# 0.5 V Versatile Voltage- and Transconductance-Mode Analog Filter Using Differential Difference Transconductance Amplifier

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## Abstract

**:**

## 1. Introduction

_{y1}, V

_{y2}, V

_{y3}at the w-terminal, namely, V

_{w}= V

_{y1}− V

_{y2}+ V

_{y3}. This terminal usually provides low-impedance level (results from negative feedback of DDA), which is suitable for output terminals of voltage-mode circuits. The o-terminal provides output current which is the product of the differential voltage V

_{w}-V

_{y4}and the transconductance g

_{m}, i.e., I

_{o}= g

_{m}(V

_{w}-V

_{y4}). Thus, an electronic tuning ability of DDTA can be obtained by tuning ${g}_{m}$. The ideal characteristics of DDTA in Figure 1a can be described by

## 2. Proposed Circuit

#### 2.1. 0.5 V DDTA

_{1}-M

_{2}[30], which is well suited for low-voltage circuits. However, the input transistors of this pair were replaced by multiple-input bulk-driven transistors (BD MI-MOST), shown in Figure 3. The multiple-input bulk-driven transistors are realized with an additional capacitive voltage divider, consisting of the capacitors C

_{B}, shunted by anti-parallel connections of the transistors M

_{L}. Note, that M

_{L}operate with V

_{GS}= 0, thus realizing a very large resistance R

_{LARGE}. Their purpose is to provide proper biasing of the input terminals for DC. For frequencies much larger than 1/R

_{LARGE}C

_{B}, the resulting impedance of the R

_{LERGE}C

_{B}connections is dominated by capacitors, and the AC voltage at the bulk terminals of the multiple input transistors can be expressed as:

_{B}are equal to each other, then β

_{i}= 1/2, for i = 1,2. Denoting the non-inverting and inverting input voltages of the input differential pair as V

_{i+}and V

_{i-}, respectively, the differential voltage at the bulk terminals of the input non-tailed pair M

_{1A}and M

_{1B}can be expressed as:

_{13}-M

_{16}, is a typical class-A common source amplifier. The capacitor C

_{C}is used for frequency compensation. The two-stage architecture allows increasing the open-loop voltage gain and consequently the accuracy of the realized circuit function. The voltage gain is further increased thanks to the partial-positive feedback (PPF) circuit, realized using two sub-circuits M

_{7}-M

_{8}and M

_{9}-M

_{10}. Each of the cross-coupled pairs of transistors generates negative resistances, thus partially increasing the resulting resistances at the gate/drain nodes of the transistors M

_{2A,B}and M

_{5,6}respectively, and consequently increasing the voltage gain of the DDA. However, PPF leads to increased sensitivity of the circuit to transistor mismatch, that can lead to stability problems. This sensitivity, like the voltage gain, increases with the amount of positive feedback, i.e., when the ratio of transconductances g

_{m7,8}/g

_{m2A,B}(g

_{m9,10}/g

_{m5,6}) increases towards unity. Therefore, there is a tradeoff between the achieved improvement of the voltage gain and the circuit sensitivity to transistor mismatch. As it was shown in [16], applying two PPF circuits with weaker positive feedback, leads to the same improvement of the voltage gain, with less sensitivity, than applying one PPF circuit providing the same improvement of the gain. Therefore, in the circuit of Figure 2 two PPF circuits have been applied, one connected directly to the input transistors and the second, applied to the load of the input pair.

_{o}is the DC open-loop voltage gain of the DDA without PPF circuits, calculated from the bulk terminals of M

_{1}, and given as:

_{1}and m

_{2}can be expressed as:

_{1}and m

_{2}can be considered as the ratios of negative to positive conductances in “bottom” and “upper” PPF. In general case, the coefficients can range from zero (lack of positive feedback) to unity (100% positive feedback). The overall voltage gain increases to infinity, as m

_{1}or m

_{2}tends to unity, i.e., as the amount of positive feedback increases. Thanks to the application of two PPFs in the proposed design with m

_{1}= m

_{2}= 0.5, a sufficient increase of the voltage gain (12 dB), with acceptable circuit sensitivity to transistor mismatch was achieved.

_{11}and M

_{12}has been applied, which allows increasing the linear range of the TA by about 3 times, as compared with the conventional BD pair used in [16]. Even better linearity could be achieved using a non-tailed pair with a linear resistor [18,31], but such a solution is not very suitable for transconductors operating in nS range, since it would require very large resistors, not practical in silicon realizations.

