A 40 μW–30 mW generated power, 280 Ω–1.68 kΩ load resistance CMOS controllable constant-power source for thermally-based sensor applications

Abstract

A controllable constant-power source in 0.35 μm CMOS technology is presented in this paper. It is based on the resistive mirror method, and suitable for thermally-based sensor applications. Two versions have been developed and fabricated: a high-voltage design with a single supply of 10 V, and a low-voltage design with a single supply of 3.6 V. The measured results for the high voltage (low voltage) design are the following: a generated power dynamic range of 57.5 dB (52.4 dB), a load resistance dynamic range of 15.6 dB (12 dB), a voltage efficiency of 0.7 (0.61), and a relative error of the generated power less than 1.8% (1.8%). The temperature compensation of the controllable constant-power source has been performed for the temperature range 0 °C ≤ T ≤ 50 °C. The stability test has been carried out using a resistive load in pulse mode operation confirming that the stability of the proposed controllable constant-power source is not dependent on either the load resistance or the generated power.

Appendix derivation of the temperature coefficients of the offset voltages

The offset voltage V_OFFOA2 of OA₂ and its temperature coefficient ∂V_OFFOA2/∂T will be derived in this appendix. The derivations of the offset voltage V_OFFOA1 of OA₁ and its temperature coefficient ∂V_OFFOA1/∂T are similar to those of OA₂. The derivations of the offset voltage V_OFFCCI of the voltage follower part of the CCI and its temperature coefficient ∂V_OFFCCI/∂T are similar to those of the differential pair M₁₅–M₁₆ within OA₂.

The offset voltage V_OFFOA2 of OA₂ is presented by the difference of the input voltages V_G24 and V_G25 of OA₂, Fig. 1, required to drive the output voltage of OA₂ to the value V_OUT2 = V_DD − V_SG19

$$ V_{OFFOA2} = V_{G24} - V_{G25} = V_{SG25} - V_{GS16} + V_{GS15} - V_{SG24} = V_{OFFSF2} + V_{OFFDA2} $$

(33)

where V_GS15 and V_GS16 are the gate-to-source voltages of M₁₅ and M₁₆, respectively, V_SG24 and V_SG25 are the source-to-gate voltages of M₂₄ and M₂₅, respectively. So, the offset voltage V_OFFOA2 of OA₂ consists of two parts: the offset voltage V_OFFSF2 caused by the imperfections of the source followers made by M₂₄ and M₂₅ given by

$$ V_{OFFSF2} = V_{SG25} - V_{SG24} $$

(34)

and the offset voltage V_OFFDA2 caused by the imperfections of the differential pair M₁₅–M₁₆ expressed as

$$ V_{OFFDA2} = V_{GS15} - V_{GS16} $$

(35)

Consequently, the temperature coefficient ∂V_OFFOA2/∂T of the offset voltage of OA₂ can be calculated as follows

$$ \frac{{\partial V_{OFFOA2} }}{\partial T} = \frac{{\partial V_{OFFSF2} }}{\partial T} + \frac{{\partial V_{OFFDA2} }}{\partial T} $$

(36)

Assuming a simple quadratic model of saturated MOSFET [18,19,20] and neglecting the channel length modulation, the drain currents I_D24 and I_D25 of M₂₄ and M₂₅, respectively, are given by

$$ I_{D24} = \frac{1}{2}\beta_{24} \left( {V_{SG24} + V_{t24} } \right)^{2} = \frac{{V_{B6} - V_{SG24} - V_{G24} }}{{R_{5} }} $$

(37)

$$ I_{D25} = \frac{1}{2}\beta_{25} \left( {V_{SG25} + V_{t25} } \right)^{2} = \frac{{V_{B7} - V_{SG25} - V_{G25} }}{{R_{6} }} $$

(38)

The following expressions for the source-to-gate voltages V_SG24 and V_SG25 of M₂₄ and M₂₅, respectively, are obtained by solving the quadratic Eqs. (37) and (38)

$$ V_{SG24} \approx - V_{t24} + \sqrt {2\frac{{V_{B6} + V_{t24} }}{{\beta_{24} R_{5} }}} $$

(39)

$$ V_{SG25} \approx - V_{t25} + \sqrt {2\frac{{V_{B7} + V_{t25} }}{{\beta_{25} R_{6} }}} $$

(40)

