and A with the same phase. Then the voltage of node A is amplified by a common-source stage M2 and node B is followed by a source-follow stage M3. Consequently, the thermal noise of M1 could be counteracted at the output due to the opposite voltage gain sign of M2 and M3. The input and output impedances can be calculated as Ri = 1/gm1 and Ro = 1/gm3, respectively. And the load impedance is RL = RS = 50 Ω. Then the impedance matching condition is gm1 = gm3 = 1/RS (1) Considering the influence of load impedance RL, the noise￾canceling condition is gm2 = (1 + RF /RS)/RS (2) where RF is the feedback resistor. Thus, under the impedance matching and noise-canceling condition, the voltage gain from node A to output is AV = vo/vA = −RF /RS (3) According to [4] , the noise figure (or noise factor) can be represented as NF = 1 + v 2 n,M1 + v 2 n,RF + v 2 n,M2,M3 + v 2 n,RL A2 V · 4kT RS (4) where v 2 n,M1 , v 2 n,RF , v 2 n,M2,M3 and v 2 n,RL are the noise contribu￾tions at the output of M1, RF , M2-M3 and RL, respectively. Substituted with their expressions under the noise-canceling condition, (4) can be calculated as NF = 1 + RF + γ(2RS + RF )/4 + RL (RF /RS) 2RS = 1 + RS RF + γ 4 RS RF  2RS RF + 1 +  RS RF 2 (5) where γ is a parameter greater than 1 for submicron MOSFET. B. Linearity Analysis Considering only the first-order deviation of the transcon￾ductance from the square law and weakly nonlinear condition, the drain current of M1 and M2 can be given by id1 = g1,1vA + g1,2v 2 A + g1,3v 3 A (6) id2 = g2,1vA + g2,2v 2 A + g2,3v 3 A = n ￾ g1,1vA + g1,2v 2 A + g1,3v 3 A
(7) where gi,j means the j th-order distortion of the transcon￾ductance of MOSFET Mi (for i = 1, 2 and j = 1, 2, 3), and n = g2,j/g1,j = 1 + RF /RS. The voltage of node B can be calculated by small-signal analysis as vB = (1 − g1,1RF )vA − g1,2RF v 2 A − g1,3RF v 3 A (8) Then the input IP3 of the first stage is AIP 3,fs = s 4 3     1 − g1,1RF g1,3RF     = s 4 3     g1,1 g1,3 − 1 g1,3RF     (9) Here, the first term is the non-linearity contribution of the common-source topology M1 and the latter is that of the feedback resistor RF . Normally g1,3 < 0 when M1 is in saturation region, therefore the IP3 of the first stage is decreased due to RF . The third-order distortion of M3 can be ignored owing to the over-driven voltage of M3 is quite large (VGS3 > 1V). Therefore, the voltage at the output can be written as vo = − (ng1,1RS + g1,1RF − 1)vA + g1,2(nRS + RF )v 2 A +g1,3(nRS + RF )v 3 A /2 (10) And the input IP3 of the total circuit is AIP 3,total = s 4 3     ng1,1RS + g1,1RF − 1 g1,3(nRS + RF )     = s 4 3     g1,1 g1,3 − 1 g1,3(RS + 2RF )     (11) From (3), (5) and (11), it can be seen that AV , NF and IP3 are only depended on the feedback resistor RF . High gain, low noise figure and high linearity can be achieved simultaneously when RF is large enough. However, a large RF needs a large gm2 to meet the noise-canceling condition (2), and this would lead much greater power consumption. The relationships of AV , NF, IIP3, Idd (total current) versus the feedback resistor RF are shown in Fig. 3. 260