Chapter 1 Introduction
1.3 Thesis Overview
Chapter 2 demonstrates some basic structures of OTAs operating with high linearity. It describes the advantages and disadvantages of these structure as well as its characteristics. Furthermore, some linearity-improved circuits are also presented in this chapter.
Chapter 3 demonstrates the proposed OTAs, which modify the basic structures to enhance the linearity. First, we discuss the operating mechanism of the OTAs, and then utilize the mathematical equations to verify the concept. Finally, the noise analysis of the OTAs is discussed.
Chapter 4 present the principle of the Gm-C filters. Furthermore, output buffers are also discussed.
In chapter 5, the simulation results and experimental results are presented.
Chapter 6 makes the conclusion to this work.
Chapter 2
Operational Transconductance Amplifiers
2.1 Introduction
Operational transconductance amplifier (OTA) is one key building block in continuous-time integrated filters. In high-frequency continuous-time filters, Gm-C filters have often been employed since OTAs provide high Gm’s and a good controllability. However, the main disadvantage of a Gm-C structure is the poor linearity caused by the openloop operation. Therefore, the linearity is a critical topic to enhance. In recent years, several techniques for improving the linearity of CMOS OTA have been proposed [2]-[6].
In the design of OTAs, the transconductance should be tuned for compensation for process tolerances and temperature variation without degrading the entire circuit performance. Moreover, with the progressing of the CMOS technology, short channel effect is another obstacle to resolve.
2.2 Basic Structures of High-linearity OTAs
In this section, we introduce several basic structures of OTA. For instance, a fully differential structure is used to suppress common-mode noise, even-order distortions and power supply noise.
2.2.1 Differential Input
One approach to maintain a constant transconductance is to apply a differential pair, as shown in Fig. 2.1.
Fig. 2.1 the Differential Pair
As M1 and M2 operate in the saturation region, the output current I1 and I2 are expressed as As a result, the output differential current can be obtained by subtracting equation (2.2) from the equation (2.1) as
)
where the value Vcm is the input common mode voltage, and it is fixed to a constant DC level. Also, the Vx can be described as The transconductance of the differential pair is constant if the Vx remains constant.
However, in practice, the Vx varies with the process and the variation of input signal.
Consequently, keeping Vx constant is one of the solutions to improve the linearity. In addition, according to the equation (2.4), tuning the tail current can adjust Vx to obtain different Gm value.
2.2.2 Pseudo-differential Input
Another approach is a pseudo-differential pair, which removes the tail current from the differential pair, as shown in Fig. 2.2. The pseudo-differential pair alleviates the influence caused by the variation of Vx. As a result, the linearity can be improved.
Moreover, the voltage headroom of the pseudo-differential pair is larger than the differential pair. Therefore, the pseudo-differential pair is adequate for low-voltage applications.
Fig. 2.2 the Pseudo-Differential Pair
The analyses of the pseudo-differential pair are as follow. Assuming the two transistors are operating in the saturation region, the output current I1 and I2 are described as The output differential current is given by
)
As we can see, the output differential current is a linear function without the factor Vx. Although the pseudo-differential pair has better linearity and larger headroom than the differential pair, the former also has several shortcomings. First, the tuning of transconductance is limited. Although tuning the body voltage [5] can vary the threshold voltage to adjust the transconductance, the tuning range of body voltage
should be restricted to avoid large leakage current. Second, the common mode gain increases without the tail current. The common mode rejection ratio (CMRR) is about 0dB. Therefore, the pseudo-differential pair requires extra circuit, common mode feedforward (CMFF), to increase the CMRR.
The concept of the common mode feedforward circuit is shown in Fig. 2.3 [7].
Fig. 2.3 the Concept of Common Mode Feedforward Circuit
The CMFF circuit generates the common mode current of output. The differential mode current will remain by cancelling the common mode current of output, which implies that the common mode signal would not be amplified. As a result, the CMRR increases.
2.2.3 Source Degeneration
A source degeneration structure is one popular method to implement the OTA.
The circuit is shown in Fig. 2.4.
Fig. 2.4 the Source Degeneration Pair
The ideal operation of the source degeneration pair is that Vi+ and Vi- perfectly follow to the ends of the resister. Thus, the voltage across the resister generates the output current. The voltage-to-current conversion is extremely linear. However, the impedance between the gate and the source of the two transistors are not zero, and the impedance varies with the transconductance of M1 and M2. Therefore, the linearity of the source degeneration pair is degraded.