The relation (39) have been derived assuming β₂₄R₅|V_t24| ≫ 1, V_B6 + V_t24 ≫ V_G24, and 2β₂₄R₅(V_B6 + V_t24) ≫ 1 (β₂₄ ≈ 0.5 mA/V², R₅ = 20 kΩ, V_t24 = − 1.35 V, V_B6 = 3.2 V, V_G24 < 100 mV for the HV CCPS, and β₂₄ ≈ 30 mA/V², R₅ = 20 kΩ, V_t24 = − 0.74 V, V_B6 = 1.4 V, V_G24 < 50 mV for the LV CCPS). The relation (40) have been derived assuming β₂₅R₆|V_t25| ≫ 1, V_B7 + V_t25 ≫ V_G25, and 2β₂₅R₆(V_B7 + V_t25) ≫ 1 (β₂₅ ≈ 0.5 mA/V², R₆ = 20 kΩ, V_t25 = − 1.35 V, V_B7 = 3.15 V, V_G24 < 100 mV for the HV CCPS, and β₂₅ ≈ 30 mA/V², R₆ = 20 kΩ, V_t25 = − 0.74 V, V_B7 = 1.39 V, V_G25 < 50 mV for the LV CCPS). Using (34), (39), and (40) the following expression can be derived for the offset voltage V_OFFSF2 caused by the imperfections of the source followers made by M₂₄ and M₂₅

$$ V_{OFFSF2} = V_{SG25} - V_{SG24} = V_{t24} - V_{t25} + \sqrt 2 \frac{{\sqrt {\left( {V_{B7} + V_{t25} } \right)\beta_{24} R_{5} } - \sqrt {\left( {V_{B6} + V_{t24} } \right)\beta_{25} R_{6} } }}{{\sqrt {\beta_{24} \beta_{25} R_{5} R_{6} } }} $$

(41)

Using the following substitutions: ΔV_t2425 = V_t24 − V_t25, V_t2425 = (V_t24 + V_t25)/2, Δβ₂₄₂₅ = β₂₄ − β₂₅, β₂₄₂₅ = (β₂₄ + β₂₅)/2, ΔR₅₆ = R₅ − R₆, R₅₆ = (R₅ + R₆)/2, ΔV_B67 = V_B6 − V_B7, V_B67 = (V_B6 + V_B7)/2, the relation (41) is transformed into the following form

$$ V_{OFFSF2} \approx \Delta V_{t2425} + \sqrt {\frac{{V_{B67} + V_{t2425} }}{{2\beta_{2425} R_{56} }}} \left( {\frac{{\Delta \beta_{2425} }}{{\beta_{2425} }} + \frac{{\Delta R_{56} }}{{R_{56} }} - \frac{{\Delta V_{B67} + \Delta V_{t2425} }}{{V_{B67} + V_{t2425} }}} \right) $$

(42)

The relation (42) have been derived using the following approximation: (1 + x)^1/2 ≈ 1 + x/2, for x ≪ 1. The temperature coefficient the offset voltage V_OFFSF2 (42) caused by the imperfections of the source followers made by M₂₄ and M₂₅ can be calculates as follows:

$$ \frac{{\partial V_{OFFSF2} }}{\partial T} \approx \frac{1}{2}\sqrt {\frac{{V_{B67} + V_{t2425} }}{{2\beta_{2425} R_{56} }}} \left( {\frac{{\Delta \beta_{2425} }}{{\beta_{2425} }} + \frac{{\Delta R_{56} }}{{R_{56} }} - \frac{{\Delta V_{B67} + \Delta V_{t2425} }}{{V_{B67} + V_{t2425} }}} \right) \cdot \left[ {\frac{1}{{V_{B67} + V_{t2425} }}\frac{{\partial \left( {V_{B67} + V_{t2425} } \right)}}{\partial T} - \frac{1}{{\beta_{2425} }}\frac{{\partial \beta_{2425} }}{\partial T} - \frac{1}{{R_{56} }}\frac{{\partial R_{56} }}{\partial T}} \right] $$

(43)

