A Low-Voltage Low-Power Voltage-to-Current Converter with Low Temperature Coefficient Design Awareness

Abstract

This paper presents a low-voltage, low-power voltage-to-current converter (V-I Converter) implemented in TSMC 40 nm CMOS technology. Operating at a supply voltage of 0.45 V with an input range of 0.1 V to 0.3 V, the proposed circuit achieves a temperature coefficient of 54.68 ppm/°C, which is at least 2× better than prior works, ensuring stable performance across a wide temperature range (−20 °C to 80 °C). The design employs a three-stage operational transconductance amplifier (OTA) with a Q-reduction frequency compensation technique to produce programmable output currents while maintaining a power dissipation of less than 2.76 μW. With a bandwidth of 34.45 kHz and a total harmonic distortion (THD) of −56.66 dB at 1 kHz and 0.1 V_PP input signal, the circuit demonstrates high linearity and low power consumption under ultra-low voltage design scenarios. These features make the proposed V-I Converter highly suitable for energy-constrained applications such as biomedical sensors, energy harvesting systems, and IoT nodes, where low power consumption and temperature stability are critical parameters.

1. Introduction

As research advances in the areas of microsensors, biomedical implantable devices, portable electronic equipment, and other applications where speed is not of concern, there is an increasing demand for ultra-low voltage and low power consumption. Furthermore, the continuous down-scaling of the process results in the gate oxide becoming thinner. It is then more susceptible to breakdown, so the supply voltage needs to drop to ensure device stability [1]. To meet this demand, designing circuits with MOS transistors operating in the subthreshold region has emerged as a widespread approach since the mid-1970s [2]. Notably, the subthreshold CMOS circuits offer a high gm/ID ratio and exceptional energy efficiency [3].

The V-I Converter converts voltage signals to current signals, and this becomes a significant analog signal-processing block for different usages in integrated circuits and systems. Based on the function of the V-I Converter, a Gm-C filter, a data converter, a multiplier and a voltage-to-frequency converter, which are the basic circuits for use in sensors, can be built [4,5,6,7]. For biomedical sensors, the V-I Converter can be applied for the analog pre-processing of very low biomedical signals. It is also suitable for use as a low-frequency analog front-end for biomedical sensor interfaces [8]. In terms of portable devices, the V-I Converter adjusts the output current by feeding back current to the input voltage of a hearing aid device [9,10]. In a MEMS gyroscope, the V-I converter is used to precisely and dynamically adjust the temperature of the heater [11]. In addition, in the application of a VCO-based sensor, a voltage-to-current converter is needed to generate the feedback current in order to achieve a small noise variation at the output of the VCO [12]. Therefore, the V-I Converter can be considered a common basic block for sensor applications.

The V-I Converter has a universal expression, as follows:

I_out = K∙V_in,

(1)

where I_out is the output current, V_in is the input voltage and K is a constant.

In the typical design of a V-I Converter, designers focus more on linearity. Turning to very low-voltage applications, circuit designers also seek low power consumption. On top of that, trade-off parameters like bandwidth (BW), power supply rejection ratio (PSRR), common mode rejection ratio (CMRR), input-referred noise, total harmonic distortion (THD), rail-to-rail input range, and power consumption are also addressed. In previous research, temperature stability has been given little consideration. In very low-voltage circuits, temperature drift can have a significant effect on the precision. For example, the variation arising from temperature drift may be fed back or amplified and processed within the sensor system, resulting in an error in the output of the sensor. Temperature instability is particularly fatal in low-frequency, low-voltage sensor circuits because it consumes the available headroom under a limited supply voltage.

This raises the motivation of this work, which aims to design a very low-voltage voltage-to-current converter with design awareness for achieving a low temperature coefficient. In order to prepare the V-I Converter for application in a variety of scenarios, the output current is made programmable so as to yield flexibility and meet different requirements in terms of precision in the context of process variation. The following section of this paper is organized as follows. Section 2 reviews the previous design of the V-I Converter. Section 3 describes the design of the proposed 0.45 V V-I Converter with temperature-compensated output currents. Section 4 presents the simulation results and discussions. Section 5 gives the conclusion.

2. Review of V-I Converters

A traditional voltage-to-current circuit is implemented by OTA, as shown in Figure 1. In this conventional structure, OTA always drives a MOSFET and the resistor in a negative feedback loop [13]. Negative feedback can make the negative input of the OTA match closely to the output voltage, achieving an output current approximately equal to V_in/R_S. In this topology, the driving transistor is either an NMOS or a PMOS. As shown in Figure 1a, the NMOS is used as the driving transistor, and the output current is copied through the PMOS current mirror. The minimum supply voltage is about V_in,max + V_DS0 + V_SG1. Instead, the supply voltage can be further reduced to V_in,max + V_SD1 if the PMOS is used as the driver transistor, as shown in Figure 1b.

The topology shown in Figure 1b is extended to another improved topology (Figure 2) [7], which is called the feedback voltage-attenuated (FBVA) V-I Converter.

In the FBVA structure, the OTA permits V_C = V_in under a high-gain feedback loop. With voltage dividing the function according to the feedback resistors R₁ and R₂ of the OTA_aux-M₁-M₄ feedback loop, V_A becomes αV_in, where α = R₂/(R₁ + R₂). Then, the output current will become I_out = αV_in∙K/R_s. As C_C2 and R_C2 create a zero to cancel a pole, this permits the FBVA V-I Converter to extend the bandwidth. Through the attenuation provided by the negative feedback loop, the FBVA V-I Converter displays low distortion levels. Despite this, this topology achieves improved temperature compensation over that of the conventional V-I Converter. However, this structure, making use of multiple OTAs, increases the complexity. Of particularly note, since it uses a cascade of transistors, it leads to an increase in supply voltage requirements.

