More Enhanced Swing Colpitts Oscillators: A Circuit Analysis

Abstract

In this paper, we show that an additional inductor–capacitor–inductor filter can increase the oscillation amplitude of the enhanced swing Colpitts oscillator (ESCO), and call this topology the more enhanced swing Colpitts oscillator (mESCO). When it is connected with a rectifier, the DC–DC boost conversion ratio can be increased, especially for low-voltage sensor ICs or energy harvesting. This paper focuses on the electrical characteristics of mESCO. The oscillation frequency was modeled as a function of the circuit parameters of mESCO. The common gate voltage gain (A_CG), defined by the ratio of the drain voltage amplitude to the source voltage amplitude of the switching MOSFET of mESCO, was also modeled under the assumption that all the circuit elements are ideal. The model was validated with a SPICE simulation. For A_CG < 1.5, the model was in good agreement with the SPICE results within 10%. In addition, the drain voltage amplitude v_da was modeled by assuming that the average transconductance of the MOSFET in a half cycle is null when the long-channel Shockley model is used. v_da needs to be sufficiently high to have a large DC–DC boost conversion ratio. The model can predict the tendency that v_da increases as A_CG approaches unity. We found that the voltage difference of the drain voltage amplitude to the source voltage amplitude is a constant even when the circuit parameters, and thereby A_CG are varied.

1. Introduction

Battery-free IoT sensor modules are required to eliminate battery replacement to reduce costs. Wireless power transfer (WPT) charges the modules with electromagnetic waves [1,2,3,4]. An energy transducer (ET) transforms environmental energy such as lights, heat flow, and vibration into electric power [5,6]. Those techniques can remove the batteries out of the modules. Circuit designs have improved the power efficiency of power converters even with low power received via WPT and ET. Multisine WPT can increase the peak power with the same average transmitter power, resulting in improvement in power efficiency [7,8,9]. Figure 1 graphically explains how it works. Each of the N multisine waves has power of 1/N to attain the same power as the conventional continuous wave. At the receiving antenna, the peak voltage amplitude can be theoretically increased by a factor of $\surd N$ with beating multisine waves when the input impedance is common for N multisine waves with different frequencies. Due to the nonlinearity of the rectifying diode, the multisine waves can output more averaged current than the continuous wave.

Figure 1. Efficiency improvement in wireless power transfer with multisine electromagnetic waves.

The Colpitts oscillator is one of the oscillator circuit topologies that provides a carrier frequency for wireless communication [10,11,12,13]. The phase noise of the oscillator is a critical factor limiting the sensitivity of wireless communication. It is important to increase the output voltage amplitude of the oscillator to reduce the phase noise. In order to increase the voltage amplitude, an enhanced swing Colpitts oscillator (ESCO) was proposed [14,15,16,17,18]. What about adding beating to the ESCO to further increase the output voltage amplitude in DC–DC boost converter applications for low-voltage sensor ICs or energy harvesting? In this paper, we show that an additional inductor–capacitor–inductor filter can increase the oscillation amplitude of the ESCO, and call this topology the more enhanced swing Colpitts oscillator (mESCO). Rectifiers were connected with the output terminals of ESCO and mESCO to output boosted voltages, called ER (ESCO followed by rectifier) and mER (mESCO followed by rectifier), respectively. A SPICE simulation was run to compare the performance of ER and mER. The result showed that mER output a higher open circuit voltage and higher output current at a certain voltage than ER.

The rest of the paper discusses a circuit analysis of the mESCO. The oscillation frequency is shown as a function of the circuit parameters of mESCO. The common gate voltage gain A_CG, defined by the ratio of the drain voltage amplitude to the source voltage amplitude of the switching MOSFET of mESCO, was modeled under the assumption that all the circuit elements are ideal. The drain voltage amplitude v_da was also modeled by assuming that an averaged transconductance of the MOSFET in a half cycle is null when the long-channel Shockley model is used. v_da needs to be sufficiently high to have a large DC–DC boost conversion ratio. The model can predict the tendency that v_da increases as A_CG approaches unity. We also found that the voltage difference of the drain voltage amplitude to the source voltage amplitude is a constant even when the circuit parameters, and thereby A_CG, are varied. The paper is organized as follows: Section 2 proposes the mESCO and mER. The performance of mER is compared with that of ER. Section 3 analyzes the oscillation frequency, common gate voltage gain, and drain voltage amplitude of mESCO. The models are validated with SPICE simulation. The conclusion is given in Section 4.

