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Modeling of Cross-Coupled AC–DC Charge Pump Operating in Subthreshold Region

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19 November 2023

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21 November 2023

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Abstract

This paper proposes a circuit model of Cross-Coupled CMOS AC – DC Charge Pump (XC – CP) operating in subthreshold region. The aim is improving the efficiency to design XC – CPs with variety of specifications, e.g., input and output voltages and AC input frequency. First, it is shown that the output resistance (Ro) of XC – CP is much higher than those of CPs with single diodes (SD – CP) and ultra-low-power diodes (ULPD – CP) as charge transfer switches (CTSs). Second, the reason behind the above feature of XC – CP is identified by a simple model that the gate-to-source voltages of CTS MOSFETs are independent of the output voltage of the CP. Third, high but finite Ro of XC – CP is explainable with a more accurate model that includes the dependence of the saturation current of MOSFETs operating in subthreshold region on the drain-to-source voltage which is a function of the output voltage of CP. The model was in good agreement with measured and simulated results of XC – , SD – and ULPD – CPs fabricated in 250 nm CMOS.

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Subject: Engineering - Electrical and Electronic Engineering

1. Introduction

AC – DC charge pumps (CPs) are used to convert AC input power to DC output power for microwave wireless power transmission (MWPT) or RF energy harvesting (RF – EH) [1,2,3,4,5,6,7,8,9,10,11], biomedical application [12,13,14,15,16] and (c) vibration energy harvesting [17,18,19,20], as shown in Figure 1 (a) – (c). Figure 1 (d) illustrates a circuit diagram of CP composed of multiple rectifiers and capacitors connected in series between the input and output terminals. The capacitors are driven by differential signals CLK and CLKB, alternately. A filtering capacitor is connected at the output terminal to generate a DC voltage. The DC voltage is used for the following circuit blocks or ICs such as sensors and RF ICs in IoT edge modules or medical devices. The rectifier used in AC – DC CPs was originally a diode-connected single MOSFET [21,22,23], as shown in Figure 2(a). When the amplitude of the input AC voltage is low in low power systems, reverse leakage becomes significant concern. To reduce the reverse leakage current, ultra-low-power diodes were proposed and used in low power AC – DC CPs [24,25,26], as shown in Figure 2(b). To boost the gate voltages of switching MOSFETs for increasing the forward current, cross-coupled CMOS or CMOS latch has been widely used [27,28,29,30,31,32,33,34], as shown in Figure 2(c).

The frequency of AC signals is spread depending on use cases. MWPT utilizes ISM bands such as 920 MHz [1 – 3], 2.4 GHz [4 – 7], 5.8 GHz [8 – 9], and 24 GHz [10 – 11]. To have low tissue attenuation, ultrasound with moderate frequencies of 6.78 MHz and 13.56 MHz is used for biomedical application [12,13,14,15,16]. Fundamental resonant frequency used for vibration energy harvesting is nominally at 1 –100Hz [17 – 20]. Thus, improving the efficiency to design AC – DC CPs is required. A circuit model plays a key role to improve the design efficiency. Models for AC – DC CPs with single diodes have been developed in [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. DC – DC CPs with cross-coupled CMOS operating in triode region was modeled in [41]. However, the output resistance of the CP (Ro) was given not specifically but by the traditional method of N/fC coth(Ton/Ron C), where N is the number of stages, f clock frequency, C stage capacitance, Ton the period that the MOSFET turns on, Ron on-resistance of the MOSFET.

To directly take a look at differences in Ro among SD – CP, ULPD – CP and XC – CP, SPICE simulation was done using the design parameters in 65nm CMOS as shown in Table 1, resulting in Figure 3. Surprisingly, Ro of XC – CP was 3X higher than those of SD – CP and ULPD – CP. As a result, the previous model [39,40] needs to be modified to predict high Ro for XC – CP.