Another reported circuit [14] consists of a differential pair with two nested feedback loops, as depicted in Figure 3. The two main loops (loop 1 and loop 2) consist of M₁ and M₂ and M₅ and M₆, used for reducing the input impedance through negative feedback. M₃, M₄ and the current source I₁ comprise a flipped voltage follower, which is used to reduce the source impedance of M₄, M₁, and M₂ such that the voltages of node X₁ and node X₂ are close to V_B1. This meets the condition of passing equal current. The third loop, which consists of M₅, M₇, and M₉, and the fourth loop, which consists of M₆, M_8, and M₁₀, are the high-gain negative feedback loops. They further reduce the impedance of nodes X₁ and X₂ whilst transferring the input current to the output without replication. Besides this, the high-gain feedback loop helps reduce the mismatch between each pair of transistors. Overall, the circuit achieves a high degree of linearity for voltage-to-current conversion. However, this circuit still relies on high supply voltage.

To allow very low-voltage operation, a bulk-driven CMOS inverter-based tunable transconductor has been proposed [15]. As depicted in Figure 4, M₅–M₁₂ form a positive feedback loop, which aims to provide a relatively large DC gain. All transistors operating in the subthreshold region permit the circuit to operate at a voltage as low as 0.25 V. Through the subthreshold current equation and the threshold voltage equation [15], the output current is derived as

$$i_o=i_p-i_n\approx {\displaystyle \frac{I_B}{2V_T}}\left({\displaystyle \frac{{\gamma }_p}{2n_p\sqrt{2\left|{\varphi }_{Fp}\right|-\left(V_{DD}-V_{CM}\right)}}}+{\displaystyle \frac{{\gamma }_n}{2n_n\sqrt{2\left|{\varphi }_{Fn}\right|-V_{CM}}}}\right)V_{in}.$$

(2)

From (2), the overall transconductance of the proposed circuit is obtained as ${\displaystyle \frac{I_B}{2V_T}}\left({\displaystyle \frac{{\gamma }_p}{2n_p\sqrt{2\left|{\varphi }_{Fp}\right|-\left(V_{DD}-V_{CM}\right)}}}+{\displaystyle \frac{{\gamma }_n}{2n_n\sqrt{2\left|{\varphi }_{Fn}\right|-V_{CM}}}}\right)$.

The advantage of this design is that it provides a good trade-off, offering ultra-low supply voltage with good linearity whilst providing tunable transconductance. However, the bulk-driven transistors also suffer from several disadvantages, such as the use of the triple-well CMOS technology in realization [15], a higher input noise, a higher offset voltage [16] and higher fabrication costs when compared to the single-well CMOS technology. Moreover, the substrate channel PN junction is likely to latch-up in cases of not being properly designed, causing reliability issues.

In order to avoid the undesirable effects of bulk driving, another reported work [17] focused on gate driving as shown in Figure 5. Due to the larger transconductance of the device, this circuit shows better noise performance when compared to that of bulk-driven transistors. It also optimizes the swing range of the output voltage. The circuit also generates sinking and sourcing currents and exhibits a large input common-mode voltage range. However, the current I₁ = (V_DDP − V_IN)/R may be subject to the offset arising from the matching between the sourcing current and the sinking current. The input impedance of the circuit depends on the effective output impedance of the current mirror.

3. Proposed V-I Converter

3.1. V-I Converter Architecture

Figure 6 illustrates the topology of the proposed circuit. The circuit operates in the subthreshold region. This circuit consists of peripheral resistors, level shifters, and the OTA_o. The resistors R₁–R₄ are formed by PMOS devices, which have small sizes and are known as pseudo-resistor (PR) devices [18]. These PR devices display very high impedance properties so that there is no loading effect on the input source and the output stage of OTA_o. The function of the level shifter is to raise the input voltages V₁ and V₂ to higher voltages V₃ and V₄ in order to satisfy the input voltage range of the OTA_o and to produce an intentional temperature coefficient in the circuit. The OTA_o consists of an OTA_i, which consists of two stages with active loads and a third stage formed by the PMOS driver transistor that drives the resistor array with the temperature compensation. The output current will be I_OUT = V_OUT/R_array. With R₁ = R₂ = R₃ = R₄ = R, V_OUT = V_IN. By choosing R₁ = R₂ = R₃ = R₄, a symmetrical circuit structure can be procued, which offers a good common mode rejection ratio.

3.1.1. Design of Pseudo Resistor (PR)

PR, which was introduced by Delbrück in 1984 [18], usually uses a PMOS transistor as a diode connection to operate in the cutoff region to provide an extremely large resistance. It is well-known and widely used for the acquisition of bioelectrical signals [19,20,21]. For the connection shown in Figure 7, if V₁ is greater than V₂, this structure is equivalent to a PMOS using a diode connection. If V₁ is less than V₂, the structure behaves like a bipolar transistor due to the positively biased PN junction formed by the drain of PMOS and the substrate. In this design, V₁ is greater than V₂, with V₁ − V₂ < |V_THP|, to allow PR to be in the cutoff region.

