Mixer Design for Variability

Abstract

This chapter discusses the process variation effect on the mixer performance. Analytical equations are dervied to provide the theoretical insight. Monte Carlo simulation results are shown to demonstrate the statistical variation. The substrate bias technique helps reduce the process variation effect on the mixer performance fluctuation.

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Both the fabrication process-induced fluctuation and time-dependent degradation cause the MOSFET model parameters to drift. The threshold voltage and mobility are the two most significant model parameters that suffer from process uncertainty and reliability degradations. Here, the most widely used double-balanced Gilbert structure [1] in Fig. 10.1 is used to evaluate the process variations and aging effects on RF mixer performance. In this figure, positive and negative RF input signals are applied to transistors M1 and M2. Local oscillator (LO) signals are applied to switching transistors M3, M4, M5, and M6. The transistor M7 provides the bias current. RF and LO multiplication produces the output signal at intermediate frequency (IF).

Fig. 10.1

Schematic of a double-balanced Gilbert mixer

The conversion gain (CG) of the mixer can be derived as

$$CG = \frac{2}{\pi }\frac{{R_{L} }}{{R_{S} + \frac{1}{{g_{m} }}}}$$

(10.1)

where R _L is the load resistance and R _S is the inductor resistance. The noise figure (NF) of the mixer is given by

$$NF = 10\log_{10} \left( F \right)$$

(10.2)

where F is the flicker noise, which is derived as

$$F = \frac{{\pi^{2} }}{4}\left( {1 + \frac{{2\gamma_{1} }}{{g_{m} R_{S} }} + \frac{2}{{g_{m}^{2} R_{L} R_{S} }}} \right)$$

(10.3)

and $\gamma_{1}$ is the noise factor.

The sensitivity of the Gilbert cell mixer can be examined. The process variation and the aging effect may degrade the mixer performance. The conversion gain variation is modeled by the fluctuation of g _m and bias current drift as

$$\Delta CG = \frac{\partial CG}{{\partial g_{m} }}\Delta g_{m} = \frac{\partial CG}{{\partial g_{m} }}\left( {\frac{{\partial g_{m} }}{{\partial V_{T} }}\frac{{\partial V_{T} }}{{\partial I_{\text{bias}} }} + \frac{{\partial g_{m} }}{{\partial \mu_{n} }}\frac{{\partial \mu_{n} }}{{\partial I_{\text{bias}} }}} \right)\Delta I_{\text{bias}}$$

(10.4)

Expanding the partial derivatives in (10.4) the conversion gain variation can be written as

$$\begin{aligned}\Delta CG & = \frac{2}{{\pi g_{m}^{2} }}\frac{{R_{L} }}{{(R_{S} + \frac{1}{{g_{m} }})^{2} }} \\ & \quad \left\{ {\frac{{I_{\text{bias}} }}{{\left( {V_{GSM1} - V_{T} } \right)^{2} }}\frac{L}{{\mu_{n} C_{ox} W_{CS} \left( {V_{GSCS} - V_{T} } \right)}} + \frac{{I_{\text{bias}} }}{{\mu_{n} \left( {V_{GSM1} - V_{T} } \right)}}\frac{2L}{{C_{ox} W_{CS} \left( {V_{GSCS} - V_{T} } \right)}}} \right\}\Delta I_{\text{bias}} \\ \end{aligned}$$

(10.5)

where V _GSM1 is the gate-source voltage to the RF transistor and V _GSCS is the gate-source voltage to the current source transistor.

Similarly, the noise figure drift is derived as

$$\begin{aligned}\Delta F & = \frac{\partial F}{{\partial g_{m} }}\Delta g_{m} = \frac{\partial F}{{\partial g_{m} }}\left( {\frac{{\partial g_{m} }}{{\partial V_{T} }}\frac{{\partial V_{T} }}{{\partial I_{\text{bias}} }} + \frac{{\partial g_{m} }}{{\partial \mu_{n} }}\frac{{\partial \mu_{n} }}{{\partial I_{\text{bias}} }}} \right)\Delta I_{\text{bias}} \\ & = \left\{ {\frac{{\pi^{2} }}{4}\left( {\frac{ - 2\gamma }{{g_{m}^{2} R_{S} }} - \frac{1}{{g_{m}^{3} R_{L} R_{S} }}} \right)} \right\} \\ & \quad \left\{ {\frac{{I_{\text{bias}} }}{{\left( {V_{GSM} - V_{T} } \right)^{2} }}\frac{L}{{\mu_{n} C_{ox} W_{CS} \left( {V_{GSCS} - V_{T} } \right)}} + \frac{{I_{\text{bias}} }}{{\mu_{n} \left( {V_{GSM} - V_{T} } \right)}}\frac{2L}{{C_{ox} W_{CS} \left( {V_{GSCS} - V_{T} } \right)}}} \right\}\Delta I_{\text{bias}} \\ \end{aligned}$$

(10.6)

Equations (10.5) and (10.6) account for process variations and aging effect of the mixer.

It is clear from (10.4) to (10.6) that the mixer performance is dependent on the drain current of current source. To maintain the mixer performance, the drain current of M7 has to be kept stable. Thus, the process invariant current source circuit shown in Fig. 10.2 is employed. In Fig. 10.2, drain currents of M8 and M9 are designed the same. Changes in M8 and M10 drain currents are negatively correlated to remain as a stable bias current (I _D8 + I _D10). For example, if the process variation increases the threshold voltage, which decreases the drain current of M8, the gate voltage of M10 increases (V _G10 = V _DD – I _D9 R). Thus, the drain current of M10 increases to compensate the loss of I _D8.

