## 1. Introduction

The transconductor is a versatile building block employed in many analog and mixed-signal circuit applications, such as continuous-time filters, delta-sigma modulators, variable gain-amplifier or data converter. The transconductor is to perform voltage-to-current conversion. Linearity is one of most critical requirements in designing transconductor. Especially in designing delta-sigma modulators for high resolution Analog/Digital converters, it needs high linearity transconductors to accomplish the required signal-to-(noise+distortions) ratio. The tuning ability of transconductor is also mandated to adjust center frequency and quality factor in filter applications.

The portable electronic equipments are the trend in comsumer markets. Therefore, the low power consumption and low supply voltage becomes the major challenge in designing CMOS VLSI circuitry. However, designing for low-voltage and highly linear transconductor, it requires to consider many factors. The first factor is the linear input range. The range of linear input is justified by the constant transconductance, *G _{m}
*. Since the distortion of transconductor is determined by the ratio of output currents versus input voltage. The second factor is the control voltage of transconductor. This voltage can greatly impact the value of transconductance, linear range, and power consumption. For example, when the control voltage increases, the transconductance also increase but the linear input range of transconductor is reduced and power consumption is increased. Hence it is critical in designing transconducotr operated at low supply voltage. The third factor is the symmetry of the two differential outputs. If the transconductance of the positive and negative output is

*G*and

_{m+}=I_{O+}/V_{i}*G*, then how close

_{m−}=I_{O−}/V_{i}*G*and

_{m+}*G*should be is a critical issue, where

_{m−}*I*is the positive output current,

_{O+}*I*is the negative output current, and

_{O−}*V*is the input differential voltage. This factor is the major cause of common-mode distortion of transconductor which occurs at outputs.

_{i}In general, the design of differential transconductor can be classified into triode-mode and saturation-mode methods depending on operation regions of input transistors. Triode-mode transconductor has a better linearity as well as single-ended performance. On the other hand, saturation-mode transconductor has better speed performance. However, it only exhibits moderate linearity performance. Furthermore, the single-ended transconductor of saturation-mode suffers from significant degradation of linearity. Several circuit design techniques for improving the linearity of transconductors have been reported in literatures. The linearization methods include: source degeneration using resistors or MOS transistors [Krummenacher & Joeh, 1988; Leuciuc & Zhang, 2002; Leuciuc, 2003; Furth & Andreou, 1995], crossing-coupling of multiple differential pairs [Nedungadi & Viswanathan, 1984; Seevinck & Wassenaar, 1987] class-AB configuration [Laguna et al., 2004; Elwan et al., 2000; Galan et al., 2002], adaptive biasing [Degrauwe et al., 1982; Ismail & Soliman, 2000; Sengupta, 2005], constant drain-source voltages [Kim et al., 2004; Fayed & Ismail, 2005; Mahattanakul & Toumazou, 1998; Zeki, 1999; Torralba et al., 2002; Lee et al., 1994; Likittanapong et al., 1998], pseudo differential stages [Gharbiya & Syrzycki, 2002], and shift level biasing [Wang & Guggenbuhl, 1990].

Source degeneration using resistors or MOS transistors is the simplest method to linearize transconductor. However, it requires a large resistor to achieve a wide linear input range. In addition, MOS used as resistor exhibits considerable varitions affected by process and temperture and results in the linearity degradation. Crossing-coupling with multiple differential pairs is designed only for the balanced input signals. The Class-AB configuration can achieve low power consumption. On the other hand, the linearity is the worst due to the inherited Class-AB structure. The adaptive biasing method generates a tail current which is proportional to the square of input differential voltage to compensate the distortion caused by input devices. However, the complication of square circuitry makes this technique hard to implement. The constant drain-source voltage of input devices is a simple structure. It can achieve a better linearity with tuning ability. However, it needs to maintain *V _{DS}
* of input devices in low voltage and triode region. Therefore, this technique is difficult to implement in low supply voltage. Hence, a new transconductor using constant drain-source voltage in low voltage application is proposed to achieve low-voltage, highly linear, and large tuning range abilities.

In section 2, basic operatrion and disadvantage of the linerization techniques are described. The proposed new transconductor is presented in section 3. The simulation results and conclusion are given in section 4 and 5.