PR has an exponential I-V characteristic, and therefore poor linearity and a strong dependence on temperature and process. Another limitation of PR is the problem of current leakage due to parasitic effects. In diode-connected PRs, intrinsic leakage currents can exist in the diode formed between the p-substrate and the n-well of the PMOS transistor, thus affecting the output offset of the OTA [22]. As shown in Figure 8, I_c represents the current during normal conduction, whereas i_l represents the reverse leakage current of the parasitic diode. If V₁ is connected to the high voltage terminal and V₂ is connected to the low voltage terminal, an extra voltage drop is generated by i_l due to the leakage current in the reverse-biased diode. This phenomenon causes the mismatch at the two input ports of the OTA, leading to an output DC voltage offset. This problem will be further pronounced if the circuit features high gain. In order to suppress such a DC offset, the symmetric topology, depicted in Figure 6, is utilized to match the resistances in the positive and negative branches of the OTA. As such, the leakage current is treated as the common-mode signal and is rejected by the converter in a form amplifier topology.

3.1.2. Design of Level Shifter

The level shifter depicted in Figure 9 contains a startup circuit that comprises M_B1, M_B2, C_B1, and C_B2, a bias circuit that comprises M_B3–M_B6, R_B1, and a source follower formed by M_S1 and M_S2.

The bias current I_BIAS is calculated as

$$I_{BIAS}={\displaystyle \frac{n{V}_{T}In\left(M\right)}{R_{B1}}},$$

(3)

where M denotes the ratio of aspect ratio of M_B3 with respect to that of M_B4 and M_B1.

When the power is on, V₅ rises to V_DD, forcing M_B1 to turn on and causing the ac shorting V₇ to GND, thus making M_B2 on. As a result, V₆ is pulled high, which in turn causes the bias circuit to generate the bias current. V₇ is then pulled high due to M_B1’s operation, which causes M_B2 to be turned off. V₅ is pulled low after the bias current has been established. Then, M_B1 is turned off. When the capacitive-based startup circuit has completed its action, it is isolated and no longer contributes any additional current, hence reducing power consumption.

M_S2 and M_S1 form a source follower with the gate-bulk driving transistor M_S2. To raise identical voltage potential, the PMOS with gate-bulk connection uses a smaller size when compared to the PMOS with source-bulk connection. It is observed that the input-referred noise decreases with the increase in bias current in the level shifter. Although it permits lower input-referred noise, it degrades the temperature coefficient. On the contrary, the gate-bulk-connected PMOS can optimize such a trade-off. This is mainly because of the change in threshold voltage, which leads to a change in temperature coefficient, which favors the temperature compensation case.

3.1.3. Design of OTA_o

Due to the low supply voltage, the OTA can only operate in the subthreshold region, and the V_DS needs to be greater than 100 mV in order to allow the MOSFETs to operate in a valid region. Besides this, the use of a cascode structure to increase the DC gain is not permitted under ultra-low-voltage design. Although the two-stage topology on the basis of a single Miller compensation capacitor is simple and efficient in terms of size, the overall DC gain may be not sufficient to meet precision requirements due to the intrinsic low output impedance of CMOS devices in the 40 nm technology node. Therefore, in order to obtain an OTA with a large gain whilst operating in the subthreshold region, this OTA design is based on the modification of the amplifier structure [23], as shown in Figure 10a. The main difference is that of the passive temperature compensation resistor array replacing the active load in this design of the final output stage.

The system block diagram is shown in Figure 11, where g_m1 is the transconductance of the first stage, contributed by M₁. g_mcf is the transconductance of M₃. g_m2 is the transconductance of the second stage, contributed by M₅. g_m3 is the transconductance of the third stage, contributed by M₉. g_mf is the transconductance of the feedforward path, used to generate the left half-plane (LHP) zero, contributed by M₁, M₃, and M₈. R_cf is the effective output resistance of M₁ and is approximately equal to 1/g_mcf. R₁ is the output resistance of the first stage. R₂ is the output resistance of the second stage. R₃ is the output resistance of the third stage, consisting of R_array and the equivalent output impedance of M₉ in parallel. C_pcf is the equivalent capacitance at the output of M₁. C_p1 is the output capacitance of the first stage. C_p2 is the output capacitance of the second stage. C₃ is the output capacitance of the third stage. C₁, together with M₃ and M₄, forms a current buffer to reduce the potential large Q generated by the non-dominant pole, thus slowing down the sharp changes in phase. C₂ is a Miller capacitor used to achieve pole splitting and increase the phase margin.

To analyze the loop stability of this OTA_o, the following assumptions are made: (1) C₁, C₂, and C₃ are larger than the internal capacitances of the circuit; (2) g_m3 is larger than g_m1, g_mcf, and g_m2; and (3) the gain of each gain stage (g_m1R₁, g_m2R₂, and g_m3R₃) > 1, except for g_mcfR_cf ≈ 1. Thus the transfer function can be approximated as follows:

$$H\left(s\right)={\displaystyle \frac{-A_{DC}\cdot \left(1+s\frac{C_2{g}_{mf}}{g_{m1}{g}_{m2}}\right)}{\left(1+p_{1}s\right)\left[1+s\cdot \frac{C_1\left(C_3+C_2{g}_{m3}{R}_{cf}\right)}{C_2{g}_{m3}}+\frac{\left(C_{p2}+C_1\right)C_3}{g_{m2}{g}_{m3}}{s}^2\right]}},$$

(4)

where the DC gain is A_DC = g_m1g_m2g_m3g_mcfR₁R₂R₃R_cf ≈ g_m1g_m2g_m3R₁R₂R₃, and the dominant pole is p₁ = 1/(C₂g_m2g_m3R₁R₂R₃). Thus the gain-bandwidth product (GBW) is obtained as

$$GBW=A_{DC}\cdot p_1={\displaystyle \frac{g_{m1}}{C_2}}$$

(5)