As shown in [8], the voltage-to-current conversion is given by
id
From the equation (2.9), the transconductance is proportional to the factor 1/R, so increasing the resistor can decrease the transconductance. By using Taylor Series, the third order harmonic distortion (HD3) can be derived as
2
increasing the transconductance or the value R, which means increasing the factor N, can improve the HD3.
Although the voltage-to-current conversions are the same in Fig. 2.4(a) and Fig.
2.4(b), the circuits have different properties. For Fig. 2.4(a), the tail currents contribute differential noise to the output, which is a dominant noise in the circuit. For Fig. 2.4(b), the voltage drop on the resistors reduces the range of the input common mode voltage.
2.2.4 Flipped Voltage Follower
In recent years, a flipped voltage follower is one popular approach for low-voltage low-power circuit design, including the OTA design. The circuit is shown in Fig. 2.5 [9].
Fig. 2.5 the Flipped Voltage Follower
Unlike the conventional voltage follower, the circuit in Fig. 2.5 is able to sink a large amount of current. However, the current source limits the sourcing capability.
The large sinking capability is due to the low impedance at the output node. By analyzing the circuit, we can derive the output impedance
approximately, where
1 2
/ 1
1 m m o
o g g r
r =
gmi and roi are the transconductance and output resistance of
transistor Mi, respectively.
2.2.5 Super Source Follower
A super source follower, Fig. 2.6, shows another method of implementing a linear OTA.
Fig. 2.6 the Super Source Follower
The properties of the super source follower are as follow. The output impedance of the super source follower is the same order as the flipped voltage follower, which is approximately . Moreover, to acquire a correct operation point for the transistors M1 and M2, the condition I
1 2
/ 1
1 m m o
o g g r
r =
B1>IB2 must be satisfied.
2.3 Linearity-improved OTAs
With the issue of the linearity becomes more and more significant, the linearity enhancement techniques are presented recently. In this section, we discuss two enhancement techniques, which have already been proposed.
2.3.1 Source Degeneration with OPAMPs
As discussion in subsection 2.2.3, the linearity degrades since the input voltages do not perfectly follow to the ends of the resistors. The idea to alleviate this phenomenon is using the operational amplifiers. The circuit is shown in Fig. 2.7.
Fig. 2.7 Improving the Linearity of a Fixed Transconductor by Using OPAMPs The virtual ground in each operational amplifier forces the source voltage of M1 and M2 to equal those of Vi+ and Vi-. Therefore, the input voltage appears directly across resistor and does not depend on the Vgs voltages of M1and M2, which obtains better linearity than the conventional circuits.
2.3.2 Source Degeneration with a Positive Feedback
Fig. 2.8 shows a source-degenerated differential pair with a positive feedback gm-boosting circuit [10].
Fig. 2.8 Source-degenerated Differential Pair with a Positive Feedback
The main boosting circuit consists of transistors M2 and M3, with gm of transistor M1 to be boosted. By choosing adequate M1 and M2 device dimensions,
the positive feedback loop reduces the high source resistance to 50Ω. The approximate expression is as
2 1
1 1
m m
s g g
R = − (2.11)
where gmi is the transconductance of transistor Mi. The smaller the resisters Rs are, the better the linearity is.
Chapter 3
Proposed OTAs for High Linearity Applications
3.1 Introduction
As mentioned in chapter 2, the main shortcoming of the OTAs is the poor linearity. Chapter 2 also presents some structures and techniques to enhance linearity.
However, it is not good enough for some applications. Therefore, we propose two modified circuits to acquire better linearity performance in this chapter. The two circuits are both based on the source degeneration structure and adding extra concepts to implement. Besides, the common mode feedback circuit, which is necessary for the fully differential circuits, is also presented in this chapter.
3.2 Proposed Flipped Voltage Follower with Input Attenuators In this section, the linearity enhancement technique which combines the flipped voltage follower with source degeneration is proposed. The FVF structure has good properties for high linearity OTA design as mentioned in subsection 2.2.4. Moreover, input attenuators are added to achieve larger input range and better linearity.
3.2.1 Characteristics and Operation of the OTA circuit The modified circuit using FVF with input attenuators is shown in Fig. 3.1.
Fig. 3.1 Modified Flipped Voltage Follower with Input Attenuators
The transistors M1~M4 are the FVF structure which implies the source of M1 and M2 are low output impedance. According to this property of the FVF, the relation between input voltages to output current can be expressed as
2 )
By comparing with the conventional source degeneration circuit in figure 2.4, which the voltage-to-current conversion can be described as 2 )
2 can identify that the non-ideal effects caused by the nonlinear transconductances can be reduced.
