Fully integrated inductive ring oscillators operating at V_DD below 2kT/q

Abstract

This paper presents two fully integrated inductive ring oscillators that operate with supply voltages below 2kT/q for energy harvesting applications. Expressions for the oscillation frequency as well as the minimum transistor gain and supply voltage required for the starting up of oscillations are derived for each topology. The experimental results for two cross-coupled oscillators, with topologies comprised of a single-inductor and two-inductors per stage, are presented. The two oscillators operate with supply voltages as low as V _DD  = 46 mV at 11.5 µW DC power and V _DD  = 31 mV at 15 µW DC power, respectively, thus confirming the extremely low voltage operation of the prototypes integrated in a 130 nm technology.

Appendices

Appendix 1—Oscillation frequency of the inductive ring oscillator with the inclusion of C _gd

The circuit of a single stage of the IRO, as well as its small-signal model with the inclusion of the gate-drain capacitance C _gd , is shown in Fig. 17, where the symbols have the same meaning as in Fig. 3.

Fig. 17

The inductive ring oscillator and the corresponding small-signal model of a single stage

The phase shift ϕ between two adjacent stages of the ring oscillator is given by ϕ = 2 kπ/N, where N is the number of stages and k is an integer. Since the node voltages in Fig. 17 are related as V ₂ = V ₁e^{j ϕ} and V ₃ = V ₂e^jϕ, we can redraw the small-signal circuit of Fig. 17 as shown in Fig. 18, with C _T  = C + 2C _gd (1−cos ϕ).

Fig. 18

Small-signal model equivalent to a single stage of the inductive ring oscillator

Thus, the transfer function of the single stage in Fig. 18 is given by

$$\frac{{I_{out} }}{{I_{in} }}(s) = - \frac{{g_{m} }}{{g_{md} + G_{P} + \frac{1}{sL} + sC_{T} }}$$

(20)

In this case, the resonant frequency ω ₀ is given by ω ²₀ LC _T  = 1. For an even number of stages ϕ = π and thus C _T  = C + 4C _gd .

Appendix 2—MOSFET model

The MOSFET model used throughout this paper is the Unified Current Control Model (UICM) [14], in which the current I _D is written as a combination of the forward (I _F ) and reverse (I _R ) currents

$$I_{D} = I_{F} - I_{R} = I_{S} \left( {i_{f} - i_{r} } \right)$$

(21)

I _S is the specific current, a parameter slightly dependent on the gate voltage, but here assumed to be independent of the gate voltage, i _f and i _r are the normalized forward and reverse currents, respectively, and

$$\frac{{V_{P} - V_{SB(DB)} }}{{\phi_{t} }} = \ln \left( {\sqrt {1 + i_{f(r)} } - 1} \right) + \sqrt {1 + i_{f(r)} } - 2$$

(22)

$$V_{P} = \frac{{V_{GB} - V_{T} }}{n}$$

(23)

where V _P is the pinch-off voltage, V _T is the threshold voltage, and n is the slope factor, assumed to be independent of the gate voltage. The differentiation of the current with respect to V _S , V _D , and V _G allows us to write

$$g_{ms} = - \frac{{\partial I_{D} }}{{\partial V_{S} }} = \frac{{2I_{S} }}{{\phi_{t} }}\left( {\sqrt {1 + i_{f} } - 1} \right)$$

(24)

$$g_{md} = \frac{{\partial I_{D} }}{{\partial V_{D} }} = \frac{{2I_{S} }}{{\phi_{t} }}\left( {\sqrt {1 + i_{r} } - 1} \right)$$

(25)

$$g_{m} = \frac{{\partial I_{D} }}{{\partial V_{G} }} = \frac{{g_{ms} - g_{md} }}{n}$$

(26)

The drain–source voltage V _DS can be expressed in terms of i _f and i _r using (22). The resulting expression for V _DS can be subsequently written in terms of the transconductances making use of (24) and (25), which yields

$$\frac{{V_{DS} }}{{\phi_{t} }} = \ln \left( {\frac{{\sqrt {1 + i_{f} } - 1}}{{\sqrt {1 + i_{r} } - 1}}} \right) + \sqrt {1 + i_{f} } - \sqrt {1 + i_{r} } = \ln \frac{{g_{ms} }}{{g_{md} }} + \frac{{\phi_{t} }}{{2I_{S} }}\left( {g_{ms} - g_{md} } \right)$$

(27)