From (4), this OTA_o yields a zero, as

$$z_1={\displaystyle \frac{g_{m1}{g}_{m2}}{C_2{g}_{mf}}}$$

(6)

which is designed for phase lead such that the circuit derives a better phase margin. The relationship in the denominator of the transfer function H(s) is

$${\left[{\displaystyle \frac{C_1\left(C_3+C_2{g}_{m3}{R}_{cf}\right)}{C_2{g}_{m3}}}\right]}^2\le 4{\displaystyle \frac{\left(C_{p2}+C_1\right)C_3}{g_{m2}{g}_{m3}}}.$$

(7)

The pair of complex poles becomes

$$\begin{array}{cc}\hfill w_O& =\left|{\displaystyle \frac{-\frac{C_1\left(C_3+C_2{g}_{m3}{R}_{cf}\right)}{C_2{g}_{m3}}\pm i\cdot \sqrt{4\frac{\left(C_{p2}+C_1\right)C_3}{g_{m2}{g}_{m3}}-{\left[\frac{C_1\left(C_3+C_2{g}_{m3}{R}_{cf}\right)}{C_2{g}_{m3}}\right]}^2}}{2\frac{\left(C_{p2}+C_1\right)C_3}{g_{m2}{g}_{m3}}}}\right|\hfill \\ & =\sqrt{{\displaystyle \frac{g_{m2}{g}_{m3}}{\left(C_{p2}+C_1\right)C_3}}}\hfill \end{array}.$$

(8)

Due to the complex conjugate poles, excessive Q values can occur. This causes the amplitude–frequency curve to display an overshoot effect. When this large Q occurs near the GBW, a sharp drop in the phase margin [24] is observed. The Q value is obtained as

$$Q=\sqrt{{\displaystyle \frac{C_{p2}{C}_3}{g_{m2}{g}_{m3}}}}\cdot {\displaystyle \frac{C_2{g}_{m3}}{C_1\left(C_3+C_2{g}_{m3}{R}_{cf}\right)}}.$$

(9)

To reduce Q, it is apparent from (9) that C₁, as well as R_cf, needs to be increased. Besides this, the phase margin [23] is given as follows:

$$\begin{array}{cc}\hfill PM& =180°-{\tan }^{-1}\left({\displaystyle \frac{GBW}{p_1}}\right)-{\tan }^{-1}\left\{{\displaystyle \frac{GBW/w_O}{Q\left[1-{\left(GBW/w_O\right)}^2\right]}}\right\}+{\tan }^{-1}\left({\displaystyle \frac{GBW}{z_1}}\right)\\ & =90°-{\tan }^{-1}\left({\displaystyle \frac{GBW/w_O}{Q\left[1-{\left(GBW/w_O\right)}^2\right]}}\right)+{\tan }^{-1}\left({\displaystyle \frac{GBW}{z_1}}\right)\hfill \end{array}.$$

(10)

To achieve programmable output current, the resistor array uses a combination of three branches, as shown in Figure 12. In the process library design file, the resistor is given by

$$R=R_0\left[1+T{C}_1\left(T-300\right)+T{C}_2{\left(T-300\right)}^2\right],$$

(11)

where TC₁ and TC₂ are the first-order temperature coefficient and the second-order temperature coefficient, respectively, and R₀ is the magnitude of the resistance at the temperature of 300 K. Based on the resistor–temperature relationship, the output current needed for this circuit, the P+ poly resistor without salicide, and the P+ poly resistor with salicide are chosen for each branch of the resistor array. The types and sizes of each device of the proposed V-I Converter are shown in

Table 1. Types and sizes of devices of proposed V-I Converter.

Device	Type	Size (W/L)	Device	Type	Size (W/L)
MB1	pch_lvt	10 μm/1 μm	M9	pch_lvt	840 μm/10 μm
MB2	pch_lvt	2 μm/1 μm	RB1	rppolywo	4 MΩ
MB3	pch_lvt	50.8 μm/20 μm	RB3	rppolywo	4.5 MΩ
MB4,S1	pch_lvt	50 μm/20 μm	R1–R4	pch_lvt	600 nm/110 nm
MB5,B6	nch_lvt	200 nm/6 μm	Rarray1a	rppoly	7.25 kΩ
MB13	nch	120 nm/1 μm	Rarray2a	rppoly	14.5 kΩ
MB14	nch	5 μm/1 μm	Rarray4a	rppoly	24.5 kΩ
MB15,B16	pch_lvt	28 μm/1 μm	Rarray1b	rppolywo	89 kΩ
MB17	nch_lvt	2.5 μm/1 μm	Rarray2b	rppolywo	178 kΩ
MB18	nch_lvt	12 μm/1 μm	Rarray4b	rppolywo	360.5 kΩ
MS2	pch_lvt	19 μm/200 nm	CB1	cap	10 pF
M0	nch_lvt	30 μm/1 μm	CB2	cap	10 pF
M1,2	nch_lvt	50 μm/1 μm	CB5,B6	cap	10 pF
M3,4	pch_lvt	8 μm/1 μm	C1	cap	1 pF
M5,8	pch_lvt	70 μm/200 nm	C2	cap	5 pF
M6,7	nch_lvt	6 μm/1 μm	C3	cap	10 pF

.

3.2. Temperature Compensation of Proposed V-I Converter

Temperature compensation is mainly realized by OTA_o and the resistor array, and in conjunction with the thermal effect contributed by the level shifter. The bias circuit is shown in Figure 10b, where M_B13, M_B14, C_B5 and C_B6 comprise the capacitive startup circuit. This constant-gm current source provides the tail current of OTA_o, and it is given as

$$I_{M0}={\displaystyle \frac{n{V}_{T}In\left[\frac{{\left(W/L\right)}_{MB18}}{{\left(W/L\right)}_{M0}}\right]}{R_{B3}}}.$$

(12)

It is noted that for a P+ poly resistor without salicide, R_B3 is given by (11), where TC₁ is negative and TC₂ is positive, with the relationship |TC₁| >> |TC₂|. Thus, R_B3 shows the CTAT effect. Then, the derivation of I_M0 is obtained as

$${\displaystyle \frac{\partial I_{M0}}{\partial T}}={\displaystyle \frac{nkIn\left[\frac{{\left(W/L\right)}_{MB18}}{{\left(W/L\right)}_{M0}}\right]}{q{R}_{B3}}}>0,$$

(13)

It can be seen that I_M0 is proportional to the absolute temperature (PTAT) current. I_M3 is equal to 0.5I_M0 and therefore also the PTAT current.

