Design of High Linearity Inductor-Less Active CMOS Mixer Based on Volterra Series Analysis

Abstract

In the present paper, a linearization technique is proposed for double-balanced active inductor-less mixer based on the interaction of two nonlinear approaches. The proposed procedure utilizes multiple-gate and second harmonic injection techniques in order to simultaneously enhance the third-order intercept point (IIP3) and second-order intercept point (IIP2). The full Volterra series analysis of transconductance stage of the proposed mixer is presented to demonstrate the effective technique in detail. Simulations are performed utilizing a TSMC $0.18\, \upmu \hbox {m}$ CMOS technology. Compared with the conventional Gilbert cell mixer, the simulations suggest improvements of 18.6 dBm and 54 dBm in IIP3 and IIP2 of the proposed mixer, respectively. The mixer has a conversion gain of 20.3 dB while 3.21 mA is drawn from a 1.8 V power supply.

Appendices

Appendices

1.1 A: Expanded Equation in Multiple-Gate Transistors

The drain current of M1 and M2 in Fig. 2 can be expressed as:

$$\begin{aligned} I_{\mathrm{D}1}= & {} \frac{1}{2}\mu _{0}C_{\mathrm{ox}}\frac{W_{1}}{L_{1}}\frac{\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2}{1 + \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{1}} + \theta \Big ) + \big ((V_{\mathrm{GS}1} + v_{\mathrm{in}}) - V_{\mathrm{th}}\big )} \end{aligned}$$

(29)

$$\begin{aligned} I_{\mathrm{D}2}= & {} \frac{1}{2}\mu _{0}C_{\mathrm{ox}}\frac{W_{2}}{L_{2}}\frac{\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2}{1 + \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{2}} + \theta \Big ) + \big ((V_{\mathrm{GS}2} + v_{\mathrm{in}}) - V_{\mathrm{th}}\big )} \end{aligned}$$

(30)

where $\mu _{0}$ denotes the zero-field mobility and $V_{\mathrm{sat}}$ and $\theta$ are the saturation velocity of carrier and the effect of the vertical field, respectively. If the second term in the denominator remains much less than unity, it is possible to rewrite Eqs. (29) and (30) as, respectively, Eqs. (31) and (32) using the approximation: $(1+\varepsilon )^{-1} \approx 1 - \varepsilon + \varepsilon ^2$ [1]

$$\begin{aligned} I_{\mathrm{D}1}= & {} \frac{1}{2}\mu _{0}C_{ox}\frac{W_{1}}{L_{1}}\Bigg [\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2 \nonumber \\&-\, \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{1}} + \theta \Big )\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^3 \nonumber \\&+\, \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{1}} + \theta \Big )\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^4\Bigg ] \end{aligned}$$

(31)

$$\begin{aligned} I_{\mathrm{D}2}= & {} \frac{1}{2}\mu _{0}C_{\mathrm{ox}}\frac{W_{2}}{L_{2}}\Bigg [\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2 \nonumber \\&-\, \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{2}} + \theta \Big )\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big ) ^3 \nonumber \\&+\, \Big (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{2}} + \theta \Big )\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^4\Bigg ] \end{aligned}$$

(32)

For simplicity, Eqs. (31) and (32) can be rewritten as Eqs. (33) and (34), respectively:

$$\begin{aligned} I_{\mathrm{D}1}= & {} k_{1}\Bigg [\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2 \nonumber \\&- \,a_{1}\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^3 \nonumber \\&+ \,a_{1}\big ((V_{\mathrm{GS}1} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^4\Bigg ] \end{aligned}$$

(33)

$$\begin{aligned} I_{\mathrm{D}2}= & {} k_{2}\Bigg [\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^2 \nonumber \\&-\, a_{2}\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^3 \nonumber \\&+\, a_{2}\big ((V_{\mathrm{GS}2} + v_{\mathrm{in}})- V_{\mathrm{th}}\big )^4\Bigg ] \end{aligned}$$