_{m}can be expressed as:

_{mb1,2}/g

_{m1,2}is the bulk to gate transconductance ratio at the operating point for the input transistors M

_{1}and M

_{2}, n

_{p}is the subthreshold slope factor for p-channel MOS transistors and U

_{T}is the thermal potential. As it can be concluded from (11), the circuit transconductance is proportional to the biasing current I

_{set}, thus it can be easily regulated using this current.

#### 2.2. Versatile Analog Filter

_{o1}, V

_{o2}, and V

_{o3}are defined at the low-impedance w-terminals while the input voltages V

_{in1}to V

_{in7}are fed to the high-impedance y-terminals of the DDTA.

_{o}) and the quality factor (Q) are given by

_{o}can be controlled electronically by g

_{m1}= g

_{m2}and, in this case, the parameter Q is given by C

_{1}/C

_{2}.

_{3}while the input voltages are same as for the VM filter. The output current of the TM filter is given by

_{o1}/V

_{in3}is used, with a gain given by

_{o1}/V

_{in3}= 1, the characteristic equation of the oscillator is

_{m1}= g

_{m2}, the condition of oscillation is given by

_{o}can be electronically controlled via g

_{m1}and g

_{m2}.

_{o1}and V

_{o2}in Figure 6, we see, that DDTA

_{2}and C

_{2}form a lossless integrator with a transfer function given by

_{o}, the phase and magnitude are given respectively by ∅ = π/2 and |g

_{m2}/C

_{2}|.

**Non-idealities analysis**

_{k1}, β

_{k2}, β

_{k3}denote the non-ideal voltage gains from y

_{1}, y

_{2}and y

_{3}terminals, respectively, to the w-terminal of the k-th DDTA (β

_{k1}= β

_{k2}= β

_{k3}= 1 in ideal case), g

_{mnk}is the frequency-dependent transconductance of the k-th DDTA. At the frequency near the cut-off frequency, g

_{mnk}can be approximated by [33]

_{k}= 1⁄ω

_{gk}, and ω

_{gk}denotes the first pole of the k-th DDTA.

_{o1}, V

_{o2}, and V

_{o3}of the VM analog filter can be rewritten as

_{out}of the TM analog filter can be rewritten as

_{o}and Q can be expressed as

_{o1}/V

_{in3}= 1, the non-ideal characteristic equation of the oscillator can be expressed by

## 3. Simulation Results

_{DD}= 0.5 V and the bias voltage V

_{B1}= −60 mV. For I

_{set}= 5 nA, the total power consumption of the DDTA was 215.5 nW (DDA = 203 nW and TA = 12.5 nW). It is worth mentioning that the DC level on the bulk terminal of the differential pair depends on the DC level of the input signals and on the shunt resistors M

_{L}that form a resistor voltage divider [34]. The simulated current of the bulk terminal of the differential pair is less than 0.8% of the input currents in the wholly input range, and hence the current of the bulk terminal could be neglected compared to the inputs one [34]. The value of this input capacitor C

_{B}was optimized, based on previous post-layout simulation, to be 0.5 pF, in order to reduce the impact of the parasitic capacitance of the MOS transistor on the circuit performance from one side and to avoid extra increase in chip area from the other side [34,35]. The performance of the MI-MOST was confirmed experimentally in [34,35].

_{set}= 5 nA. While the transconductance varies by about 10% over the nominal value for the input voltage range of ±40 mV for TA without SD, it is up to ±160 mV for the proposed TA with the SD. Note the improved linearity, obtained thanks to the SD technique.

_{1}= C

_{2}= 20 pF and the setting current I

_{set}= 5 nA. The −3 dB cut-off frequency for the LP filter was 323.3 Hz.

_{set}= 2.5 nA, 5 nA, 10 nA, 20 nA the −3 dB frequency of the LPF was 162 Hz, 323.3 Hz, 650.2 Hz and 1.333 kHz, respectively. This, for example, covers a wide spectrum of biosignal filtering applications.

_{DD}. The variation of these characteristics is in acceptable range.

_{set}.

_{pp}@ 10 Hz is shown in Figure 12. The total harmonic distortion (THD) was 0.8%. The output noise of the LPF is shown in Figure 13. The integrated noise value was 108 µV, that results in a dynamic range of 53.2 dB for 1% THD.