Using (42), the relation (43) becomes

$$ \frac{{\partial V_{OFFSF2} }}{\partial T} \approx \frac{1}{2}\left( {V_{OFFSF2} - \Delta V_{t2425} } \right)\left[ {\frac{1}{{V_{B67} + V_{t2425} }}\frac{{\partial \left( {V_{B67} + V_{t2425} } \right)}}{\partial T} - \frac{1}{{\beta_{2425} }}\frac{{\partial \beta_{2425} }}{\partial T} - \frac{1}{{R_{56} }}\frac{{\partial R_{56} }}{\partial T}} \right] $$

(44)

The derivation of the offset voltage V_OFFDA2 caused by the imperfections of the differential pair M₁₅–M₁₆ is similar to that in [20]. However, because the channel length modulation of M₁₅, M₁₆, M₁₉, and M₂₀ can be neglected thanks to the telescopic design, this derivation is slightly different compared to [20], where a differential pair with simple current mirror load is discussed.

With sufficiently small biasing voltage V_B2 = V_B5, drain-to-source voltages V_DS15 and V_DS16 of M₁₅ and M₁₆, respectively, are small enough to neglect the influence of the channel length modulation in these MOSFETs. So, assuming a simple quadratic model, the gate-to-source voltages V_GS15 and V_GS16 of M₁₅ and M₁₆, respectively, are given by

$$ V_{GS15} \approx \sqrt {\frac{{2I_{D15} }}{{\beta_{15} }}} + V_{t15} $$

(45)

$$ V_{GS15} \approx \sqrt {\frac{{2I_{D15} }}{{\beta_{15} }}} + V_{t15} $$

(46)

Using (35), (45), and (46) the following expression can be derived for the offset voltage V_OFFDA2 caused by the imperfections of the differential pair M₁₅–M₁₆

$$ V_{OFFDA2} = V_{GS15} - V_{GS16} = V_{t15} - V_{t16} + \sqrt 2 \frac{{\sqrt {\beta_{16} I_{D15} } - \sqrt {\beta_{15} I_{D16} } }}{{\sqrt {\beta_{15} \beta_{16} } }} $$

(47)

Using the following substitutions: ΔV_t1516 = V_t15 − V_t16, Δβ₁₅₁₆ = β₁₅ − β₁₆, β₁₅₁₆ = (β₁₅ + β₁₆)/2, ΔΙ_D1516 = I_D15 − I_D16, I_D1516 = (I_D15 + I_D16)/2, the relation (47) is transformed into the following form

$$ V_{OFFDA2} \approx \Delta V_{t1516} + \sqrt {\frac{{I_{D1516} }}{{2\beta_{1516} }}} \left( {\frac{{\Delta I_{D1516} }}{{I_{D1516} }} - \frac{{\Delta \beta_{1516} }}{{\beta_{1516} }}} \right) $$

(48)

The relation (48) have been derived using the following approximation: (1 + x)^1/2 ≈ 1 + x/2, for x ≪ 1. With sufficiently large biasing voltage V_B4, source-to-drain voltages V_SD19 and V_SD20 of M₁₉ and M₂₀, respectively, are small enough to neglect the influence of the channel length modulation in these MOSFETs. So, assuming a simple quadratic model, the source-to-gate voltages V_SG19 and V_SG20 of M₁₉ and M₂₀, respectively, are given by

$$ V_{SG19} \approx \sqrt {\frac{{2I_{D19} }}{{\beta_{19} }}} - V_{t19} $$

(49)

$$ V_{SG20} \approx \sqrt {\frac{{2I_{D20} }}{{\beta_{20} }}} - V_{t20} $$

(50)

Because V_SG19 = V_SG20, the following relation can be derived using (49) and (50)

$$ V_{t19} - V_{t20} \approx \sqrt 2 \frac{{\sqrt {\beta_{20} I_{D19} } - \sqrt {\beta_{19} I_{D20} } }}{{\sqrt {\beta_{19} \beta_{20} } }} $$

(51)

Using the following substitutions: ΔV_t1920 = V_t19 − V_t20, Δβ₁₉₂₀ = β₁₉ − β₂₀, β₁₉₂₀ = (β₁₉ + β₂₀)/2, ΔΙ_D1920 = I_D19 − I_D20, I_D1920 = (I_D19 + I_D20)/2, the relation (51) is transformed into the following form

$$ \Delta V_{t1920} \approx \sqrt {\frac{{I_{D1920} }}{{2\beta_{1920} }}} \left( {\frac{{\Delta I_{D1920} }}{{I_{D1920} }} - \frac{{\Delta \beta_{1920} }}{{\beta_{1920} }}} \right) $$