The current equation of PMOS operating in the subthreshold region is

$$\begin{array}{cc}\hfill I_S& ={\mu }_p\left(T_0\right){\left({\displaystyle \frac{T_0}{T}}\right)}^{m_p}{C}_{ox}{V}_T^2\left({\displaystyle \frac{W}{L}}\right)e^{\left({\displaystyle \frac{V_{SG}\left(T\right)+V_{THP}\left(T\right)}{n{V}_T}}\right)}\left(1-e^{{\displaystyle \frac{-V_{SD}\left(T\right)}{V_T}}}\right)\left[1-\lambda V_{SD}\left(T\right)\right]\\ & \approx {\mu }_p\left(T_0\right){\left({\displaystyle \frac{T_0}{T}}\right)}^{m_p}{C}_{ox}{V}_T^2\left({\displaystyle \frac{W}{L}}\right)e^{\left({\displaystyle \frac{V_{SG}\left(T\right)+V_{THP\left(T\right)}}{n{V}_T}}\right)}\left[1-\lambda V_{SD}\left(T\right)\right]\hfill \end{array},$$

(14)

with the electron mobility given as

$${\mu }_p\left(T\right)={\mu }_p\left(T_0\right)\cdot {\left({\displaystyle \frac{T_0}{T}}\right)}^{m_p}$$

(15)

and the threshold voltage given as

$$V_{THP}\left(T\right)=V_{THP0}+{\kappa }_P\left(T-T_0\right)$$

(16)

where the characteristic current in weak inversion I_S = μ_p(T)C_oxV_T², V_THP0 is the threshold voltage at T₀ = 300 K, μ_p(T₀) is the carrier mobility at T₀ = 300 K, C_ox is the capacitance of the gate oxide, W is the channel width, L is the channel length, V_T = kT/q is the thermal voltage, k is Boltzmann’s constant, q is the electronic charge, and the subthreshold slope n is a constant between 1 and 3 [25,26]. n can be estimated to be 1.3 using this technology, which is convenient for theoretical calculation. m_p is approximately 1.16, which is the exponential coefficient of μ_p(T) with respect to temperature, and κ_P is approximately 0.614 mV/K, which is the magnitude of the slope of V_THP(T). m_p and κ_P are obtained from PMOS device simulation. Since V_SD > 3 V_T, this means that the exp[−V_SD(T)/V_T] term is small enough to be ignored, while λ is the channel length modulation factor. According to (14), the voltage V_A in the first stage of OTA_o (Figure 10) is

$$\begin{array}{cc}\hfill V_A=& V_{DD}-V_{SG3}\hfill \\ \hfill =& V_{DD}-\left\{{\displaystyle \frac{nkT}{q}}\left[In\left({\displaystyle \frac{I_{M3}\cdot T^{m_p-2}}{{\mu }_p\left(T_0\right)T_0^{m_p}{C}_{ox}{\left(k/q\right)}^2{\left(W/L\right)}_{M3}}}\right)-{\displaystyle \frac{q{\kappa }_P}{nk}}\right]-V_{THP0,M3}+{\kappa }_P{T}_0\right\}\hfill \\ & \propto -{\alpha }_{1}T\left[In\left({\displaystyle \frac{{\alpha }_2{T}^{1.84}}{{\alpha }_3+{\alpha }_{4}T{C}_1\left(T-300\right)+{\alpha }_{5}T{C}_2{\left(T-300\right)}^2}}\right)-{\alpha }_6\right]\hfill \end{array},$$

(17)

where ɑ_n (n = 1, 2, 3…) is the lumped constant. With MATLAB (version: R2023a) verification, V_A can be considered as PTAT voltage. Due to the symmetrical structure of the first stage of OTA_o, V_O1 is equal to V_A, which implies V_O1 is also a PTAT voltage. Figure 13a shows the plot of V_O1 against temperature.

This PTAT voltage couples to the input of the second stage in the OTA_o. It generates the current through M₅. Since M₃ and M₅ have different channel lengths, this results in different threshold voltages. According to the BSIM3v3 model [27], V_THP0 in Equation (16) can be written as

$$V_{TH0}=V_{TH0}^{\prime }+K_1\left(\sqrt{{\varphi }_s-V_{bs}}-\sqrt{{\varphi }_s}\right)-K_2{V}_{bs}+K_1\left(\sqrt{1+{\displaystyle \frac{NLx}{L_{eff}}}}-1\right)\sqrt{{\varphi }_s}-\Delta V_{TH},$$

