A Compact Wideband Two-Stage LNA with Multiple Notches for Out-of-Band Filtering Using Center-Tap Inductors

Abstract

In order to improve the out-of-band interference filtering in the front-end of a wideband radio frequency system, this paper proposes a two-stage low-noise amplifier (LNA) that has three notches in its transfer function. In the proposed LNA, two center-tap inductors are utilized, where the center-tap inductor is modeled as two equal inductors with a mutual coupling. Here, by properly designing these two inductors and their coupling, in the utilized two center-tap inductors, three notches and proper peaking in the transfer function of the LNA, as well as input matching, are simultaneously realized. The proposed LNA is fabricated in 65 nm CMOS, and its active area is 0.13 mm². The LNA shows three notches at 2.9 GHz, 13.1 GHz and 17.2 GHz. Moreover, the 3 dB BW is from 5.4 GHz to 8.7 GHz, sufficiently wide to operate in the worldwide ultra-wideband (UWB) regulated spectrum. The proposed LNA power consumption is 9.9 mW from a 1.1 V supply, with a minimum noise figure of 3.4 dB and a gain of 24.8 dB.

Appendix A

This appendix shows the details of the KCL equations to derive the input–output transfer function of the 1^st stage of the proposed LNA. Considering the circuit model of the 1^st LNA stage, shown in Fig. 3, the following can be written:

$$ k_{1} = M_{1} /L_{1} $$

$$ v_{1} = L_{1} \cdot s \cdot i_{1} + M_{1} \cdot s \cdot i_{2} $$

$$ v_{2} = M_{1} \cdot s \cdot i_{1} + L_{1} \cdot s \cdot i_{2} $$

$$ v_{1} = v_{i} - v_{gs1} $$

$$ v_{2} = v_{gs1} - v_{x} $$

$$ i_{1} = i_{2} + C_{gs1} \cdot s \cdot v_{gs1} + C_{gd1} \cdot s\left( {v_{gs1} - v_{o1} } \right) $$

$$ i_{2} = \left( {v_{x} - v_{o1} } \right)/R_{F} + C_{ind1} \cdot s\left( {v_{x} - v_{i} } \right) + C_{B1} \cdot s \cdot v_{x} $$

$$ g_{m1} v_{gs1} + \left( {C_{L1} .s + r_{o1}^{ - 1} } \right)v_{o1} + \left( {v_{o1} - v_{x} } \right)/R_{F} + C_{gd1} .s\left( {v_{o1} - v_{gs1} } \right) = 0 $$

$$ v_{in} = R_{S} \cdot i_{in} + v_{i} $$

$$ i_{in} = i_{1} + C_{ind} \cdot s\left( {v_{i} - v_{x} } \right) + C_{p} \cdot s \cdot v_{i} $$

Note that, here, k₁ is the coupling factor between the two inductors in L₁ and L₂, and M₁ and M₂ are the resulting mutual inductances. Moreover, R_S is the source impedance of 50 Ω. The resulting transfer functionV_o1/V_in has 6 poles and 4 zeros. Considering a third-order s-domain transfer function for the numerator and denominator results in:

$$ A_{V1} = \frac{{v_{o1} }}{{v_{in} }} = \frac{{a_{3} s^{3} + a_{2} s^{2} + a_{1} s + a_{0} }}{{b_{3} s^{3} + b_{2} s^{2} + b_{1} s + b_{0} }} $$

where

$$ a_{3} = R_{F} r_{o1} \left( {2C_{gd1} C_{ind1} + C_{gd1} C_{B1} } \right)\left( {L_{1} + M_{1} } \right) $$

$$ \begin{aligned} a_{2} & = r_{o1} \left[ L_{1} \left( {C_{gd1} + 2C_{ind1} } \right) + M_{1} \left( {2C_{ind1} - C_{gs1} } \right) \right. \\ &\quad \left. - g_{m1} R_{F} \left( {L_{1} + M_{1} } \right)\left( {2C_{ind1} + C_{B1} } \right) \right] \end{aligned} $$

$$ a_{1} = r_{o1} \left( {R_{F} C_{gd1} - g_{m1} M_{1} + g_{m1} L_{1} } \right) $$