_{1}= C

_{2}= 20 pF and I

_{set}= 5 nA. The low frequency current gain of the LP filter was −150 dB, which corresponds to a transconductance 31.6 nS, and the −3 dB for the LPF was 323.3 Hz.

_{set}= 5 nA and capacitor values C

_{1}= C

_{2}= 20 pF. Figure 15 shows the starting oscillation (a) and the steady state (b), respectively. The frequency is 253 Hz and the THD was around 1%.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 9.**The simulated frequency characteristics showing the tuning capability of the LP (

**a**) and BP filters (

**b**).

**Figure 10.**The simulated frequency characteristic of the LP (

**a**) and BP (

**b**) filters with PVT corners.

**Figure 11.**The histogram of the −3 dB cut-off frequency (

**a**) and low frequency gain (

**b**) of the LPF with 200 MC runs.

Output | Input | Filtering Function | Transfer Functions |
---|---|---|---|

${V}_{o1}$ | ${V}_{in2}={V}_{in5}={V}_{in}$ | Non-inverting LP | ${g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ |

${V}_{in3}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in4}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in5}={V}_{in}$ | Inverting BP | $-\mathrm{s}{C}_{1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in6}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in}$ | Inverting HP | $-{s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in2}={V}_{in5}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}$ | |

${V}_{in2}={V}_{in5}={V}_{in6}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}$ | |

${V}_{o2}$ | ${V}_{in5}={V}_{in}$ | Non-Inverting LP | ${g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ |

${V}_{in1}={V}_{in}$ | Inverting BP | $-\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in2}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in6}={V}_{in}$ | Inverting BP | $-\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in3}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in4}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in3}={V}_{in6}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in4}={V}_{in6}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in3}={V}_{in5}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in4}={V}_{in5}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in3}={V}_{in5}={V}_{in6}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in4}={V}_{in5}={V}_{in6}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in3}={V}_{in5}={V}_{in1}={V}_{in6}={V}_{in}$ | Non-inverting AP | ${s}^{2}{C}_{1}{C}_{2}-\mathrm{s}{C}_{2}{g}_{m1}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{o3}$ | ${V}_{in3}={V}_{in}$ | Non-inverting LP | ${g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ |

${V}_{in4}={V}_{in}$ | Non-inverting LP | ${g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in5}={V}_{in}$ | Inverting LP | $-{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in2}={V}_{in}$ | Inverting BP | $-\mathrm{s}{C}_{2}{g}_{m1}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in5}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in3}={V}_{in6}={V}_{in}$ | Inverting HP | -${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in}$ | Non-inverting BS | ${s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in6}={V}_{in}$ | Inverting BS | $-\left({s}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}\right)/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in2}={V}_{in7}={V}_{in}$ | Non-inverting AP | ${s}^{2}{C}_{1}{C}_{2}-\mathrm{s}{C}_{2}{g}_{m1}+{g}_{m1}{g}_{m2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in6}={V}_{in}$ | Inverting AP | $-\left({s}^{2}{C}_{1}{C}_{2}-s{C}_{2}{g}_{m1}+{g}_{m1}{g}_{m2}\right)/\mathrm{D}\left(\mathrm{s}\right)$ |

Output | Input | Filtering Function | Transfer Functions |
---|---|---|---|

${I}_{out}$ | ${V}_{in3}={V}_{in}$ | Non-inverting LP | ${g}_{m1}{g}_{m2}{g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ |

${V}_{in4}={V}_{in}$ | Non-inverting LP | ${g}_{m1}{g}_{m2}{g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in5}={V}_{in}$ | Inverting LP | $-{g}_{m1}{g}_{m2}{g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in}$ | Non-inverting BP | $\mathrm{s}{C}_{2}{g}_{m1}{g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in2}={V}_{in}$ | Inverting BP | $-\mathrm{s}{C}_{2}{g}_{m1}{g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in5}={V}_{in}$ | Non-inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in3}={V}_{in6}={V}_{in}$ | Inverting HP | ${s}^{2}{C}_{1}{C}_{2}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in6}={V}_{in}$ | Inverting BS | $\left\{\right(s{\}}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2})/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in7}={V}_{in}$ | Non-inverting BS | $-\left\{\right(s{\}}^{2}{C}_{1}{C}_{2}+{g}_{m1}{g}_{m2}){g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in2}={V}_{in7}={V}_{in}$ | Non-inverting AP | $\left\{\right(s{\}}^{2}{C}_{1}{C}_{2}-\mathrm{s}{C}_{2}{g}_{m1}+{g}_{m1}{g}_{m2}){g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ | |

${V}_{in1}={V}_{in6}={V}_{in}$ | Inverting AP | $-\left\{\right(s{\}}^{2}{C}_{1}{C}_{2}+\mathrm{s}{C}_{2}{g}_{m1}+{g}_{m1}{g}_{m2}){g}_{m3}/\mathrm{D}\left(\mathrm{s}\right)$ |

DDA | W/L (µm/µm) | TA | W/L (µm/µm) |
---|---|---|---|

M_{1A}, M_{2A}, M_{1B}, M_{2B} M_{14}, M_{15} | 16/3 | M_{1}, M_{2} | 5 × 15/1 |

M_{3}-M_{8}, M_{11}-M_{12}, M_{B} | 8/3 | M_{3}-M_{6} | 2 × 10/1 |

M_{9}, M_{10} | 4/3 | M_{3c}-M_{6c} | 10/1 |

M_{16} | 6 × 16/3 | M_{8}, M_{9}, M_{B1}, M_{11}, M_{12} | 2 × 15/1 |

M_{13} | 6 × 8/3 | M_{8c}, M_{9c}, M_{B1c} | 15/1 |

M_{L} | 4/5 | M_{7} | 2 × 30/1 |

MIM capacitor: C_{B} = 0.5 pF, C_{c} = 6 pF | M_{7c} | 30/1 |

Factor | [11] | [15] | [16] | [18] | Proposed |
---|---|---|---|---|---|

Number of active devices | 3 DDCC | 5-DDTA | 3 DDTA | 2 DDTA | 3 DDTA |

Realization | 130 nm | 180 nm | 130 nm | 130 nm | 180 nm |

Number of passive devices | 2 R, 2 C | 2 C | 2 C | 2 C | 2 C |

Type of filter | MISO | MIMO | MIMO | MIMO | MIMO |

Operation mode | VM | VM/CM/TAM/TIM | VM | VM | VM/TIM |

Number of offered responses | 5 (VM) | 36 (VM/CM/TAM/TIM) | 23 (VM) | 22 (VM) | 34 (VM) 11 (TIM) |

Active device offers electronic control | No | Yes | Yes | Yes | Yes |

High-input and low-output impedance of VM | Yes | No | No | No | Yes |

High-input and high-output impedance of TM | - | Yes | - | - | Yes |

Orthogonal control of ${\omega}_{o}$ and $Q$ | Yes | Yes | Yes | Yes | Yes |

Electronic control of ${\omega}_{o}$ | No | Yes | Yes | Yes | Yes |

Offer modified into oscillator | No | No | No | Yes | Yes |

Orthogonal control of CO and FO | - | - | - | Yes | Yes |

Natural frequency (kHz) | 6370 | 1.04 | 0.254 | 0.08147 | 0.323 |

Simulated power supply (V) | $\pm $0.75 | 1.2 | 0.5 | 0.3 | 0.5 |

Power dissipation ($\mu W$) | 3650 | 330 | 0.616 | 0.715 | 0.646 |

THD (%) | 3 @120 m V_{pp} | 1.09@650 mV_{pp} | 0.62 @100 m V_{pp} | 0.5 @100 m V_{pp} | 0.8@100 mV_{pp} |

Dynamic range (dB) | - | - | 49.7 | - | 53.2 |

Verification of result | Sim/Exp | Sim/Exp | Sim | Sim | Sim |

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## Share and Cite

**MDPI and ACS Style**

Kulej, T.; Kumngern, M.; Khateb, F.; Arbet, D.
0.5 V Versatile Voltage- and Transconductance-Mode Analog Filter Using Differential Difference Transconductance Amplifier. *Sensors* **2023**, *23*, 688.
https://doi.org/10.3390/s23020688

**AMA Style**

Kulej T, Kumngern M, Khateb F, Arbet D.
0.5 V Versatile Voltage- and Transconductance-Mode Analog Filter Using Differential Difference Transconductance Amplifier. *Sensors*. 2023; 23(2):688.
https://doi.org/10.3390/s23020688

**Chicago/Turabian Style**

Kulej, Tomasz, Montree Kumngern, Fabian Khateb, and Daniel Arbet.
2023. "0.5 V Versatile Voltage- and Transconductance-Mode Analog Filter Using Differential Difference Transconductance Amplifier" *Sensors* 23, no. 2: 688.
https://doi.org/10.3390/s23020688