This paper proposes a circuit model of cross-coupled CMOS AC – DC charge pump (XC – CP) operating in subthreshold region. The aim is improving the efficiency to design XC – CPs with variety of specifications, e.g., input and output voltages and AC input frequency. First, it is shown that the output resistance (Ro) of XC-CP is much higher than those of CPs with single diodes (SD – CP) and ultra-low-power diodes (ULPD – CP) as charge transfer switches (CTSs). Second, the reason behind the above feature of XC-CP is identified by a simple model that the gate-to-source voltages of CTS MOSFETs are independent of the output voltage of the CP. Third, high but finite Ro of XC – CP is explainable with a more accurate model that includes the dependence of the saturation current of MOSFETs operating in subthreshold voltage on the drain-to-source voltage which is a faction of the output voltage of CP. The model was in good agreement with measured and simulated results of XC –, SD – and ULPD – CPs fabricated in 250 nm CMOS.

This paper is composed of the following sections. Section 2 shows the characteristics and schematics of rectifiers composing each stage to be optimized. Section 3 presents fabricated circuits and measurement results. Section 4 summarizes this research.

2. Modeling of cross-coupled CMOS AC – DC charge pump (XC – CP) operating in subthreshold region

This section starts with the review of the previous AC – DC CP model [39], [40] and proposes a new one for XC – CP operating in subthreshold region. Table 1 summarizes the circuit parameters of the XC – CP.

f

I_{S}

V_{i n}

V_{d d}

N

V_{P P}

R_{O}

I_{P P}

V_{T}

(2-1) Previous model of AC – DC CP [39], [40]

Figure 4a illustrates a sub-circuit of XC – CP to define the nodal voltages V_n and V_n+1. CLK and CLKB have the voltage amplitude of V_dd/2. Figure 4b shows the waveform of V_n and V_n+1.

The charge transfer switch (CTS) or rectifying diode D1 operates forward bias regime between T1 and T3 and reverse bias one after T3. The peak forward voltage appears around the middle of the time period between T1 and T3, namely T2. V₁ and V₂ at T2 can be expressed by V_dd/2 – I_pp/fC and V_dd/2 + I_pp/fC + V_pp/(N+1), respectively, where V_dd/2 is the clock amplitude of each of CLK and CLKB when the top plate parasitic capacitance is ignored for simplicity, I_pp is the output current, I_pp/fC is a voltage shift due to the amount of transferred charges I_pp/f in steady state, and V_pp/(N+1) is the DC offset between the next neighbor capacitor nodes. As a result, the voltage difference at the peak points of V_n and V_n+1 at T2 (V_d,) is given by (1).　

V_{d} = V_{d d} - \frac{2 I_{p p}}{f C} - \frac{V_{p p}}{N + 1}

(1)

When the CTS MOSFET operates under subthreshold region, the drain-to-source current I_d is expressed by (2), where I_S is the saturation current, V_gs the gate-to-source voltage, and V_T the effective thermal voltage.

I_{d} = I_{s} e^{\frac{V_{g s}}{V_{T}}}

(2)

With a conduction angle γ defined by Ton / Tc, i.e., (3), the average output current I_pp can be expressed by (4).

γ = \cos^{- 1} (1 - \frac{2 I_{p p}}{f C V_{d d}})

(3)

I_{p p} = \frac{γ}{π} I_{d}

(4)

When V_gs is equal to V_d as in a single-MOSFET CTS, from (1), (2) and (4), one can result in V_pp – I_pp relationship (5).

I_{p p} = \frac{2}{π} \sqrt{\frac{I_{p p}}{f C V_{d d}}} I_{s} e^{\frac{V_{d d} - \frac{2 I_{p p}}{f C} - \frac{V_{p p}}{N + 1}}{V_{T}}}

(5)

“Previous model” in Figure 3 is given by (5). One can numerically calculate V_pp with a certain value input to I_pp.