By analyzing the harmonic distortion of the circuit, we discover that the small input signal can obtain good linearity, such as equation (2.10). Therefore, using the transistors M5~M8 as the input attenuators is one approach to execute it. The attenuate ratio is determined by the aspect ratio of transistors, which is given by
5 respectively. From equations (3.1) and (3.2), the transconductance can be derived as
2
From the equation (3.3), there is a trade-off between the transconductance and the linearity.
In the OTA designs, the tuning circuit is not only used to alleviate the influences resulting from the process and temperature variations, but also applied to implement the multi-band filters. In this case, the transistor M23 operating in the triode region can replace the resistor Rtune, which is given by
Tuning the gate voltage of M23 can adjust the value of resistor, thereby varying the transconductance. Furthermore, a regulated cascade output stage, the transistors M9~M16, is used to enhance the output resistance.
3.2.2 Non-ideality Analysis
While designing the OTA, a fully differential structure is used to suppress the even-order distortions, ideally. However, the mismatch caused by the process variations is unavoidable. Consequently, the critical paths should be designed carefully.
First of all, the mismatch of the input pair has a significant influence on the
linearity. In addition, it also impacts the value of the transconductance directly.
Second, the mismatch caused by the current mirror leads to the even-order distortion.
In other words, the precise current mirror can alleviate the distortion. Finally, the body effect should be taken into consideration for the distortion.
In figure 3.1, M1~M8 are the input pair and the layout should be symmetry for low even-order distortion. Furthermore, by connecting the bulk and source terminals together, the body effect would be minimized.
3.2.3 Noise Analysis
In the communication systems, the noise is a critical issue for transmitting the signal. While designing the devices, the noise should be taken into consideration to ensure that the signal can be transmitted correctly. The device electronic noise is separated into two different types: the flicker noise and the thermal noise. The flicker noise, also called 1/f noise, is the dominant noise when the frequency is less than the corner frequency. On the contrary, the dominant noise is the thermal noise.
Since flicker noise is related to the level of DC, if the current is kept low, thermal noise will be the predominant effect. The thermal noise can be modeled by a current source connected between the drain and the source with a special density as:
m
n
kT g
I
2= 4 δ
(3.5) where k is the Boltzmann constant, T is the absolute temperature, gm is the source conductance, and the device noise parameter δ depends on the bias condition. We have defined , where n equals to the odd number (ex: ).Thus, the thermal noise density evaluated at the output node is derived as
) 1 ( )
(n = m n+
m g
g gm1 =gm2
( ) ( )
From the equation (3.6), decreasing the attenuate ratio will cause the additional noise at the output. This is a tradeoff between the linearity and the noise. Also, to reduce the thermal noise, the transconductance of input transistors should be maximized and the transconductance of tail current should be minimized.
3.3 Proposed Super Source Follower OTA with a Positive Feedback As discussion in subsection 2.3.2, the source degeneration with a positive feedback can obtain low source resistance
2 process and temperature variations, the transconductance of M1 and M2 does not perfectly match. Therefore, the super source follower with a positive feedback is proposed to alleviate the non-ideal effects and achieves better linearity than conventional one.
3.3.1 Characteristics and Operation of the OTA circuit The proposed transconductor circuit is shown in Fig. 3.2
Fig. 3.2 Super Source Follower OTA with a Positive Feedback
The transistors M1~M4 and M5~M8 are the two pairs of super source follower. The negative feedback loops, loop 1~4, make the source of M1, M2, M5 and M6 to be the low impedance nodes. Moreover, with the same aspect ratio of M9~M12, the positive feedback loops, loop 5 and 6, reduce the output impedance of X and Y. The output impedance of X and Y is given by
This result is derived through several steps. At first, we transfer Fig. 3.2 into small signal model. And then, assuming the body effect is ignored for simplicity.
Furthermore, the equations can be expressed by using the Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL). Finally, the result is derived from the equations.
The value of RX could approximate to zero by choosing appropriate M1~M8 device dimension. By comparing with equation (2.11), the first and second terms of RX are quite less than RS. Therefore, the mismatch caused by process variation could be minimized. From the equations (3.7) and (3.8), the transconductance can be presented as nonlinearity to acquire better total harmonic distortion (THD). As mentioned before, tuning the gate voltage of M25 can vary the transconductance.
