Regimes of operation

Opamp based

[1] OTA based

[This work]

RB

2.RB= _g¹_m+_g^R_m^F_ro ro

R_F R_F

A gm

1. Requires low impedance 1. Inverters can be used [3]

3. Two Regimes:

(a) Linearity limited if _g¹_m ≪ _g^R_m^F_ro (b) Noise limited if _g¹_m ≫ _g^R_m^F_ro output stage.

2.RB= _1+A^RF 3. Linearity limited

Figure 4.2: Comparison between Opamp based TIA and OTA based TIA.

and

R_sh =











(R_s+ R_sw)_1−4γ^4γ for N=4, (Rs+ Rsw)_1−8γ^8γ for N=8.

(4.3)

The analytical expression in (4.1) is verified by the design and simulation of two N-path mixer-first receivers operating at fs = 1 GHz. Ideal switches with 5 Ω ON-resistance and a resistance RBare used in simulations. Fig. 4.1(c) shows the input resistance of 4-path and 8-path mixer-first receivers for different values of R_B. From Fig. 4.1(c), the R_B values should be close to 260 Ω and 370 Ω for N=4 and N=8 respectively for 50 Ω impedance matching.

4.4 Regimes of operation

From (4.1), R_B or the TIA input impedance plays a significant role in input matching. In passive mixer-first receivers, a shunt-feedback TIA is typically used¹. The main amplifier in a shunt-feedback TIA can be an operational amplifier or an operational transconductance amplifier (OTA). These two TIA implementations are shown in Fig. 4.2. In [59], an opamp based TIA is considered². An OTA based TIA is more frequently found in recent implementations of highly linear mixer-first receivers [56, 58,61,62]. In this work, a single stage OTA based TIA such as the one shown in Fig. 4.2 is considered.

1Other possibility is a common-gate amplifier based TIA.

2A detailed comparison between Opamp based TIA and OTA based TIA is discussed in Appendix A

In the OTA based TIA model shown in Fig. 4.2, RF is the feedback resistance, gm represents the transconductance of the main amplifier, and ro represents the output impedance of the OTA. The input resistance of this OTA based TIA is given by

RB= 1

R_F − 1

R_F(r_o||R_F) +gm(ro||RF) R_F

−1

≈ R_F

g_mr_o||R_F ≈ 1

g_m + R_F

g_mr_o (4.4)

The specific RB value required for input matching can be realized either by using only _g¹

m or by using only _g^RF

mro or by using both _g¹

m and _g^RF

mro. Based on this choice of g_m, the receiver operates in one of the two possible regimes. If we choose r_o R_F so that R_B ≈ _g¹_m, then the receiver operates in the noise-limited regime. In the noise-limited regime, the receiver exhibits poor NF but a relatively good IB-IIP3 which can be tuned by adjusting the R_F value. If we choose _g¹

m _g^R_m^F_ro so that R_B≈ _g^R_m^F_ro, then the receiver operates in the linearity-limited regime. In the linearity-limited regime, the receiver exhibits poor IB-IIP3 but a relatively good NF which can be tuned by adjusting the gm value.

4.4.1 Noise-limited regime

Fig. 4.3(a) shows an equivalent LTI model of the N-path mixer-first receiver for noise analysis. The main sources of noise in the model shown in Fig. 4.3(a) are switch resistance Rsw, shunt resistance R_sh, feedback-resistance R_F and noise of the OTA. Here the effect of r_o is neglected as _g¹

m _g^R_m^F_ro. The noise factor of the receiver can be computed by calculating the noise contribution of each source at the output. The noise factor is given by

NF = 1 + R_sw Rs

+R_sh Rs

R_a Rsh

2

+ γR_F Rs

1 G²_N +γi²n,gm

R_F+^R_γ^p 1−gmRF

2 1 4KTRs

, (4.5)

where, Ra = Rs+ Rsw, Rp= Ra||R_sh, i²n,gm is the noise current at the output of the OTA, and GN is the transfer function of the receiver which is given by