(34)

where $k_{1}=\frac{1}{2}\mu _{0}C_{\mathrm{ox}}\frac{W_{1}}{L_{1}}$, $k_{2}=\frac{1}{2}\mu _{0}C_{\mathrm{ox}}\frac{W_{2}}{L_{2}}$, $a_{1}=(\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{1}} + \theta )$ and $a_{2}= (\frac{\mu _{0}}{2V_{\mathrm{sat}}L_{2}} + \theta )$. Now, by applying KCL at the drains of transistors, one obtains:

$$\begin{aligned} I_{\mathrm{D}} = I_{\mathrm{D}1} + I_{\mathrm{D}2} \end{aligned}$$

(35)

By summing $I_{\mathrm{D}1}$ and $I_{\mathrm{D}2}$, and classification of $I_{\mathrm{D}}$, the coefficients A1, A2, and A3 can be given as follows as well as what is shown in Eqs. (5)–(7) in Sect. 3.

$$\begin{aligned} I_{\mathrm{D}}= & {} I_{\mathrm{DC}} + v_{\mathrm{in}}\Bigg [2k_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}}) - 3k_{1}a_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}})^2 \nonumber \\&+\, 4k_{1}a_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}})^3 + 2k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}}) \nonumber \\&-\, 3k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}})^2 + 4k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}})^3\Bigg ] \nonumber \\&+\, v_{\mathrm{in}}^2\Bigg [k_{1} - 3k_{1}a_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}}) + 6k_{1}a_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}})^2 \nonumber \\&+\, k_{2} - 3k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}}) + 6k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}})^2\Bigg ] \nonumber \\&+\, v_{\mathrm{in}}^3\Bigg [-k_{1}a_{1} + 4k_{1}a_{1}(V_{\mathrm{GS}1} - V_{\mathrm{th}}) \nonumber \\&-\, k_{2}a_{2} + 4k_{2}a_{2}(V_{\mathrm{GS}2} - V_{\mathrm{th}})\Bigg ] \end{aligned}$$

(36)

1.2 B: Expanded Volterra Series Analysis of Transconductance Stage

The drain current of $M_{1b}$ and $M_{2b}$ in Fig. 3 can be expressed as:

$$\begin{aligned} I_{\mathrm{D}_{1b}}= & {} g_{\mathrm{m}_{1b}}\big (0 - v_{\mathrm{in}}\big ) + g_{\mathrm{m}_{1b}}^{'}\big (0 - v_{\mathrm{in}}\big )^2 + g_{\mathrm{m}_{1b}}^{''}\big (0 - v_{\mathrm{in}}\big )^3 \end{aligned}$$

(37)

$$\begin{aligned} I_{\mathrm{D}_{2b}}= & {} g_{\mathrm{m}_{2b}}\big (0 - (-v_{\mathrm{in}})\big ) + g_{\mathrm{m}_{2b}}^{'}\big (0 - (-v_{\mathrm{in}})\big )^2 \nonumber \\&+\, g_{\mathrm{m}_{2b}}^{''}\big (0 - (-v_{\mathrm{in}})\big )^3 \end{aligned}$$

(38)

Now, by assuming $g_{\mathrm{m}_{1b}} = g_{\mathrm{m}_{2b}}$, $g_{\mathrm{m}_{1b}}^{'} = g_{\mathrm{m}_{2b}}^{'}$, and $g_{\mathrm{m}_{1b}}^{''} = g_{\mathrm{m}_{2b}}^{''}$, and applying KCL at the drain node of transistors, $I_{\mathrm{Dt}}$ (as shown in Fig. 3) can be obtained as:

$$\begin{aligned} I_{\mathrm{Dt}} = I_{\mathrm{D}_{1b}} + I_{\mathrm{D}_{2b}} = 2g_{\mathrm{m}_{1b(2b)}}v_{\mathrm{in}}^2 \end{aligned}$$

(39)

Now, by injecting $I_{\mathrm{Dt}}$ current into the RLC network, $V_{\mathrm{D}}$ is obtained as:

$$\begin{aligned} V_{\mathrm{D}} = I_{\mathrm{D}} \times Z_{\mathrm{D}} \end{aligned}$$