(2-2) Proposed model of XC – CP

Figure 5 (a) illustrates next neighbor two-stages of XC – CP. Because each stage has two capacitors, the capacitance of each capacitor is designed to be C/2 so that the total capacitance per stage is the same as the other CPs. The peak voltage difference V_d between the next neighbor capacitors is the sum of V_ds of PMOSFET and NMOSFET, i.e., 2V_ds. Figure 5 (b) shows V_gs and V_d at the peak points. When the stage at the right half of Figure 5 (a) is the second stage, V_2D and V_2U are given by –V_dd/2 + I_pp/fC + V_pp/(N+1) and V_dd/2 – I_pp/fC + V_pp/(N+1), respectively.

Thus, V_gs = V_2U –V_2D is given by (6).

V_{g s} = V_{d d} - \frac{2 I_{p p}}{f C}

(6)

V_d is given by (1) as well as SD – CP. Assuming NMOSFET has the drain-to-source current as large as PMOSFET does, the peak current from one capacitor to the next is given by (7).

I_{d} = I_{s} e^{\frac{V_{g s}}{V_{T}}} (1 - e^{- \frac{V_{d s}}{V_{T}}})

(7)

Charge transfer occurs in every half cycle, resulting in (8).

I_{p p} = \frac{2 γ}{π} I_{d}

(8)

From V_d = 2V_ds and (6) – (8), a subthreshold XC – CP model is given by (9).

I_{p p} = \frac{2 γ I_{s}}{π} e^{\frac{V_{d d} - \frac{2 I_{p p}}{f C}}{V_{T}}} (1 - e^{- \frac{V_{d d} - \frac{2 I_{p p}}{f C} - \frac{V_{p p}}{N + 1}}{V_{T}}})

(9)

When

V_{d s} ≫ V_{T}

, (9) is reduced to be (10). Because it has no V_pp term, it indicates that a subthreshold XC – CP is a current source with an infinite output resistance. This fact is originated by (6) which has no dependency on V_pp.

I_{p p} = \frac{2 γ I_{s}}{π} e^{\frac{V_{d d} - \frac{2 I_{p p}}{f C}}{V_{T}}}

(10)

(2-3) More accurate model with a finite output resistance

Even though (10) can express a large output resistance, it needs modification to have a finite output resistance rather than an infinite one. Figure 6(a) shows I_ds – V_gs of an NMOSFET in 250 nm CMOS, which suggests that I_s has dependency on V_ds, namely drain-induced barrier lowering. Based on the data of Figure 6(a), I_s – V_ds was plotted in Figure 6(b). The curve is well fit with I_S = 18.4 e^{10.7 Vds} in nA.

Assuming the I_s – V_ds curve can be generally described by (11), (10) and (9) can be revised as (12) and (13), respectively.

I_{s} = I_{s 0} e^{\frac{V_{d s}}{η V_{T}}}

(11)

I_{p p} = \frac{2 γ I_{s 0}}{π} e^{\frac{{(1 + \frac{1}{2 η}) (V}_{d d} - \frac{2 I_{p p}}{f C}) - \frac{V_{p p}}{2 η (N + 1)}}{V_{T}}}

(12)

I_{p p} = \frac{2 γ I_{s 0}}{π} e^{\frac{{(1 + \frac{1}{2 η}) (V}_{d d} - \frac{2 I_{p p}}{f C}) - \frac{V_{p p}}{2 η (N + 1)}}{V_{T}}} (1 - e^{- \frac{V_{d d} - \frac{2 I_{p p}}{f C} - \frac{V_{p p}}{N + 1}}{V_{T}}})

(13)

Figure 7 compares V_pp – I_pp between SPICE and models (10), (12), (13) at Vdd = 200 mV when XC – CP is designed with the parameters in Table 3. The three CPs were fabricated with those design parameters which will be discussed in Section 3. Even though the short circuit current at Vpp = 0V had discrepancies from the SPICE result, the output resistance of Model (13) was in better agreement with SPICE than Model (12).