G_N=

1−gmR_F 1 +^gm_γ^Rp

R_sh R_sh+ R_a

≈

1−_R^RF_B 1 +_γR^Rp

B

R_sh Rsh+ Ra

as RB≈ 1 gm

. (4.6)

For amplification, we need g_mR_F>1. The last two terms of (4.5) represent the noise contribution

4.4 Regimes of operation

0.5 1.5 2.5 3.5 4.5 3

6 9 12 15 18 21 24

0.5 1.5 2.5 3.5 4.5 5

8 11 14 17 20

R_F (kΩ) RF (kΩ)

NF(dB) NF(dB) ^{vn,gm= 3}

√nV Hz

vn,gm= 4^√nV_Hz

v_n,gm= 2√^nV Hz

vn,gm= 3^√nV

Hz

vn,gm= 4^√nV

Hz

vn,gm= 2^√nV_Hz

(a)

g_m γ

γRF v²_n,RF

i²_n,gm γ

v²_n,sh R_sh v²_n,s

Rs v²_n,sw Rsw

Rp= Rsh||Ra

ZB =γRB = _g^γ

m

(b) (c)

RF< 2.5 k

RF< 3.7 k

noise-limited noise-limited

regime regime

Figure 4.3: (a) Equivalent LTI model of the receiver for noise analysis in the noise-limited regime. Variation of NF with RF for different input referred noise voltage of transconductor (b) in a 4-path mixer-first receiver, and (c) in an 8-path mixer-first receiver (Solid lines indicate simulation, while dashed lines indicate (4.5)).

of the TIA to the overall receiver noise. (4.5) is verified by the design and simulation of 4-path and 8-path mixer-first receivers. Ideal switches with 5 Ω ON-resistance, and a VCCS with an appropriate input referred noise voltage v²n,gm = ^i2n,gm_g2

m to model the OTA are used in simulations. Moreover, the OTA is assumed to have g_mr_o = 100 ³. Fig. 4.3(b) and Fig. 4.3(c) show the variation of NF with feedback resistance R_F for different v²_n,gm in 4-path and 8-path receivers respectively. The NF of the receiver in this noise limited regime depends on v²_n,gm and R_F. The g_m required for input matching is≈4 mfin a 4-path receiver and is≈2.7 mfin an 8-path receiver. The noise-limited regime in the plots shown in Fig. 4.3 is separated by a dashed line where _g¹

m is 10 times more than _g^RF

mro.

3In this work, a single stage OTA is considered where gmro is a constant. A two-stage OTA restricts the receiver to the linearity-limited regime only. [56, 58, 61, 62] use single stage OTAs.

20 100 140 180 0.8

1 1.2 1.4

0.6 60

g_m (mS)

NF(dB)

20 60 100 140 180 1.6

2 2.4 2.8

1.2

g_m (mS)

NF(dB)

g_m>37 mS

g_m>25 mS

linearity-limited regime linearity-limited regime

Analytical NF Simulated NF without TIA noise

Simulated NF Analytical NF

Simulated NF without TIA noise

Simulated NF

v²n,sh

R_sh v²_n,s

Rsv²_n,sw Rsw

R_p= R_sh||R_a

gm

γ i²_n,gm

γ

v²_n,RF

r_oγ

RF

gmro

(a)

(b) (c)

Figure 4.4: (a) Equivalent LTI model of the receiver for noise analysis in the linearity-limited regime. Variation of NF with gm with and without TIA noise (b) in a 4-path mixer-first receiver, and (c) in a 8-path mixer-first receiver (Solid lines indicate simulation, while dashed lines indicate (4.7)).

