Chapter 4 -5dBm IIP3 UWB Low Noise Amplifier Using
4.5 Analysis of Linearity
In UWB RF transceiver design, linearity requirement becomes more and more challenging. Circuit nonlinearity results in various system distortions associated with the even and odd order nonlinearities. Of these distortions, the third-order intermodulation is one of the most critical terms responsible for linearity degradation in general RF systems. Due to the fact, how to improve the linearity of RF circuits without
Fig. 4.8 Gain response
Fig. 4.7 A 3–11GHz UWB LNA as a design example of dual reactive feedback
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extra power consumption becomes an important topic to be studied.
As far as cascade amplifier stages are concerned, system design calls for high linearity in the LNA to alleviate the distortion issues. Linearity is most limited by the transconductance amplifier, hence we will discuss the nonlinear effect of the common source amplifier in section 4.5.1.
In recent years, several techniques have been proposed to improve the linearity of RF circuits by linearization of the nonlinear transconductance, such as degeneration feedback. Another scheme is the superposition of auxiliary transistors operated in different bias conditions to null the derivative of device transconductance. Combined with the technique of out-of-band impedance termination, circuit linearity can be further enhanced, as indicated by the Volterra series analysis. The scheme, named as derivative superposition or multiple gated transistors (MGTR), offers a good opportunity to extend linearity without increasing power consumption. Further, we try to improve the linearity of broadband LNA using MGTR.
It is proposed that a complex transconductance shall be employed to search for the optimal design parameters for MGTR design consideration. Therefore we propose a compact equivalent circuit for the design of the multiple gated transistors technique in section 4.5.2. In section 4.5.3, we improve UWB LNA using multiple gated transistor technique by complex transconductance analysis.
4.5.1 Nonlinear Effects of Common Source Amplifier
For the common-source amplifier, the nonlinear effect is dominated by the transconductance and the output conductance of the device. The nonlinear effects of the capacitances and substrate can be neglected, and they can be considered as linear elements. Therefore we only consider the transconductance and the output
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conductance as the nonlinear source for the following analysis.
Fig. 4.9 shows a common-source amplifier, where Z1 is the input impedance, and Z2 is the output impedance. From the above-mentioned introduction and assuming that the common-source amplifier works in the weakly nonlinear region, the equivalent circuit of the common-source amplifier can be shown as Fig. 4.9, where the transconductance (gm) and the output conductance (ro) are the nonlinear elements.
Therefore the I-V curve of the device can be expressed as
....
The nonlinear distortion can be obtained by calculating the Volterra kernels of order one, two and three of voltages.
First-order kernels
In order to obtain the first-order Volterra kernels, the nonlinear elements must be replaced with its linearized equivalent, as shown in Fig. 4.10. Applying Kirchoff’s current law in Fig. 4.10 yields:
Fig. 4.9 The equivalent circuit of the common-source amplifier
44 transfer functions indicates the order of the transfer function, whereas the second subscript corresponds to the numbering of the node voltages. Then the first-order kernels H11(s) and H12(s) can be expressed as
For computing second-order kernels, the input signal v is replaced by a short in circuit, and the second-order nonlinear current sources are applied to the linearized circuit, as shown in Fig 4.11. Applying Kirchoff’s current law in Fig. 4.11 yields:
Fig. 4.10 Linearized equivalent of the circuit
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Just as second-order kernels, the third-order ones are computed as the response to the third-order nonlinear current sources, as shown in Fig. 4.12. Applying Kirchoff’s current law in Fig. 4.12 yields:
⎥⎦
Fig. 4.11 The equivalent circuit for the computation of the second-order kernels
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Of the nonlinear distortions, the third-order intermodulation is one of the most critical terms responsible for linearity degradation in general RF systems. In order to obtain the third-order intermodulation distortion, input signal is replaced by a two-tone test signal (V1=Asin(ω1t), V2 =Asin(ω2t)), and the fundamental signal at ω and the distortion at 1 2ω1−ω2 must be computed by (4-14) and (4-29). By assuming s1 =s2 = jω1 =s , s3 =−jω2 ≈−s , and jω1− jω2 =Δs , then the third-order intermodulation distortion (IM ) can be expressed as 3
Fig. 4.12 The equivalent circuit for the computation of the third-order kernels
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From (4-31). The equation provides good agreement with harmonic-balance simulation. The nonlinear distortions result from the nonlinear effects of the transconductance and the output conductance and vary with different load impedance.
For our case, the linearity is dominated by the device transconductance in the low load impedance.
4.5.2 Multiple Gated Transistors Method Using Complex Transconductance Analysis
The MGTR method improves linearity by cancellation of these effects due to auxiliary transistor (AT) as shown in Fig. 4.13. This negative peak of the main transistor (MT) can be cancelled by the positive peak value of a properly bias and size of AT. Because AT is biased in the sub-threshold region, this linearization method does not consume much extra power.
For typical MGTR, using DC transconductance the conventional analysis lacks of accuracy to predict the high-frequency operating condition. Essentially nonlinear distortion is frequency-dependent. An effective method using complex AC transconductance is therefore used to achieve optimized device size and bias for the auxiliary transistor.
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Without loss of generality in this nonlinear analysis, the superposed configuration in MGTR can be simply represented by the transconductance element Gm(ω). The transconductance Gm(ω)is defined in the same way as the ratio of the output current to the input voltage, including all the intrinsic and extrinsic frequency-dependency of MOSFET devices. It could be a complex value at high frequencies. Its nonlinearity shall be related to the load impedance and the operation frequency. Consequently the equivalent circuit model of a transconductance amplifier is shown in Fig. 4.14.