(40)

where $Z_{\mathrm{D}} = R_{\mathrm{D}} \parallel j\omega L_{\mathrm{D}} \parallel \frac{1}{j\omega C_{\mathrm{D}}}$, thus:

$$\begin{aligned} V_{\mathrm{D}} = \frac{2g_{\mathrm{m}_{1b}}^{'} \times j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}}R_{\mathrm{D}}}{R_{\mathrm{D}} + j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}} - (\pm \omega _{1} \mp \omega _{2})^2R_{\mathrm{D}}L_{\mathrm{D}}C_{\mathrm{D}}} \times v_{\mathrm{in}}^2 \end{aligned}$$

(41)

The voltage at the output node of subsystem B is related to the input voltage signal. Accordingly, it can be expressed based on Volterra series as follows:

$$\begin{aligned} V_{\mathrm{out}} = B_{1}v_{\mathrm{in}} + B_{2}v_{\mathrm{in}}^{2} + B_{3}v_{\mathrm{in}}^{3} \end{aligned}$$

(42)

where $B_{1}$, $B_{2}$, and $B_{3}$ are Volterra kernels of $V_{\mathrm{out}}$. The goal is to obtain $B_{1}$, $B_{2}$, and $B_{3}$. Hence by applying KCL at the output node ($I_{3} = I_{1} - I_{2}$), these currents can be obtained as:

$$\begin{aligned} I_{1}= & {} g_{\mathrm{m}_{3b}}\big (0 - v_{\mathrm{in}}\big ) + g_{\mathrm{m}_{3b}}^{'}\big (0 - v_{\mathrm{in}}\big )^2 + g_{\mathrm{m}_{3b}}^{''}\big (0 - v_{\mathrm{in}}\big )^3 \end{aligned}$$

(43)

$$\begin{aligned} I_{2}= & {} g_{\mathrm{m}_{5b}} \times V_{\mathrm{D}} \nonumber \\= & {} \frac{ g_{\mathrm{m}_{5b}} \times 2g_{\mathrm{m}_{1b}}^{'} \times j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}}R_{\mathrm{D}}}{R_{\mathrm{D}} + j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}} - (\pm \omega _{1} \mp \omega _{2})^2R_{\mathrm{D}}L_{\mathrm{D}}C_{\mathrm{D}}} \times v_{\mathrm{in}}^2 \end{aligned}$$

(44)

$$\begin{aligned} I_{3}= & {} \frac{V_{\mathrm{out}}}{Z_{\mathrm{L}}} \end{aligned}$$

(45)

where $Z_{\mathrm{L}} = R_{\mathrm{out}} \parallel \frac{1}{j\omega C_{\mathrm{P}}}$ and $R_{\mathrm{out}} = r_{\mathrm{o}_{3b}} \parallel r_{\mathrm{o}_{5b}}$. Substituting of (43)–(45) in $I_{3} = I_{1} - I_{2}$ and grouping the variables, the coefficients of $v_{\mathrm{in}}$, $v_{\mathrm{in}}^{2}$, and $v_{\mathrm{in}}^{3}$, i.e., $B_{1}$, $B_{2}$, and $B_{3}$, can be expressed according to Eqs. (15)–(17).

The output current of subsystem A is related to the input voltage signal. Thus, it can be expressed based on Volterra series as follows:

$$\begin{aligned} I_{\mathrm{out}} = H_{1}v_{\mathrm{in}} + H_{2}v_{\mathrm{in}}^2 + H_{3}v_{\mathrm{in}}^3 \end{aligned}$$

(46)

where $H_{1}$, $H_{2}$, and $H_{3}$ are Volterra kernels of output current. $I_{\mathrm{out}}$ can be obtained as:

$$\begin{aligned} I_{\mathrm{out}} = A_{1}v_{\mathrm{out}} + A_{2}v_{\mathrm{out}}^2 + A_{3}v_{\mathrm{out}}^3 \end{aligned}$$

(47)