3. Validation of the proposed model

XC – CP, SD – CP and ULPD – CP were designed in 250nm CMOS with the parameters shown in Table 3. Figure 8, Figure 9 and Figure 10 respectively show simulated waveforms at 12^th and 13^th stages when Vdd is 400 mV and Vpp is 3.0V. Note that V3, N- and P-well voltage, of XC – CP in Figure 9, and V2 and V4, N- and P-well voltages, of ULPD – CP in Figure 10, are much more stable than V1 – V3 of SD – CP as shown in Figure 8. I1 and I2 of ULPD – CP become negative in a cycle time, but those are due to AC current to the gate of CTS transistors, not actual leakage current.

XC – CP, SD – CP and ULPD – CP were fabricated in 250nm CMOS to validate the proposed model. Figure 11, Figure 12 and Figure 13 respectively illustrate layout design for those three CPs.

NMOSFETs were formed in the triple well to isolate their P-well from P-substrate. Deep N-well of the NMOSFET is shared with N-well for PMOSFET to minimize the parasitic capacitance for XC and ULPD CPs. To route wires for CLK and CLKB with minimal length, adjacent two stages are laid out by placing two capacitors at top and bottom of the cell. The CTS is placed in the middle. The cell width is determined by the CTS and the cell height is determined by the two caps and the CTS.

Figure 14 shows die photo. Each CP has 24 stages with a stage capacitor of 10 pF. Because of difference in CTS size of three CPs, XC, ULPD and SD CPs have entire size of 0.48mm², 0.49mm² and 0.44mm², respectively.

Figure 15 shows measured waveforms of the three CPs with Vdd of 200 mV, f of 1 MHz and a load resistance of 10 MΩ. The ripple voltages were below 6mV with a filtering capacitance of 22 pF.

Vpp was measured with various Vdd and load resistance. Figure 16 compares Vpp – Ipp curves between XC, ULPD and SD with SPICE, measurement and model. Hereinafter, the models for XC and SD indicate (13) and (5), respectively.

Figure 16 (a) – (c) show the comparison of Vpp – Ipp curves given by SPICE simulation between XC, ULPD and SD at Vdd of 400 mV (a), 200 mV (b), and 50 mV (c). At Vdd of 400 mV or 200 mV, “Latch” or XC – CP had the highest Ro in a lower Vpp range, but Ipp suddenly collapsed at certain Vpp. The reason of that behavior has not been identified in this work, which will be determined in the following researches. Isc of SC was the highest among the three CPs at Vdd of 400 mV, 200 mV and 50 mV. At Vdd of 50 mV, Ipp of XC is as low as that of ULPD. At such a low Vdd, Vds of each CTS transistor may play a main role. Vds of each CTS transistor in XC and ULPD is half of that in SD. When Vds goes below V_T, (7) indicates that Ids is reduced as Vds decreases.

Figure 16 (d) – (f) show the comparison of Vpp – Ipp curves given by measurement between XC, ULPD and SD at Vdd of 400 mV (d), 200 mV (e), and 50 mV (f). Vpp – Ipp characteristics at Vdd of 400mV or 200mV between XC, ULPD and SD were very similar to those of SPICE results. Unlike SPICE results, Ipp values of the three CPs were about the same at Vdd of 50 mV.

Figure 16 (g) – (i) show the comparison of Vpp – Ipp curves given by the models between XC and SD at Vdd of 400 mV (g), 200 mV (h), and 50 mV (i). Ipp of XC was larger than SD when Vdd was 400mV or 200mV whereas Ipp of XC was smaller than SD when Vdd was 50mV. Those trends were similar to SPICE results.