$$ a_{0} = r_{o1} \left( {1 - g_{m1} R_{F} } \right) $$

$$ b_{3} = \left( {L_{1} ^{2} - M_{1} ^{2} } \right)\left( {C_{{gd1}} + C_{{gs1}} + C_{{gd1}} r_{{o1}} g_{{m1}} } \right) + ~L_{1} r_{{o1}} R_{S} [C_{{L1}} \left( {C_{{g~d1}} + c_{{gs1}} + 2C_{{ind1}} + 2C_{P} } \right) + C_{{gd1}} \left( {C_{{gs1}} + C_{P} + C_{{B1}} } \right) + C_{{gs1}} \left( {2C_{{ind1}} + C_{P} + C_{{B1}} } \right) + 2C_{{ind1}} \left( {C_{P} + C_{{B1}} } \right) + 2C_{P} C_{{B1}} ] + L_{1} r_{{o1}} R_{F} \left[ {C_{{L1}} (C_{{gd1}} + C_{{gs1}} + 2C_{{ind1}} + 2C_{{B1}} ) + C_{{gd1}} \left( {C_{{gs1}} + 2C_{{ind1}} + 2C_{{B1}} } \right)} \right] + L_{1} R_{S} R_{F} \left[ {(2C_{{ind1}} + C_{P} + C_{{B1}} )\left( {C_{{gd1}} + C_{{gs1}} } \right) + 2C_{{ind1}} \left( {C_{P} + C_{{B1}} } \right) + 2C_{P} C_{{B1}} } \right] + 2M_{1} r_{{o1}} R_{S} \left[ {C_{{L1}} \left( {C_{{ind1}} + C_{P} } \right) + C_{{ind1}} \left( {C_{{gs1}} + C_{P} + C_{{B1}} } \right) + C_{P} C_{{B1}} } \right] + 2M_{1} r_{{o1}} R_{F} \left[ {C_{{L1}} \left( {C_{{ind1}} + C_{{B1}} } \right) + C_{{gd1}} (C_{{ind1}} + C_{{B1}} )} \right] + 2M_{1} R_{S} R_{F} \left[ {C_{{ind1}} \left( {C_{{gd1}} + C_{{gs1}} + C_{P} + C_{{B1}} } \right) + C_{P} C_{{B1}} } \right] + ~2M_{1} r_{{o1}} R_{S} R_{F} g_{{m1}} C_{{gd1}} C_{{ind1}} + L_{1} r_{{o1}} R_{S} R_{F} g_{{m1}} C_{{gd1}} \left( {2C_{{ind1}} + C_{{B1}} + C_{P} } \right) $$

$$ \begin{aligned} b_{2} = & L_{1} r_{o1} \left( {2C_{L1} + C_{gd1} + C_{gs1} + 2C_{ind1} + 2C_{B1} } \right) \\ &\quad + L_{1} R_{S} \left( {C_{gd1} + C_{gs1} + 2C_{ind1} + 2C_{P} } \right) \\ & + L_{1} R_{F} \left( {C_{gd1} + C_{gs1} + 2C_{ind1} + 2C_{B1} } \right) + 2M_{1} r_{o1} \left( {C_{L1} + C_{ind1} + C_{B1} } \right) \\ & + 2R_{S} M_{1} \left( {C_{ind1} + C_{P} } \right) + 2R_{F} M_{1} \left( {C_{ind1} + C_{B1} } \right) \\ & + L_{1} r_{o1} R_{S} g_{m1} \left( {C_{gd1} + 2C_{ind1} + C_{P} } \right) + L_{1} r_{o1} R_{F} g_{m1} C_{gd1} \\ & + M_{1} r_{o1} R_{S} g_{m1} \left( {2C_{ind1} + C_{P} } \right) + r_{o1} R_{S} R_{F} C_{L1} \left( {C_{gd1} + C_{gs1} + C_{P} + C_{B1} } \right) \\ & + r_{o1} R_{S} R_{F} C_{gd1} \left( {C_{gs1} + C_{P} + C_{B1} } \right) \\ \end{aligned} $$

$$ \begin{aligned} b_{1} = & 2\left( {L_{1} + M_{1} } \right) + r_{o1} g_{m1} \left( {L_{1} + M_{1} } \right) \\ &\quad + r_{o1} R_{S} \left( {C_{L1} + C_{gs1} + C_{P} + C_{B1} } \right) + r_{o1} R_{F} \left( {C_{L1} + C_{gd1} } \right) \\ & + R_{S} R_{F} \left( {C_{gd1} + C_{gs1} + { }C_{P} + C_{B1} } \right) + R_{S} R_{F} r_{o1} g_{m1} C_{gd1} \\ \end{aligned} $$

$$ b_{0} = R_{F} + R_{S} + r_{o1} + R_{S} r_{o1} g_{m1} $$