(18)

where K₁, K₂, NLx, and ∆V_TH are parameters given in [27], V’ TH0 is the threshold voltage at zero substrate bias, and ϕ_s is the surface potential. As the channel length becomes shorter, the threshold voltage shows a greater dependence on the channel length. In this design, M₅ has a smaller channel length than M₃, causing the threshold voltage V_THP0,M5 to be less than V_THP0,M3. The current I_M5 becomes

$$\begin{array}{cc}\hfill I_{M5}& =\left[{\mu }_p\left(T_0\right)T_0^{m_p}{C}_{ox}{\displaystyle \frac{k^2}{q^2}}{\left({\displaystyle \frac{W}{L}}\right)}_{M5}{e}^{{\displaystyle \frac{q{\kappa }_p}{nk}}}\right]e^{{\displaystyle \frac{q\left(V_{DD}-V_{O1}+V_{THP0,M5}-{\kappa }_P{T}_0\right)}{nkT}}}{T}^{2-m_p}\hfill \\ & ={\displaystyle \frac{{\left(W/L\right)}_{M5}}{{\left(W/L\right)}_{M3}}}\cdot I_{M3}\cdot e^{-{\displaystyle \frac{q{\kappa }_P}{nk}}}\cdot e^{{\displaystyle \frac{q\left(V_{THP0,M5}-V_{THP0,M3}\right)}{nkT}}}\hfill \end{array}.$$

(19)

In (19), V_THP0,M5 − V_THP0,M3 is negative, resulting in the term $e^{q\left(V_{THP0,M5}-V_{THP0,M3}\right)/\left(nkT\right)}$, showing a PTAT effect. Since I_M3 also has a PTAT effect, I_M5 becomes a PTAT current. This current passes through M₆ and generates the V_C in Figure 10. V_C can be represented as

$$\begin{array}{cc}\hfill V_C=& {\displaystyle \frac{nkT}{q}}\left[In\left({\displaystyle \frac{I_{M5}\cdot T^{m_n-1}}{{\mu }_n\left(T_0\right)T_0^{m_n}{C}_{ox}{\left(k/q\right)}^2{\left(W/L\right)}_{M6}}}\right)-{\displaystyle \frac{q{\kappa }_N}{nk}}\right]+V_{THN0,M6}+{\kappa }_N{T}_0\hfill \\ & \propto {\alpha }_7\cdot T\left[In\left({\alpha }_8\cdot I_{M5}{T}^{0.5}\right)-{\alpha }_9\right]+{\alpha }_{10}\hfill \end{array},$$

(20)

where m_n is approximately 1.5, and κ_N is approximately 1.589 mV/K. They are characterized by simulation in this technology. According to the analytical result, V_C is also a PTAT voltage. Due to the balanced design, V_O2 is approximately equal to V_C and it displays the PTAT effect. This is verified with the plot of V_O2 against the temperature, as illustrated in Figure 13b.

Assuming that the resistor array R_array of the third stage has a temperature-independent property, the output current I_M9 can be expressed as

$$\begin{array}{cc}\hfill I_{M9}=& \left[{\mu }_p\left(T_0\right)T_0^{m_p}{C}_{ox}{\displaystyle \frac{k^2}{q^2}}{\left({\displaystyle \frac{W}{L}}\right)}_{M9}{e}^{{\displaystyle \frac{q{\kappa }_P}{nk}}}\right]e^{{\displaystyle \frac{q\left(V_{DD}-V_{O2}+V_{THP0,M9}-{\kappa }_P{T}_0\right)}{nkT}}}{T}^{2-m_p}\\ \hfill =& {\mu }_p\left(T_0\right){\mu }_n\left(T_0\right)T_0^{m_p+m_n}{C_{ox}}^2{\displaystyle \frac{k^4}{q^4}}{\left({\displaystyle \frac{W}{L}}\right)}_{M9}{\left({\displaystyle \frac{W}{L}}\right)}_{M6}{e}^{{\displaystyle \frac{q\left({\kappa }_P+{\kappa }_N\right)}{nk}}}\cdot \hfill \\ & e^{{\displaystyle \frac{q\left(V_{DD}-V_{THN0,M6}+V_{THP0,M9}-{\kappa }_N{T}_0-{\kappa }_P{T}_0\right)}{nkT}}}\cdot {I_{M5}}^{-1}\cdot T^{0.34}\hfill \\ & \propto {\alpha }_{11}{e}^{{\displaystyle \frac{{\alpha }_{12}}{T}}}\cdot T^{0.34}\cdot {I_{M5}}^{-1}\hfill \end{array},$$

(21)

where ɑ₁₂ is negative and it causes the term $e^{{\alpha }_{12}/T}$ to display a PTAT effect. Although ${I_{M5}}^{-1}$ shows a CTAT effect, the term $e^{{\alpha }_{12}/T}{T}^{0.34}$ is dominant, resulting in I_M9 as a PTAT current. This PTAT current passes through R_array and will generate the output voltage V_OUT with PTAT characteristics. In order to ensure I_M9 is temperature independent, R_array needs to be positively temperature dependent so that V_OUT/R_array becomes temperature independent and achieves a low temperature coefficient for the output current. This can be synthesized using two types of poly resistors with opposite temperature coefficients. For the complete temperature compensation, the level shifter allows some modulation of the temperature coefficient to the output current. As depicted in Figure 9, the source follower M_S2 uses a gate-bulk connection instead of a source-bulk connection. The use of gate-bulk connection reduces the |V_TH| of PMOS and slightly modifies the trend of |V_TH| with temperature, as shown in Figure 14.

Compared to PMOS with source-bulk connection, the trend of |V_TH| has relatively low T.C. using the gate-bulk driven transistor in the source follower of the level shifter. According to (14), if the bias current is fixed, a more slowly decreasing |V_TH| will improve the CTAT effect of the output voltage of the level shifter. This voltage entering into the OTA_o will cause the PTAT effect of the output voltage of OTA_i shown in Figure 6, which offers better temperature compensation. It allows the driven transistor to receive more CTAT compensation, which compensates for the PTAT effect brought in by the resistor array and optimizes the temperature coefficient. On the whole, the gate-bulk connection improves the operation headroom and mitigates excessive PTAT effects due to the resistor array. It makes compensation of the temperature coefficient easier.