Figure 17 compares Vpp – Ipp curves between SPICE, measurement and model. Figure 17 (a) – (c) show the comparison of Vpp – Ipp curves of XC – CP at Vdd of 400 mV (a), 200 mV (b), and 50 mV (c). The proposed model (13) was in good agreement with SPICE and measured within a factor of 3 in the swept ranges of Vpp and Vdd. The model was not succeeded to show sudden collapse at a certain Vpp with Vdd of 400mV. Figure 17 (d) – (f) show the comparison of Vpp – Ipp curves of ULPD – CP at Vdd of 400 mV (d), 200 mV (e), and 50 mV (f). SPICE and measured results were well matched in terms of the open circuit voltage and within a discrepancy of 20% in terms of Ro. Figure 17 (g) – (i) show the comparison of Vpp – Ipp curves of SD – CP at Vdd of 400 mV (g), 200 mV (h), and 50 mV (i). Ro had discrepancies from SPICE and measured by a factor of two at Vdd of 400mV and was in good agreement with measured at Vdd of 200mV and 50mV. The discrepancy in Ro between SPICE and measured increases as Vdd decreases in comparison with XC and ULPD.

Figure 18 (a) – (c) compare Ro (a), Isc (b) and Voc (c) of XC – CP between SPICE, measured and model (13) at three conditions of (Vdd, Vpp) = (400 mV, 2.5 V), (200 mV, 1.0 V), and (50 mV, 0.2 V). Voc of the model was closer to that of SPICE and measured than Ro and Isc of the model were. This means that the discrepancy in Ro of the model from SPICE and measured is as large as that in Isc. Figure 18 (d) – (f) compare Ro (d), Isc (e) and Voc (f) of SD – CP between SPICE, measured and model (5) at three conditions of (Vdd, Vpp) = (400 mV, 2.5 V), (200 mV, 1.0 V), and (50 mV, 0.2 V). Like XC – CP, Voc of the model was closer to that of SPICE and measured than Ro and Isc of the model were.

Figure 19 (a) – (c) compare Ro (a), Isc (b) and Voc (c) of XC – CP normalized by those of SD – CP between SPICE, measured and model at the three conditions of (Vdd, Vpp) = (400 mV, 2.5 V), (200 mV, 1.0 V), and (50 mV, 0.2 V). From Figure 19 (a), except for the measured result at (Vdd, Vpp) = (50 mV, 0.2 V), Ro of XC – CP was larger than that of SD – CP by a factor of 2.5 or more. When the condition of (Vdd, Vpp) moved from (400 mV, 2.5 V) to (200 mV, 1.0 V) and from (200 mV, 1.0 V) to (50 mV, 0.2 V), the Ro ratio increased with SPICE whereas didn’t with measured and model except for the measured result from (200 mV, 1.0 V) to (50 mV, 0.2 V). Figure 19 (b) shows that the Isc ratio decreased with SPICE and model whereas didn’t with measured. Figure 19 (c) shows that the Voc ratios were in good agreement between SPICE, measured and model in those three conditions even though the Ro ratio and the Isc ratio had different tendency on the operation condition.

6. Conclusions

In this paper, a circuit model of AC – DC charge pumps with subthreshold operation cross-coupled CMOS as charge transfer switch (CTS), namely XC – CP, was developed for circuit designers to use the model for determining the initial condition for SPICE simulation. It was observed that XC – CP had much higher output resistance than charge pumps with single NMOSFETs as CTS, namely SD – CP, did. The reason is that the gate-to-source voltage of the MOSFETs in XC – CP doesn’t depend on the output voltage unlike SD – CP and charge pumps with ultra-low-power diodes (ULPDs) do. Very high but finite output resistance is resulted from weak but finite dependence of the drain-to-source voltage on the subthreshold current through drain-induced barrier lowering. In both SPICE simulation and measured results, the output current of XC– CP collapses at a certain output voltage. The developed model (13) doesn’t predict that behavior. Further research is required to identify what made such behavior. To validate the proposed model, XC -, SD -, ULPD - CPs were fabricated in 250 nm CMOS. The output resistance predicted by the model was in good agreement with SPICE and measured within a factor of two at Vdd of 400 mV, 200 mV and 50 mV.

Author Contributions

Conceptualization, T.T.; methodology, R.K. and T.T.; software, R.K.; validation, R.K. and T.T.; formal analysis, R.K. and T.T.; investigation, R.K. and T.T.; writing—original draft preparation, R.K.; writing—review and editing, T.T. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data is contained within the article or supplementary material.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Applications of AC – DC/RF – DC CPs ((a) Microwave wireless power transmission or RF energy harvesting, (b) biomedical application, (c) vibration energy harvesting) and (d) circuit diagram of CP.