Fig. 10.2

Process insensitive current source

ADS simulation is used to compare the mixer performance using the single transistor current source versus process invariant current source [2]. The RF mixer is operated at 900 MHz with an intermediate frequency of 200 MHz. In the circuit design, CMOS 0.18 µm mixed-signal technology node is used. R _L1 is 210 Ω and R _L2 is 190 Ω. The transistor channel width of M3–M6 is 200 µm. The channel widths of M1 and M2 are 190 and 210 µm, respectively. L _s1 and L _s2 are chosen at 2 nH. The width of M7 is 250 µm. The gate resistor size of the current source is 400 Ω. The mixer sets the gate biasing voltage at the current source at 0.62 V. In the current source, the transistor M8 and M9 match each other as 100 µm. The width of M10 is 600 µm. The supply voltage V _DD is 1.8 V.

For the process variation effect, the conversion gain of the mixer is evaluated using different process corner models due to inter-die variations. The simulation result of the fast–fast, slow–slow, slow–fast, fast–slow, and normal–normal models is shown in Fig. 10.3a. It is clear from Fig. 10.3a that the mixer with the invariant current source shows robust conversion gain against different process variations.

Fig. 10.3

a Conversion gain predicted by different process models. b Conversion gain versus threshold voltage. c Conversion gain versus electron mobility

The conversion gain is also evaluated using different threshold voltage and mobility degradations resulting from aging (hot carrier effect) as shown in Fig. 10.3b, c. The hot carrier injection increases the threshold voltage, but decreases the electron mobility. The conversion gain decreases with an increased threshold voltage or decreased mobility due to reduced transconductance. Again, the mixer with process invariant current source exhibits more robust performance against threshold voltage increase and mobility degradation.

In addition, the noise figure of the mixer using the process invariant current source is compared with that using the single transistor current source. The noise figure versus different process models is displayed in Fig. 10.4a. It is clear from Fig. 10.4a that the noise figure is more stable over different corner models for the mixer using the current invariant current source. The noise figure also shows less threshold voltage and mobility sensitivity as evidenced in Fig. 10.4b, c. In Figs. 10.4b and 10.5c, the noise figure increases with increased threshold voltage and decreased mobility due to reduced drain current and transconductance in the mixer.

Fig. 10.4

a Noise figure predicted using different process models. b Noise figure versus threshold voltage. c Noise figure versus electron mobility

Fig. 10.5

a Predicted mixer out power using different process models. b Output power versus threshold voltage. c Output versus electron mobility

The output power of the mixer has been evaluated using different process corner models as well. As shown in Fig. 10.5a the output power of the mixer using the process invariant current source demonstrates robust performance against process variations. In Fig. 10.5b, c the output power decreases with increased threshold voltage and decreased mobility due to reduced drain current in the mixer. The output power in Fig. 10.5b, c also shows less sensitivity against aging effect, which increases the threshold voltage and decreases the electron mobility.

The output power of the mixer has been evaluated using different process corner models as well. As shown in Fig. 10.5a the output power of the mixer using the process invariant current source demonstrates robust performance against process variations. In Fig. 10.5b, c the output power decreases with increased threshold voltage and decreased mobility due to reduced drain current in the mixer. The output power in Fig. 10.5b, c also shows less sensitivity against aging effect which increases the threshold voltage and decreases the electron mobility.

To further examine the process variation and reliability impact on RF mixer, Monte Carlo (MC) circuit simulation has been performed. In ADS, the Monte Carlo simulation [3] assumes statistical variations (Gaussian distribution) of transistor model parameters such as the threshold voltage, mobility, and oxide thickness. In the Monte Carlo simulation, a sample size of 1000 runs is adopted. Figure 10.6a, b display the histograms of conversion gain using single transistor current source (original) and using the process invariant current source (after compensation). For the mixer using the traditional current source, the mean value of conversion gain is −6.608 dB and its standard deviation is 3.18 %. When the process invariant current source is used, the mean value of conversion gain changes to −6.324 dB and its standard deviation reduces to 2.08 %.

Fig. 10.6

a Conversion gain statistical distribution without compensation. b Conversion gain statistical distribution after process compensation effect

The noise figure after 1000 runs of Monte Carlo simulation is dialyzed in Fig. 10.7a, b. For the mixer using the single transistor current source, the mean value of noise figure is 11.667 dB and its standard deviation is 2.49 %. When the process invariant current source is adopted, the mean value of noise figure changes to 11.159 dB and its standard deviation reduces to 1.29 %. Clearly, the mixer using the process invariant current source shows better stability against statistics process variations.

Fig. 10.7

a Noise figure statistical distribution without current compensation. b Noise figure statistical distribution after process compensation effect

In addition, the output power of the mixer is examined in Monte Carlo simulation. Figure 10.8a, b demonstrates an improvement of output power for the mixer using the process invariant current source over that using the traditional current source. In Fig. 10.8, the mean value of output power changes from −16.608 to −16.324 dB and its standard derivation reduces from 3.81 to 2.08 % once the process invariant current source is used.

Fig. 10.8

a Output power statistical distribution without current compensation. b Output power statistical distribution after process compensation effect