4. Results and Discussions

Realized using TSMC 40 nm CMOS technology, the proposed V-I Converter circuit operates at a supply voltage of 0.45 V.

The stability analysis of the OTA_o is first carried out. Its Bode plot at the TT process corner is shown in Figure 15. The DC open-loop gain is 64.79 dB, the phase margin (PM) is 80.07°, the bandwidth (BW) is 25.1 Hz, and the unity gain bandwidth (UGB) is 48.13 kHz. The stability analysis of OTA_o, considering TT, SS, and FF process corners and fluctuations in supply voltage, is summarized in

Table 2. Stability of OTA_o in PVT simulation.

VDD (V)	Process Corner	DC Gain (dB)	PM (°)	BW (Hz)	UGB (kHz)
0.4275	TT 1	66.6	82.29	19.95	65.45
	SS 2	67.75	79.06	13.94	51.58
	FF 3	65.12	85.52	30.96	86.37
0.45	TT	67.97	79.12	17.69	70.04
	SS	69.2	75.34	12.23	55.13
	FF	66.53	82.76	27.43	93.07
0.4725	TT	68.94	75.39	16.42	74.99
	SS	70.2	71.08	11.29	58.84
	FF	67.49	79.55	25.45	100.1

¹ Typical-Typical corner, representing nominal N transistors and nominal P transistors. ² Slow-Slow corner, representing slow N transistors and slow P transistors. ³ Fast-Fast corner, representing fast N transistors and fast P transistors.

. It can be observed that the circuit sustains the function regardless of the PVT variation.

For the closed-loop configuration with the capacitive load capacitor C₃ = 10 pF, the closed-loop gain (Figure 16) is −5.92 mdB, whereas the closed-loop bandwidth is 34.45 kHz. This indicates that the gain of the entire closed-loop system is about 1, achieving an output voltage equal to the input voltage.

The input DC voltage of the design is in the range of 0.1–0.3 V, and the programmable binary-weighted output currents at T = 300 K, V_DD = 0.45 V and the TT process corner are shown in Figure 17. The minimum output current is 260 nA, and the maximum output current is 5.44 μA.

Monte-Carlo simulations have bene used to evaluate the accuracy of the output currents under process variations and mismatch, which are presented in Figure 18. These Monte-Carlo simulations were performed at V_DD = 0.45 V and the input DC voltage V_IN = V_DD/2 = 0.225 V.

The results are summarized in

Table 3. Output currents in Monte-Carlo simulation.

	1	2	3	4	5	6	7
μ (μA)	2.335	1.17	3.503	0.586	2.92	1.755	4.087
σ (nA)	107.9	54.09	162.1	27.5	135.5	81.53	189.4
Sensitivity of Output currents [(σ/μ)%]	4.62	4.62	4.63	4.69	4.64	4.65	4.63

, showing the mean and the standard deviation of seven different output current levels, as well as the sensitivity of the respective output currents. It has been confirmed that the variation is around a few percentage points, which is acceptable in the context of process variation. Due to the programmable feature in this design, the variation can be compensated for by tuning in applications that require higher precision.

For each of the seven sets of output currents, their temperature coefficients at three different process corners are shown in Figure 19, where the temperature range is −20–80 °C.

The simulation results indicate that the T.C. ranges from 13.80 ppm/°C to 54.68 ppm/°C at a 0.45 V supply voltage and TT process corner. Considering TT, SS, and FF process corners, the T.C. ranges from 9.43 ppm/°C to 78.78 ppm/°C at 0.45 V supply voltage. When the voltage varies by ±5%, the T.C. ranges from 12.60 ppm/°C to 62.11 ppm/°C at the TT corner. The maximum and minimum values of T.C. are shown in

Table 4. T.C. summary.

VDD(V)	0.4275		0.45		0.4725		Overall
T.C. (ppm/°C)	Min.	Max.	Min.	Max.	Min.	Max.	Min.	Max.
TT 1	13.56	53.83	13.80	54.68	12.60	62.11	12.60	62.11
SS 2	36.72	79.48	34.97	78.78	35.88	81.85	34.97	81.85
FF 3	9.33	64.95	9.43	65.39	8.88	68.03	8.88	68.03
Min.	9.33		9.43		8.88		NA 4
Max.	79.48		78.78		81.85		NA

¹ Typical-Typical corner, representing nominal N transistors and nominal P transistors. ² Slow-Slow corner, representing slow N transistors and slow P transistors. ³ Fast-Fast corner, representing fast N transistors and fast P transistors. ⁴ NA means Not-Applicable.

. It can be observed that the variation in T.C. maintains reasonable values regardless of supply voltages and different corner variations.

In order to observe the change in T.C. under process mismatch and variations in statistical approach, seven sets of Monte-Carlo simulations, with each comprising 100 random sample points, are illustrated in Figure 20. These Monte-Carlo simulations are performed at V_DD = 0.45 V, and the input DC voltage V_IN = V_DD/2 = 0.225 V.

The results show that the mean value of the temperature constant is less than 52 ppm/°C, with a standard deviation of less than 13 ppm/°C. The obtained statistical performance parameters are listed in

Table 5. T.C. in Monte-Carlo simulation.

	1	2	3	4	5	6	7
μ (ppm/°C)	50.54	49.92	51.10	30.08	46.81	43.58	48.30
σ (ppm/°C)	12.74	12.96	12.36	8.33	11.68	11.46	11.69
Sensitivity of T.C. [(σ/μ)%]	25.20	25.96	24.19	27.69	24.95	26.30	24.20

.