Figure 2. AC – DC CPs with diode-connected MOSFETs (a), ultra-low-power diodes (ULPDs) (b) and cross-coupled CMOS or CMOS latch (c).

Figure 3. Output voltage Vpp – output current Ipp under the condition of Table 1.

Figure 4. Nodal voltages V_n and V_n+1 of neighboring stages (a) and their waveform (b).

Figure 5. Sub-circuit of XC – CP (a) and waveform of nodal voltages (b).

Figure 6. I_ds – V_gs (a) and an extracted I_S – V_ds (b) of an NMOSFET in 250 nm CMOS.

Figure 7. Comparison of V_pp – I_pp between SPICE and models (10), (12), (13).

Figure 8. 12^th and 13^th stages of SD – CP (a) and their waveforms (b).

Figure 9. 12^th and 13^th stages of XC – CP (a) and their waveforms (b).

Figure 10. 12^th and 13^th stages of ULPD – CP (a) and their waveforms (b).

Figure 11. Layout unit of SD – CP (a) and its layout design (b).

Figure 12. Layout unit of XC – CP (a) and its layout design (b).

Figure 13. Layout unit of ULPD – CP (a) and its layout design (b).

Figure 14. Die photo.

Figure 15. Measured waveform with Vdd of 200 mV and f of 1 MHz.

Figure 16. Vpp – Ipp characteristics: SPICE results with Vdd of 400 mV (a), 200 mV (b) and 50 mV (c); measured results with Vdd of 400 mV (d), 200 mV (e) and 50 mV (f); models with Vdd of 400 mV (g), 200 mV (h) and 50 mV (i).

Figure 17. Vpp – Ipp characteristics: XC – CP (Latch) with Vdd of 400 mV (a), 200 mV (b) and 50 mV (c); ULPD – CP with Vdd of 400 mV (d), 200 mV (e) and 50 mV (f); SD – CP (Single) with Vdd of 400 mV (g), 200 mV (h) and 50 mV (i).

Figure 18. Ro (a), Isc (b) and Voc (c) of XC – CP and Ro (d), Isc (e) and Voc (f) of SD – CP between SPICE, measured and model (13) at three conditions of (Vdd, Vpp) = (400 mV, 2.5 V), (200 mV, 1.0 V), and (50 mV, 0.2 V).

Figure 19. Comparison of Ro (a), Isc (b) and Voc (c) of XC – CP normalized by those of SD – CP between SPICE, measured and model at the three conditions of (Vdd, Vpp) = (400 mV, 2.5 V), (200 mV, 1.0 V), and (50 mV, 0.2 V).

Table 1. Design parameters for Figure 3.

Parameter	Symbol	Value
Clock frequency	f	1 GHz
Number of stages	N	32
Stage capacitance	C	100 fF
Clock amplitude	Vdd	400mV

Table 2. Definition of design parameters.

Parameter	Description	Parameter	Description
$f$	Frequency of input power	$I_{S}$	Saturation current of MOSFET operating in subthreshold region
$V_{i n}$	Input AC voltage of XC–CP	C	Stage capacitor (capacitance per stage)
$V_{d d}$	Amplitude of V_in	$N$	Number of stages
$V_{P P}$	Output DC voltage of XC–CP	$R_{O}$	Output resistance of CP
$I_{P P}$	Average output current of XC–CP	I_SC	Short circuit current of CP
$V_{T}$	Effective thermal voltage	V_OC	Open circuit voltage defined by R_O I_SC

Table 3. Design parameters for Figure 7 and used for the fabricated CPs which will be discussed in Section 3.

Parameter	Symbol	Value
Clock frequency	f	1 MHz
Number of stages	N	24
Stage capacitance	C	10 pF
Clock amplitude	Vdd	400mV, 200mV, 50mV

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