2. More Enhanced Swing Colpitts Oscillator (mESCO), and DC–DC Converter with mESCO and Rectifier (mER)

Let us consider on-chip DC–DC boost converters composed of an oscillator whose peak output voltage is higher than the input DC voltage V_IN and a rectifier to supply power to building blocks in sensor/RF IC, as shown in Figure 2. When the decoupling capacitor C_DEC is too large to stabilize the input power rails, the oscillator runs with the DC supply voltage, as shown in Figure 2a. On the other hand, if C_DEC is set to a relatively small value, the AC component adds to the DC supply voltage to create beat tones into the oscillator, as shown in Figure 2b.

Figure 2. (a) Relatively large decoupling capacitor to filter out the AC component; (b) relatively small decoupling capacitor to create beat tones into the oscillator.

Figure 3a illustrates a boost converter (ER) with an enhanced swing Colpitts oscillator (ESCO) followed by a rectifier to drive the load R_L. The drain voltage of M1 exceeds V_IN. M2 transfers the current when the peak drain voltage of M1 is higher than the target voltage V_OUT. To increase the V_SS of the ESCO, a small decoupling capacitor is connected between the V_IN and V_SS through the parasitic inductance of bonding wires, as shown in Figure 3b. Additional oscillation at V_SS increases the drain voltage of M1, resulting in a higher output current at the same V_OUT as ER.

Figure 3. (a) Enhanced swing Colpitts oscillator (ESCO) followed by rectifier (ER); (b) ER with LC filter (more enhanced swing Colpitts oscillator followed by rectifier (mER).

A SPICE simulation was run with V_IN = 0.3 V, L_BW = 2 nH, C_DEC = 25 pF, L₁ = 0.239 nH, L₂ = 2.93 nH, C₁ = 4 pF, and C₂ = 1.5 pF. The parameters L₁, L₂, C₁, and C₂ are common to both circuits. The SPICE model in 65 nm CMOS was used in this study. Figure 4a shows the waveform of V_D and V_S of ER and mER when the outputs are open. The highest peak voltages of V_D were 1.08 V with ER and 2.01 V with mER. In mER, the peak voltages were alternately high and low. Figure 4b shows the V_OUT vs. I_OUT of ER and mER. In addition to an increase in the maximum attainable output voltage, mER can have a larger output current at any operating output voltage than ER. In Section 3, a circuit analysis for mESCO is conducted.

Figure 4. Comparison of mESCO with ESCO; (a) waveform of V_D and V_S when the outputs are open, (b) V_OUT − I_OUT.

3. Circuit Analysis of mESCO

Let us define the notation of voltages in this work, as shown in Figure 5. The DC offset and voltage amplitude are described by V_X and v_xa, respectively. The voltage differences from the ground and DC offset are described by V_x and v_x as a function of phase θ, respectively.

Figure 5. Notation of voltages in this work.

3.1. Oscillation Frequency

The oscillation frequency of ESCO was modeled in [16] by ignoring any real part of impedance and conductance, assuming the ESCO operates in steady state. We can similarly formulate the oscillation frequency of mESCO. Figure 6 shows the circuit reduction of mESCO from Figure 6a through Figure 6g. A small-signal equivalent circuit of Figure 6a is reduced to Figure 6b. Introducing L_tb for the series connection of L_t and L_b with L_tb = L_t + L_b, and eliminating M₁ results in Figure 6c under the assumption that any real part of impedance and conductance is omitted for a simple model. When the impedances of the parallel connections of L_tb and C_IN and of L₂ and C₂ are capacitive, they can be replaced with single capacitors C_eq1 and C_eq2, respectively, by using the formula shown in Figure 6d, where ω is the angular velocity of a signal of interest. Likewise, when the impedance of the series connection of L₁ and C₁ is inductive, we can replace it with a single inductor L_eq, by using the formula shown in Figure 6e. Thus, the circuit shown in Figure 6c is reduced to Figure 6f using L_eq, C_eq1 and C_eq2. Finally, we can obtain an LC circuit as shown in Figure 6g, where C_eq = C_eq1 + C_eq2. Thus, (1)–(4) hold for mESCO.