Now, by substituting Eq. (42) in (47) as well as using Eqs. (15)–(17), and (5)–(7) instead of $A_{1}$, $A_{2}$, and $A_{3}$, Eq. (47) can be rewritten as:

$$\begin{aligned} \begin{aligned} I_{\mathrm{out}} =&g_{\mathrm{m}_{T1}}(B_{1}v_{\mathrm{in}} + B_{2}v_{\mathrm{in}}^2 + B_{3}v_{\mathrm{in}}^3) \\&+ \,g_{\mathrm{m}_{T1}}^{'}(B_{1}v_{\mathrm{in}} + B_{2}v_{\mathrm{in}}^2 + B_{3}v_{\mathrm{in}}^3)^2 \\&+ \,g_{\mathrm{m}_{T1}}^{''}(B_{1}v_{\mathrm{in}} + B_{2}v_{\mathrm{in}}^2 + B_{3}v_{\mathrm{in}}^3)^3 \end{aligned} \end{aligned}$$

(48)

Now, all Volterra kernels obtained in Eqs. (5)–(7) and (15)–(17) should be substituted into Eq. (48) in order to achieve Volterra kernels of the output current, as given in Eqs. (19)–(21).

$M_{1c}-M_{4c}$ are added to the circuit to enhance IIP2.These transistors affect $H_{2}$ (second coefficient of Volterra kernels of $I_{\mathrm{out}}$). Figure 4 shows these added transistors. $I_{\mathrm{out}}^{'}$ can be expressed as:

$$\begin{aligned} \begin{aligned} V_{\mathrm{D}}^{'}&= -g_{\mathrm{m}_{1c}} \times V_{\mathrm{D}} \times \frac{R_{\mathrm{C}}}{1 + j(\pm \omega _{1} \mp \omega _{2}) C_{p2}R_{\mathrm{C}}} \\&= \frac{ g_{\mathrm{m}_{1c}} \times 2g_{\mathrm{m}_{1b}}^{'} \times j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}}R_{\mathrm{D}}}{R_{\mathrm{D}} + j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}} - (\pm \omega _{1} \mp \omega _{2})^2R_{\mathrm{D}}L_{\mathrm{D}}C_{\mathrm{D}}} \\&\quad \times \, \frac{R_{\mathrm{C}}}{1 + j(\pm \omega _{1} \mp \omega _{2}) C_{p2}R_{\mathrm{C}}} v_{\mathrm{in}}^2 \end{aligned} \end{aligned}$$

(49)

where $C_{p2}$ is the total parasitic capacitance of the drain and gate of $M_{1c}$ and $M_{3c}$. Thus, $I_{\mathrm{out}}^{'}$ can be expressed as:

$$\begin{aligned} \begin{aligned} I_{\mathrm{out}}^{'}&= -g_{\mathrm{m}_{3c}} \times V_{\mathrm{D}}^{'} \\&= \frac{-g_{\mathrm{m}_{3c}} g_{\mathrm{m}_{1c}} \times 2g_{\mathrm{m}_{1b}}^{'} \times j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}}R_{\mathrm{D}}}{R_{\mathrm{D}} + j(\pm \omega _{1} \mp \omega _{2})L_{\mathrm{D}} - (\pm \omega _{1} \mp \omega _{2})^2R_{\mathrm{D}}L_{\mathrm{D}}C_{\mathrm{D}}} \\&\quad \times \,\frac{R_{\mathrm{C}}}{1 + j(\pm \omega _{1} \mp \omega _{2}) C_{p2}R_{\mathrm{C}}} v_{\mathrm{in}}^2 \end{aligned} \end{aligned}$$

(50)

By applying KCL at the output node of subsystem B, the total current of transconductance stage is obtained by summing $I_{\mathrm{out}}$ and $I_{\mathrm{out}}^{'}$ as in:

$$\begin{aligned} I_{\mathrm{out}_{\mathrm{total}}} = I_{\mathrm{out}} + I_{\mathrm{out}}^{'} \end{aligned}$$

(51)

Substituting Eqs. (48) and (50) in equation (51), Volterra kernels of total current can be obtained. As seen, $H_{1}$ and $H_{3}$ do not change but $H_{2}$ changes as shown in Eq. (25).