Although the circuit has six loops, the stability of the circuit is not an issue. The reason is that the output impedance and capacitance are larger than other nodes. As a
result, the dominant pole is located at the output. Because the impedance and intrinsic capacitance of the other nodes are much lower than the output node, the second pole is in high frequency without affecting the stability.
While designing the Gm-C filters, the input and output of OTAs normally connect together. As a result, to confirm the correct common mode voltage is significant. The transistors M13~M16 are the source follower for dc level shifting to assure function work. In addition, the transistors M17~M24 are the output stage.
3.3.2 Non-ideality Analysis
As discussion in subsection 3.2.2, we can suppress the non-ideality effects by using several methods. In figure 3.2, the source followers, M13 and M14, may slightly influence the linearity. Moreover, the bias currents of the super source followers are too vital to neglect. Also, the bulk and source terminals connect together in the critical paths.
3.3.3 Noise Analysis
As discussion in subsection 3.2.3, the thermal noise is the dominant noise in high frequency. Also, we have definedgm(n) = gm(n+1), where n equals to the odd number
(ex: ). As a result, the output-referred noise density of the super source follower OTA with a positive feedback is derived as
2
where gmT1 is the transconductance of the tail current transistors. From the equation (3.10), the source follower adds the input-referred noise while providing a voltage
gain less than unity. The increase in degeneration factor, Rtotal, increases the noise contribution of the tail current transistors since it is split in an unbalanced way causing differential output noise. Moreover, it can be the most significant noise component for large degeneration factors.
3.4 Common Mode Feedback Circuit
In the filter design, the output of OTA generally connects to the input of next stage OTA. Consequently, the common mode feedback circuit is necessarily needed to obtain the correct input and output common mode voltage of the OTA. The two circuits are presented as below to interpret the necessity of common mode feedback circuit [11].
A simple differential amplifier which the inputs and outputs are short is shown in Fig. 3.3. The common mode voltage of the inputs and output can be easily derived as
D 2
SS
DD I R
V − .
Fig. 3.3 the Simple Differential Amplifier
The other circuit is shown in Fig. 3.4. In ideality, the currents through M3 and M4 are equivalent to ISS/2. Nevertheless, because of the fabrication process, the mismatches in the current mirror cause the difference between ID3,4 and ISS/2. If ID3,4 is
slightly greater than ISS/2, M3 and M4 will operate in the triode region to reduce the drain current to ISS/2. On the contrary, if ISS/2is slightly greater than ID3,4, M5 will operate in the triode region to make ISS/2 equal to ID3,4. The non-well defined output common mode voltage would make the transistors operate in the unwanted regions.
Therefore, the common mode feedback circuit is indispensable for the fully differential circuits to fix the output common mode voltage at the expected value.
Fig. 3.4 the High-gain Differential Pair
The CMFB circuit is shown in Fig. 3.5, and the operational mechanisms are described as follows [12].
Fig. 3.5 the Common Mode Feedback Circuit for Both Proposed OTA
The input transistors MF1~MF4 is utilized to detect the common mode voltage and compare with the reference voltage. If the common mode of the OTA output signal is equal to the desired voltage VREF, the current through MF8 will keep constant and thus the voltage VCM is fixed. Nevertheless, the common mode of the OTA output signal is not the same as VREF all the time. The voltage difference between them is mirrored through MF8 to vary VCM, thus making the output common mode voltage to the desired value. For example, if the output common mode voltages are larger than the VREF, the drain current of MF8 will increase. The current mirror also makes the current through MF10 increase, thereby VCM increasing. In the output stage of Fig.
3.1 and Fig. 3.2, increasing VCM leads to decreasing the output common mode voltage.
This mechanism of negative feedback loop makes the output common mode voltage equal to VREF.
When the OTA operates at high frequency, the CMFB circuit must be stable as well. The open loop gain of the CMFB circuit is
(
L out)
where CA and CB are the total capacitance at the points A and B, respectively. From the equation (3.11), the dominant pole is at 1/(CL×Rout)and the non-dominant poles are at and . The non-dominant poles should be designed far away from the unit gain frequency to increase the phase margin of the OTA.
A
mf C
g 8/ gmf13/CB
Chapter 4
Transconductor-C Filter
4.1 Introduction
As mentioned in the chapter 1, the sampled-data analog filters, the active RC filters and MOSFET-C filters are restricted for high-frequency operation. On the contrary, Gm-C filters are aimed specifically at high-frequency integrated filters.
Although high-frequency filters are the main aim of this design method, Gm-C filters
Although high-frequency filters are the main aim of this design method, Gm-C filters