The third-order intermodulation product IMD3 in a two-tone test is derived by Volterra series analysis and expressed as
· ·
| | |·| |·| | |, (4.32) where|iNL G | |H | · |ε , ∆ , 2 |, (4.33) Fig. 4.14 The compact box-type equivalent circuit model
Fig. 4.13 MGTR architecture
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|ε , ∆ , 2 | G" ! ·G ! 2Z ∆ H ∆ G ! Z 2 H 2 G ! , (4.34)
Lower IMD3 can be obtained by reducing ε , ∆ , 2 in Equation (4.32). Similarly the major effort is cancellation of G”m(ω), which is conducted in the complex domain. For the value of G”m for MT is negative in the gate bias voltage vgs_MT=0.77V and the device sizes of MT is NF=13. It can be cancelled by the positive value of AT with a proper bias voltage. This complex transconductance analysis actually suggests that the proper bias voltage vgs_AT=0.46V, and the device sizes of AT are chosen as NF=11, as can be seen, G”m appears close to zero at the gate bias voltage vgs_MT=0.77V at 7GHz in the polar plot as shown in Fig. 4.15. From Fig. 4.16, we can get the magnitude of complex transconductance to matching the traditional DC transconductance analysis.
For the optimal device size and bias voltage of AT minimizes the value of G”m. Using the IIP3 contour as shown in Fig.4.17, we observe the value of IIP3 in the CS amplifier by sweeping the device size and bias voltage of AT for the condition which is the gate bias voltage vgs_MT=0.77V for MT, and the device sizes of MT is finger numbers of 13. Final, we choose the proper bias voltage vgs_AT=0.46V, and the device sizes of AT is finger numbers of 11 for the MGTR optimal parameters.
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single transistor MGTR
0.4 0.5 0.6 0.7 0.8 0.9 0.000
0.005 0.010 0.015 0.020
|G
m"| ( A/ V
3)
Main transistor V
gs(V)
Fig. 4.16 Cancellation of AC gm” in MGTR configuration Fig. 4.15 Cancellation of complex AC Gm” in polar plot
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4.8 6.4
8.0 13
9.6
4.8 8.0
3.2 6.4
1.6
14 14
14
0.30 0.35 0.40 0.45 0.50 0.55 0.60 6 14 8
10 12 14 16 18 20
Au xi liary tran sistor si ze (n r)
Auxiliary transistor V
gs(V)
Fig. 4.17 Search for the optimal device size and bias voltage using IIP3 contour
4.5.3 Broadband Linearity Improvement By Using MGTR
For a single transistor, the third-order intermodulation of the transconductance element G”m(ω) in a two-tone test is derived by small signal circuit analysis and expressed as"
" 3
2 2 2
( ) 3( ) ( )
4 3! (1 )(1 )(1 )
m L
m gs
gs s gs L gs s
g Z
G v
C Z j C Z j C Z
ω = − +ω + ω + ω , (4.35)
Similarly the major effort is canceling the IMD3 of MT, which is conducted in the complex domain. In order to improve the linearity of broadband LNA, lower output IMD3 can be obtained by reducing the negative IMD3 of MT using the positive IMD3 of AT. From Equation (4.35), we get the IMD3 frequency response of a transistor. To keep the perfect cancellation effect in broadband condition, we choose the similar
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device sizes of MT and AT to get the phases of IMD3 which are always differential in operation frequency, and the magnitude of IMD3 in MT, we choose a properly bias to get the same magnitude of IMD3 in AT for broadband cancellation.
In Equation (4.35), we assume the transconductance element G”m(ω) has magnitude and phase function depends on frequency, the IMD3 has the different magnitude variation and phase variation in operation frequency as shown in Fig. 4.18 We can get the IMD3 variation between 3GHz and 11GHz, the magnitude of third order intermodulation is similar, and the phase of third order intermodulation is always differential. Therefore, MGTR linearization method can provide good linearity improvement in broadband application. From the bias voltage vgs_MT=0.77V, and the device sizes of MT and AT are chosen as NF=13 and 11, we can observe the value of IIP3 in the CS amplifier by sweeping transistor size and bias voltage of AT in difference frequency as shown in Fig. 4.19. Using the IIP3 contour, we can choose the optimal bias voltage to get the perfect cancellation in broadband condition.
0 2 4 6 8 10 12
3.0n 4.0n 5.0n 6.0n 7.0n
MT_mag AT_mag MT_phase AT_phase
Frequency (GHz) The magnitude of G
m"
-150 -100 -50 0 50 100 150 200 T h e phase of G
m"
Fig. 4.18 The current of third order intermodulation of MT and AT in broadband condition
sugg
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0 2 4 6 8 10 12 14 16
-15 -10 -5 0 5 10 15 20
IIP3 ( d B m )
Frequency (GHz)
Single transistor MGTR
Fig. 4.20 The UWB LNA IIP3 be improved in broadband condition
-8.0n -4.0n 0.0 4.0n 8.0n
-4.0n -2.0n 0.0 2.0n
4.0n MT
AT
MT+AT(MGTR)
-1.0n 0.0 1.0n
-1.0n 0.0 1.0n
Frequency@(1~12GHz) Im(Gm ) (A)
Re(Gm ) (A)
Fig. 4.21 The MGTR cancellation effect in broadband condition
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