For the benefits of using gate-bulk connections in level shifters, a comparison of the temperature coefficients of the output currents using gate-bulk connection and source-bulk connection is carried out at a supply voltage of 0.45 V and the TT process corner. This comparison is shown in Figure 21. The use of gate-bulk connection exhibits smaller and more stable temperature coefficients. Thus, the use of gate-bulk connections in level shifters helps optimize the temperature coefficient of the output current.

To evaluate the PSRR, the resistive test load is connected in parallel with C₄ = 100 pF in Figure 6. From Figure 22, we see that the PSRR at low frequency is 44.96 dB.

The CMRR of this circuit is shown in Figure 23. From Figure 23, we see that the CMRR at low frequency is 47.44 dB.

Figure 24 illustrates the input-referred noise spectrum. The input-referred noise is 4.93 µV/sqrt(Hz) at an input signal frequency of 1 kHz.

Figure 25 illustrates the relationship between THD and the peak-to-peak value of the input signal for different input frequencies. In this evaluation, the input signal has a DC quiescent bias voltage of 0.225 V, and the V-I Converter is powered at 0.45 V supply.

With THD = 1%, i.e., −40 dB as an upper limit, the input peak-to-peak value V_pp can reach a maximum of 0.36 V when the input frequency is 1 kHz. When V_pp is 0.225 V, the maximum input frequency can reach 2 kHz.

Table 6. Simulated performance comparison with previously reported works.

	2001 [29]	2007 [14]	2013 [7]	2019 [30]	2020 [17]	This Work
Technology (μm)	NA 1	0.5	0.18	0.18	0.13	0.04
Supply voltage (V)	+1/−2	±1.5	1.2	0.3	±0.2	0.45
Input range (V)	NA	0–3	0–1.1	0–0.3	−0.1–0.1	0.1–0.3
Temperature range (°C)	0–70	NA	−40–120	NA	NA	−20–80
T.C. (ppm/°C)	276	NA	113	NA	NA	54.68
Power consumption (μW)	NA	3000	85	0.01–0.1	0.36	0.36–2.76
BW (Hz)	NA	90 M	14.1 M	50–334	1.1 M	34.45 k
PSRR (dB)	46.2	35/43	47.8	50.2	52	45.58
CMRR (dB)	NA	62	NA	54.9	70	47.35
Input referred noise [µV/sqrt(Hz)]	NA	1.7	0.26	1.33	0.99	4.93
THD (dB@Vpp@kHz)	NA	−60@6@100	−44.2@1@100	−46@0.1@0.1	−41.61@0.1@10	−56.66@0.1@1
FOM {µW·[µV/sqrt(Hz)]/Hz}	NA	5.67 × 10−7	1.56 × 10−6	3.98 × 10−4	3.24 × 10−7	5.15 × 10−5

¹ NA means Not-Applicable.

shows a comparison in terms of performance with the previously reported designs. The figure of merit (FOM) is introduced to evaluate the quality of the work on a comparative basis. FOM is given by the following equation [28]:

$$FOM={\displaystyle \frac{P_w\times Input Referred Noise}{Bandwidth}}\mu W\cdot \left(\mu V/\sqrt{Hz}\right)/Hz$$

(22)

where P_w is the power consumption of the system. The smaller the FOM, the better the overall performance of the circuit.

Referring to Table 6, we see that the conventional FOM does not account for temperature stability—a critical metric for precision analog circuits. The proposed circuit has demonstrated several key advantages for very low-voltage applications. First, this design achieves a temperature coefficient of 54.68 ppm/°C, which is 2× better than in [7] (113 ppm/°C) and 5× better than in [29] (276 ppm/°C). This ensures stable operation across a wide temperature range (−20 °C to 80 °C), making it suitable for precision applications in the context of temperature variation. Additionally, the ability to program output currents enhances its flexibility, enabling the circuit to be adapted to diverse scenarios, from biomedical sensors to energy harvesting systems. With a power consumption of only 0.36–2.76 μW, the circuit is ideal for battery-powered or energy-constrained applications, such as wearable devices and IoT nodes.

The proposed circuit also achieves a CMRR of 47.35 dB and a PSRR of 45.58 dB, which are comparable to those in the representative works. To optimize the trade-off between input-referred noise and temperature stability, a gate-bulk connection in the level shifter, which replaces the conventional source-bulk connection, is suggested. Although the circuit operates in the subthreshold region with Q-factor frequency compensation for stabilizing multiple gain stages in exchange for reduced bandwidth (34.45 kHz), the key objectives for attaining low-power consumption as well as the low temperature coefficient of the output currents have been confirmed. Future work can explore advanced compensation techniques to further improve bandwidth without compromising these critical advantages.

5. Conclusions

This paper describes a V-I Converter embedded with a three-stage OTA, which is based on a modified conventional structure. The circuit is designed using the TSMC 40 nm process and functions at a supply voltage of 0.45 V, enabling operation in the subthreshold region. The circuit has a programmable resistor array for a wide range of output currents when compared to previous designs, and guarantees a low temperature coefficient when pushing for the precision requirement. The low temperature coefficient is a critical parameter for very low-voltage cells. In addition, it has achieved lower power consumption and reasonable PSRR and CMRR. Therefore, the design can be adapted to a variety of application scenarios, and is useful for low-voltage, low-power analog signal processing and sensor signal processing applications. Compared to the previously reported designs, this design has lower bandwidth, which is due to the complicated frequency compensation allowed by its three-stage structure. This results in a restricted operating frequency range, leading to one of the limitations of the design. To this end, the design of the V-I Converter with low temperate coefficient awareness gives the extra benefit of larger operation headroom in the context of temperature variation.