## 2. Linearization techniques

In this section, reviews of common linearization techniques reported in literatures are presented. The first one is the transconductor using constant drain-source voltage. The second one is using regulated cascode to replace the auxiliary amplifier. The third one is transconductor with source degeneration by using resistors and MOS transistors. The last one is the linear MOS transconductor with a adaptive biasing scheme. Besides introducing their theories and analyses, the advantages and disadvantages of these linearization techniques are also discussed.

### 2.1. Transconductor using constant drain-source voltage

The idea of transconductors using constant drain-source voltages is to keep the input devices in triode region such that the output current is linearized. The schematic of this method is shown in Figure 1. Considering that transistors M_{1}, M_{2} operate at triode region, M_{3}, M_{4} are biased at saturation region, channel length modulation, body effect, and other second-order effects are ignored, the drain current of M_{1} and M_{2} is given by

where *β =μ _{n}C_{OX}
*(W/L),

*V*is the gate-to-source voltage,

_{GS}*V*is the threshold voltage, and

_{T}*V*is the drain-to-source voltage. If the two amplifiers in Figure 1 are ideal amplifiers, then

_{DS}The transfer characteristic of this transconductor is given by

The transconductance value is

In fact, it is difficult to design an ideal amplifier implemented in this circuits. However, it can force *V _{DS1}
* =

*V*by using two auxiliary amplifiers controlled with the same

_{DS2}=V_{DS}*V*to keep

_{C}*V*at the constant value. Therefore, the transfer characteristic of this transconductor is changed as follows:

_{DS}where V_{GS1}= V_{in1} and V_{GS2}= V_{in2}.

Therefore, the new transconductance value is

The linearity of this transconductor is moderated. It is also easy to implement in circuit. However, *V _{DS}
* of the input devices must be small enough to keep transistors in triode region. The following condition has to be satisfied:

On the other hand, the auxiliary amplifiers need to design carefully to reduce the overhead of extra area and power.

### 2.2. Transconductor using regulated cascode to replace auxiliary amplifier

In Figure 2(a) regulating amplifier keeps *V _{DS}
* of M

_{1}at a constant value determined by

*V*. It is less than the overdrive voltage of M

_{C}_{1}. The voltage can be controlled from

*V*so as to place M

_{C}_{3}in current-voltage feedback, thereby increasing output impedance. The concept is to drive the gate of M

_{3}by an amplifier that forces

*V*to be equal to

_{DS1}*V*. Therefore, the voltage variations at the drain of M

_{C}_{3}affect

*V*to a lesser extent because amplifiers “regulate” this voltage. With the smaller variations at

_{DS1}*V*the current through M

_{DS1}_{1}and hence output current remains more constant, yielding a higher output impedance [Razavi, 2001]

It is one of solutions using regulated cascode to replace the auxiliary amplifier in order to overcome restrictions on Figure 1. The circuit in Figure 2(b) proposed in [Mahattanakul & Toumazou, 1998] uses a single transistor, M_{5}, to replace the amplifier in Figure 2(a). This circuit called regulated cascode which is abbreviated to RGC. The RGC uses M_{5} to achieve the gain boosting by increasing the output impedance without adding more cascode devices. *V _{DS1}
* is calculated by follows: Assuming M

_{5}is in saturation region in Figure 2(b). It can be shown that

From (6)
*G _{m}
* can be tuned by using a controllable voltage source

*V*or current source

_{C}*I*. However, it is preferable in practice to use a controllable voltage source

_{C}*V*for lowering power consumption since

_{C}*V*only varies as a square root function of

_{DS1}*I*.

_{C}Simple RGC transconductor using a single transistor to achieve gain boosting can reduce area and power wasted by the auxiliary amplifiers. However, it still has some disadvantages. First, it will cause an excessively high supply-voltage requirement and also produce an additional parasitic pole at the source of transistors. Therefore, it can not apply to the low-supply voltage design. Second, the tuning range of* V _{DS1}
* is restricted. The smallest value of

*V*is

_{DS1}*V*= 0. In other words,

_{C}*V*can not be set to zero. Owing to the restriction of (7),

_{DS1}*V*is as low as possible and the best value is zero. Third,

_{DS}*V*dependent

_{T}*G*may be a disadvantage due to the substrate noise and

_{m}*V*mismatch problems [Lee et al., 1994].