4.4.2 Linearity limited regime

A noise equivalent LTI model of the receiver under study is shown in Fig. 4.4(a). Using Miller theorem, the feedback resistor R_Fcan be modeled by resistive terminations at the input and output of the OTA. Since R_Fr_o, only r_o is shown at the output of the OTA in Fig. 4.4(a). The noise voltage due to R_F is added to the OTA output. From Fig. 4.4(a), the noise factor can be computed as

NF = 1 +Rsw

Rs

+Rsh

Rs

Ra

Rsh

2

+γRF

Rs

1 G²_L +γi²_n,gmr²_o

G²_L

1 4KTRs

, (4.7)

where GL=

γgmroRshRF

γR_FR_sh+γR_aR_F+ g_mr_oR_shR_a

≈

gmro

1 +_R^Ra

sh +_γR^Ra

B

as RB≈ RF

gmro

. (4.8)

4.4 Regimes of operation

2 4 6 8 10 12 14 16 18

0 4 8 12 16 20 24 28 32

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

NF(dB) Gain(dB)andIIP3(dBm) NF(dB)

Gain NF IB-IIP3

RF (kΩ)

0 3 6 9 12 15 18 21

0 4 8 12 16 20 24 28

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 RF<2.6 kΩ

Gain NF IB-IIP3

Gain(dB)andIIP3(dBm)

R_F (kΩ) noise-limited

regime

(a) (b)

RF<1.9 kΩ noise-limited

regime

Figure 4.5: Variation of NF, gain and IB-IIP3 of the receiver (implemented with inverter based TIA) with feedback resistance RF in (a) a 4-path mixer-first receiver, and (b) in an 8-path mixer-first receiver.

Using RshRa from (4.3), the above expression can be further simplified to GL≈

gmro

1 +_γR^Ra

B

. (4.9)

(4.7) is verified by the design and simulation of 4-path and 8-path mixer-first receivers. Ideal switches with 5 Ω ON-resistance, and a VCCS to model the OTA are used in the simulations. The noise current of an OTA depends on the gm value. For simulation and verification purposes, i²n,gm is assumed to be equal to 4KTγMgm, where γM ≈ ²₃ is the noise proportionality constant. Fig. 4.4(b) and Fig. 4.4(c) show the variation of NF with the gm value in 4-path and 8-path mixer-first receivers respectively. The R_F required for input matching in a 4-path, and an 8-path mixer-first receivers respectively are 27 kΩ and 39 kΩ. The linearity-limited regime is separated in the plots of Fig. 4.4 by a dashed line where _g¹

m is 10 times less than _g^RF

mro. From Fig. 4.4(b) and Fig. 4.4(c), one can infer that the NF of the receiver in a linearity-limited regime can be brought closer to the NF of the mixer alone. The cost of reducing the NF is the increase in power consumption to increase the gm value.

By substituting γ, R_sh and RB values from (4.2) and (4.3) in (4.6) and (4.9), one can deduce that GNGL. For a given output swing, a system with lower gain will support higher input amplitude and hence will have higher IB-IIP3. Thus, the IB-IIP3 of a receiver operating in the linearity-limited regime is smaller than the IB-IIP3 of a receiver operating in the noise-limited regime.

0.8 1 1.2 1.4 1.6 1.8 2 2.2

-20 -10 0 10 20 30 40 50

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

-30 -20 -10 0 10 20 30 40 50

20 60 100 140 180 20 60 100 140 180

Gain(dB)andIIP3(dBm)

NF(dB) Gain(dB)andIIP3(dBm)

NF(dB)

g_m(mS) g_m>44 mS g_m>30 mS

gm (mS)

(a) (b)

Gain NF IB-IIP3 Gain

NF IB-IIP3

linearity-limited regime linearity-limited

regime

Figure 4.6: variation of NF, gain and IB-IIP3 of receiver (implemented with inverter based TIA) with transconductancegm in (a) a 4-path mixer-first receiver, and (b) in an 8-path mixer-first receiver

20 60 100 140 180 58

62 66 70 74 78

0.4 1.2 2 2.8 3.6 4.4 5.2 6 58

62 66 70 74 78

RF (KΩ)

SFDR(dB) SFDR(dB)

gm (mS)

gm>44mS gm>30mS RF<1.9KΩ

RF<2.6KΩ

linearity-limited regime N=8

N=4

N=8 N=4

(a) (b)

noise-limited regime

Figure 4.7: (a) variation of SFDR of a 4-path and a 8-path mixer-first receiver (implemented with inverter based TIA) withRF, (b) variation of SFDR of 4-path and 8-path mixer-first receiver (implemented with inverter based TIA) withgm