(52)

The relation (52) have been derived using the following approximation: (1 + x)^1/2 ≈ 1 + x/2, for x ≪ 1. Taking into account the fact that the drain currents of M₁₅ and M₁₉ are equal, I_D15 = I_D19, and that the drain currents of M₁₆ and M₂₀ are equal, I_D16 = I_D20, the following equalities are valid: ΔI_D1516 = ΔI_D1920, I_D1516 = I_D1920. Hence, the following expression can be derived from the relation (52)

$$ \frac{{\Delta I_{D1516} }}{{I_{D1516} }} = \frac{{\Delta I_{D1920} }}{{I_{D1920} }} \approx \frac{{\Delta \beta_{1920} }}{{\beta_{1920} }} + \sqrt {\frac{{2\beta_{1920} }}{{I_{D1920} }}} \Delta V_{t1920} $$

(53)

The offset voltage V_OFFDA2 caused by the imperfections of the differential pair M₁₅–M₁₆ is obtained using (48) and (53)

$$ V_{OFFDA2} \approx \Delta V_{t1516} + \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \Delta V_{t1920} + \sqrt {\frac{{I_{D1516} }}{{2\beta_{1516} }}} \left( {\frac{{\Delta \beta_{1920} }}{{\beta_{1920} }} - \frac{{\Delta \beta_{1516} }}{{\beta_{1516} }}} \right) $$

(54)

The temperature coefficient the offset voltage V_OFFDA2 (54) caused by the imperfections of the differential pair M₁₅–M₁₆ can be calculates as follows:

$$ \begin{aligned} \frac{{\partial V_{OFFDA2} }}{\partial T} & \approx \frac{1}{2}\Delta V_{t1920} \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \left( {\frac{1}{{\beta_{1920} }}\frac{{\partial \beta_{1920} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ & \quad + \frac{1}{2}\left( {\frac{{\Delta \beta_{1920} }}{{\beta_{1920} }} - \frac{{\Delta \beta_{1516} }}{{\beta_{1516} }}} \right)\sqrt {\frac{{I_{D1516} }}{{2\beta_{1516} }}} \left( {\frac{1}{{I_{D1516} }}\frac{{\partial I_{1516} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ \end{aligned} $$

(55)

Using (54), the relation (55) becomes

$$ \begin{aligned} \frac{{\partial V_{OFFDA2} }}{\partial T} & \approx \frac{1}{2}\Delta V_{t1920} \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \left( {\frac{1}{{\beta_{1920} }}\frac{{\partial \beta_{1920} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ & \quad + \frac{1}{2}\left( {V_{OFFDA2} - \Delta V_{t1516} - \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \Delta V_{t1920} } \right)\left( {\frac{1}{{I_{D1516} }}\frac{{\partial I_{1516} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ \end{aligned} $$

(56)

Finally, the temperature coefficient ∂V_OFFOA2/∂T = ∂V_OFFSF2/∂T + ∂V_OFFSDA2/∂T of the offset voltage of OA₂ is obtained using the relations (36), (44), and (56)

$$ \begin{aligned} \frac{{\partial V_{OFFOA2} }}{\partial T} & \approx \frac{1}{2}\left( {V_{OFFSF2} - \Delta V_{t2425} } \right)\left[ {\frac{1}{{V_{B67} + V_{t2425} }}\frac{{\partial \left( {V_{B67} + V_{t2425} } \right)}}{\partial T} - \frac{1}{{\beta_{2425} }}\frac{{\partial \beta_{2425} }}{\partial T} - \frac{1}{{R_{56} }}\frac{{\partial R_{56} }}{\partial T}} \right] \\ & \quad + \frac{1}{2}\Delta V_{t1920} \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \left( {\frac{1}{{\beta_{1920} }}\frac{{\partial \beta_{1920} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ & \quad + \frac{1}{2}\left( {V_{OFFDA2} - \Delta V_{t1516} - \sqrt {\frac{{\beta_{1920} }}{{\beta_{1516} }}} \Delta V_{t1920} } \right)\left( {\frac{1}{{I_{D1516} }}\frac{{\partial I_{D1516} }}{\partial T} - \frac{1}{{\beta_{1516} }}\frac{{\partial \beta_{1516} }}{\partial T}} \right) \\ \end{aligned} $$

(57)