_{T}In Figure 3, another RGC transconductor that can apply to the low-voltages applications is proposed in [Likittanapong et al., 1998]. The circuit overcomes the disadvantages mentioned above is to utilize PMOS transistor that can operate in saturation region as gain boosting. The use of this PMOS gain boosting in the feedback path can result in a circuit with a wide transconductance tuning range even at the low supply voltage. In [Likittanapong et al., 1998], it mentions that at the maximum input voltage, M_{3} may be forced to enter triode region, especially if the dimension of M_{2} is not properly selected, resulting in a lower dynamic range. Besides, *β _{2}
* may be chosen to be larger for a very low distortion transconductor. It means that the tradeoff between linearity and bandwidth of transconductor is controlled by

*β*Therefore,

_{2}*β*should be selected to compromise these two characteristics for a given application.

_{2}
*V _{DS1}
* is calculated by follows. Assuming M

_{3}is in saturation region in Figure 3.

From (6)
*V _{DS1}
* can be set to zero when

*V*is set to zero. The minor drawback is that

_{DS1}*V*depends on the

_{T}*G*. It also may cause substrate noise and

_{m}*V*mismatch problems [Lee et al., 1994].

_{T}### 2.3. Transconductor using source degeneration

A simple differential transconductor is shown in Figure 4(a). Assuming that M_{1} and M_{2} are in saturation and perfectly matched, the drain current is given by

The transfer characteristic using (5) is given by

where V*
_{i}
* = (V

_{ in1}−V

_{ in2})

If *V _{GS}
* is large enough, the higher linearity can be achieved. Unfortunately, it can not be used in the low-voltage application and the linear input range is limited. Simplest techniques to linearize the transfer characteristic of MOS transconductor is the one with source degeneration using resistors as shows in Figure 4(b). The circuit is described by

A transfer characteristic derived from (13) is given by

The transconductance *G _{m}
* is

where *g _{m}
* is the transconductance of transistor M

_{1}and M

_{2}.

We should notice that in (14), the nonlinear term depends on *V _{i} − RI_{out}
* rather than

*V*. Higher linearity can be achieved when

_{i}*R >> 1/g*. The disadvantage of this transconductor is that large resistor value is needed in order to maintain a wider linear input range. Owing to

_{m}*G*≈

_{m}*1/R*, the higher transconductance is limited by the smaller resistor. Hence, there is a tradeoff between wide linear input range and higher transconductance which is mainly determined by a resistor.

Another method to linearize the transfer characteristic of MOS transconductor is using source degeneration to replace the degeneration resistor with two MOS transistors operating in triode region. The circuit is shown in Figure 5. Notice that the gates of transistor M_{3} and M_{4} connect to the differential input voltage rather than to a bias voltage. To see that M_{3} and M_{4} are generally in triode region, we look at the case of the equal input signals (*V _{in1}=V_{in2}
*), resulting in

Therefore, the drain-source voltages of M_{3} and M_{4} are zero. However, *V _{DS}
* of M

_{3}and M

_{4}equal those of M

_{1}and M

_{2}. Owing to (7), M

_{3}and M

_{4}are indeed in triode region. Assuming M

_{3}, M

_{4}are operating in triode region, the small-signal drain-source resistance of M

_{3}, M

_{4}is given by

It must be noted that in this circuit the effect of varying *V _{DS}
* of M

_{1}and M

_{2}can not be ignored since the drain currents are not fixed to a constant value. The small-signal source resistance of M

_{1}, M

_{2}is given by

Using small-signal T model, the small-signal output current, *i _{o1}
*, is equal to

Assuming M_{1} is in saturation region, the drain current of M_{1} is given by

Using (20) substitutes for (19), that leads to

The transconductance* G _{m}
* is

Linearity can be enhanced (assuming *r _{ds3}>> r_{s1}
*) compared to that of a simple differential pair because transistors operated in triode region exhibits higher linearity than the source resistances of transistors operated in saturation region. When the input signal is increased, the small-signal resistance in one of two triode transistors in parallel, M

_{3}or M

_{4}, is reduced. Meanwhile, the reduced resistance results in the lower linearity and the larger transconductance. As discussed in [Krummenacher & Joeh, 1988], if the proper size ratio of

*β*is chosen, the balance between higher linearity and stable transconductance can be achieved. How to choose the optimum size ratio of

_{1}/β_{3}*β*for the best linearity performance becomes slightly dependent on the quiescent overdrive voltage,

_{1}/β_{3}*V*The size ratio of

_{GS}−V_{T}.*β*=6.7 is used to achieve the best linearity performance.

_{1}/β_{3}According to (22), the transconductance can be tuned by changing *I _{SS}
* and size ratio of

*β*. Nevertheless, the nonlinearity error is up to 1% for

_{1}/β_{3}*I*< 80%. It is required to have a better linearity so as to achieve a THD of -60 dB or less in some filtering applications [Kuo & Leuciuc, 2001].

_{out}/I_{SS}### 2.4. Transconductor using adaptive biasing

The transconductor using adaptive biasing is shown in Figure 6. All transistors are assumed to be operated in saturation region, neglecting channel lengh modulation effect. First, transistor M_{3} is absent, and output current as a function of two input voltages *V _{in1}
* and

*V*is obtained as

_{in2}where *I _{SS}
* is a tail current and equals

*I*.

_{B}An adaptive biasing technique is using a tail current containing an input dependent quadratic component to cancel the nonlinear term in (23). Consequently, the circuit in Figure 6 changes the tail current by adding transistor M_{3}. The tail current will be changed by

where *I _{B}
* is tail current of differential pair and

*I*is the compensating tail current that cancel nonlinear term.

_{C}Therefore, the transfer characteristic is changed by

## 3. New transconductor

The conventional structure which uses the constant drain source-voltage such as RGC with NMOS or PMOS can not operate at 1.8V or below. The main reason is that auxiliary amplifier under the low supply voltage can’t provide enough gain to keep the constant drain-source voltage. Therefore, we propose a triode transconductor which uses new structure to replace the auxiliary amplifier. Figure 7 shows the proposed triode transconductor structure.

MOS M_{5}, M_{7}, M_{9} and M_{11} are made up a two-stage amplifier to replace the auxiliary amplifier. The two-stage amplifier is implemented using M_{9} with the active loads M_{11} formed the first stage and M_{5} with the active load M_{7} formed the second stage. The first and second stages exhibit gains equal to

Therefore, the overall gain is

The proposed transconductor is shown in Figure 8.

Considering that the large gain is achieved and is able to keep transistors M_{1} and M_{2} in triode region, the drain current of M_{1} and M_{2} is given by

The transfer characteristic is given by

where* β _{1} = β_{2}
*,

*V*=

_{T1}*V*, and

_{T2}*V*=

_{DS1}*V*. Assuming that current I

_{DS2}_{9}flows from M

_{11}through M

_{9}and MOS M

_{9}is in saturation region,

*V*can be found in (33)

_{DS1}According to (32)

The transconductance G_{m} is

From (35), the transconductance can be tuned by control voltage *V _{C}
* To keep M

_{1}and M

_{2}in triode region, the relation (36) needs to be satisfied.

Using (33) to substitute (36)

The proposed transconductor is suitable for low supply voltage and we choose 1.8V to achieve a wide linear range. Moreover, M_{9} is needed to obtain a negative feedback to keep the drain-source voltage of M_{1}, V_{DS1}, constant. This new structure can provide enough gain to keep V_{DS1} constant at 1.8V supply voltage. It has a low control voltage V_{C} between 0.69V~0.72V and the large transconductance tuning range depending on applications. Besides, it has a simple structure so as to save area.

## 4. Simulation results

The circuits in Figure 8 have been designed by using TSMC CMOS 0.18μm process with a single 1.8V supply voltage and simulated by Hspice. Figure 9. shows the curve of input voltage transferring to the output current at *V _{C}
* = 0.7V. The slope of the curve is linear when the input voltage varies from −1V to 1V. The slope in Figure 9. is equal to the transconductance in Figure 10. In order to verify the performance of the proposed transconductor, we define transconductance error (Equation 39) as the linearity of the transconductance’s output current. The transconductance error is less than 1% among ±0.9V input voltage, so the input linear range is up to 1.8V.

In Figure 11. it shows the drain-source voltage of the input transistors M_{1} and M_{2}, V_{DS1} and V_{DS2}, changes with the input voltage. Within ±1V input voltage, V_{DS1} and V_{DS2} are very small. According to equation (40), V_{DS1} and V_{DS2} are too small such that transistors M_{1} and M_{2} can be set in triode region. Once the input voltage exceeds ±1V, V_{DS1} and V_{DS2} will increase rapidly. It results in that transistors M_{1} and M_{2} enter in saturation region. In other words, when M_{1} and M_{2} entering saturation region the proposed transconductor can not maintain the high linearity.

When V_{C} is set between 0.69V and 0.72V, the linear input range is up to 2.6V and the transconductance error is less than 1%. The smallest transconductance is 3.4μs and linear input range is 1.2V when *V _{C}
* is 0.720V. The highest transconductance is 542μs and linear input range is 1.4V when

*V*is 0.690V. Table 1 shows the linear input range and the transconductance tuned by different

_{C}*V*. Therefore, the proposed transconductor achieve a large tuning range.

_{C}V _{C } (V) | Linear input range (V) | Transconductance (µS) |

0.690 | 1. 4 | 542 |

0.695 | 1.8 | 434 |

0.700 | 1.8 | 326 |

0.705 | 2.2 | 219 |

0.710 | 2.4 | 122 |

0.715 | 2.6 | 42 |

0.720 | 1.2 | 3.4 |

In Figure 12., the simulated THD as a function of the input frequency and input signal amplitude is plotted. The best THD is achieved at the low input voltage and the low frequency. When *V _{C}
* is 0.7V, the linearity of the proposed transconductor is less than −60dB for 0.7Vpp at 100KHz.

Figure 13. shows the linearity of transconductor in three linearization techniques. The transconductor using source degeneration with resistor is shown in Figure 4(b), and the transconductance in Figure 13(a) is tuned by different resistors. The transconductor using source degeneration with MOS transistors is shown in Figure 5, and the transconductance in Figure 13(b) is tuned by the different size ratio of *β _{1}
*/

*β*. The transconductor using adaptive biasing is shown in Figure 6, and the transconductance in Figure 13(c) is tuned by the different compensating tail current,

_{3}*I*. Figure 14. Shows the simulation result of the proposed technique and other techniques. Figure 14(a) is the full plot of the different linearization techniques. From Figure 14(b) it can be easily seen that the linearity achieved by the newly proposed technique is better than all other implementations.

_{C}The simulated THD of the output differential current versus the input signal amplitude for the four linearized transconductors is plotted in Figure 15. The proposed transconductor achieves THD less than −61dB for the 0.7Vpp input voltage, 11dB better than the one using source degeneration using resistor, 24dB better than the one using source degeneration using MOS, and 31dB better than the one using adaptive biasing, at the same input range.

Table 2. shows the power consumption of the four linearized transconductors at the same transconductance. Power consumption changes with the different transconductances. Therefore, the same transconductance is chosen to be compared in each configuration. Table 3. shows different power consumption at the different transconductance of the proposed transconductor.

Source degeneration using MOS | Source degeneration using resistor | Adaptive biasing | Proposed | |

Power (mW) | 1.3 1 | 1.19 | 1.38 | 1. 58 |

V C (V) | Power (mW) | G m (µA/V) |

0.690 | 1.759 | 542 |

0.695 | 1.7 14 | 434 |

0.700 | 1.5 86 | 326 |

0.705 | 1.4 42 | 219 |

0.710 | 1.2 63 | 122 |

0.715 | 0. 9 54 | 42 |

0.720 | 0.733 | 3.4 |

Table 4. shows the comparison of performance with other transconductors at the low supply voltage (under 2V). The transconductor in [Fayed & Ismail 2005] also uses constant drain-source voltage. It modifies the basic structure of constant drain source voltage and uses the moderate amplifier. The proposed transconductor modifies the auxiliary amplifiers to obtain high gain under low supply voltage.

The layout including proposed transconductor, Common Mode Feedback, and bandgap is shown in Figure 16. The proposed transconductor uses STC pure 1.8V linear I/O library in 0.18μm CMOS process. The chip area is 0.516mm^{2}.

## 5. Conclusion

The proposed low-voltage, highly linear, and tunable triode transconductor achieves the wide linear input range up to 2.4V. The total harmonic distortion is −60dB with a 0.7V_{pp} input voltage. The design uses TSMC 0.18μm CMOS technology and supply voltage is 1.8V. Moreover, it exhibits a large G_{m} tuning range from 3.4μS to 542μS and also keeps a wide linear input range. Finally, the performance comparison with other linear techniques shows that the proposed technique achieves better linearity, wider tuning range, and wider linear input range.