$$C_{eq1}=C_{in}-\frac{1}{{\omega }^2{L}_{tb}}$$

(1)

$$C_{eq2}=C_2-\frac{1}{{\omega }^2{L}_2}$$

(2)

$$L_{eq}=L_1-\frac{1}{{\omega }^2{C}_1}$$

(3)

$$C_{eq}=\frac{C_{eq1}{C}_{eq2}}{C_{eq1}+C_{eq2}}$$

(4)

$$f_{}^{}=\omega /2\pi =1/2\pi \surd L_{eq}{C}_{eq}$$

(5)

Figure 6. Circuit reduction of mESCO from (a) through (g); (a) original mESCO, (b) small-signal equivalent circuit of (a), (c) omitting M1, (d) equivalent capacitance of L and C connected in parallel, (e) equivalent inductance of L and C connected in series, (f) circuit reduced from (c) with (d) and (e), and (g) circuit reduced from (f).

A fundamental frequency is given by the solution in cubic Equation (5) in terms of f², and thereby can be resolved with three different values. To validate the models proposed in Section 3 with SPICE simulation, the nominal condition shown in Table 1 was used. The values of the inductors and capacitors were much larger than the ones fabricated in a single chip because the SPICE model in 180 nm CMOS was used for M₁. For parameter response on electrical characteristics, the remaining ones except for the sweeping parameter were set as shown in Table 1.

Table 1. Nominal condition to validate the models proposed in Section 3 with SPICE simulation.

Parameter	Default Value
L1	5 μH
L2	16 μH
Lt (=Lb)	5 μH
C1	3 nF
C2	8 nF
CIN	1 nF
VIN	0.8 V

The numerical solutions under wide ranges of circuit parameters were in good agreement with the SPICE results within 10% at most, as shown in Figure 7. Among the three solutions, the highest one was selected because the other two frequencies are too low to have L_eq and C_eq positive values.

Figure 7. Operation frequency as a function of each circuit parameter. (a) L₁ vs. fo, (b) C₁ vs. fo, (c) C_IN vs. fo, (d) L_t(=L_b) vs. fo.

3.2. Common-Gate Voltage Gain

For DC–DC boost converters to have large boost ratios, a large voltage amplitude at the drain of M1 is required. Let us determine a common gate voltage gain A_CG as a function of the circuit parameters first. As performed for modeling the operation frequency of mESCO, a circuit transformation was produced from an original circuit diagram of mESCO as shown in Figure 8a to its small-signal equivalent circuit as shown in Figure 8b, and then a circuit model of mESCO as shown in Figure 8c.

Figure 8. Circuit transformation: (a) an original circuit diagram of mESCO, (b) its small-signal equivalent circuit, and (c) a circuit model of mESCO to extract its common-gate voltage gain A_CG.

When v_d is forced, what voltage appears at v_s? Seven variables (four voltages and three currents) among six equations were applied, as shown by (6)–(11).

$$v_{ssi}=j\omega L_b\left(i_1+i_2-i_3\right)$$

(6)

$$v_s-v_{ssi}=i_1/j\omega C_{eq2}$$

(7)

$$v_d-v_{ddi}=j\omega L_1{i}_2$$

(8)

$$v_{ddi}-v_{ssi}=\left(i_2-i_3\right)/j\omega C_{IN}$$

(9)

$$v_{ddi}=j\omega L_t{i}_3$$

(10)

$$v_d-v_s=i_1/j\omega C_1$$

(11)

Therefore, an equation can relate any two variables. After some calculations, (12) was derived for the common gate voltage gain A_CG. With (13)–(18), (12) provides a value for A_CG when the circuit parameters are given.

$$A_{CG}=v_d/v_s=T_{SQ1}/T_{SQ2}$$

(12)

$$T_{SQ1}=\left(1+\beta \right)(T_{SQ3}-L_X{C}_X)+L_X{C}_{eq2}$$

(13)

$$T_{SQ2}=\beta (T_{SQ3}-L_X{C}_X)+L_t{C}_{IN}$$

(14)

$$T_{SQ3}={\omega }_{}^{ 2}{C}_{IN}^{ 2}{L}_1{L}_t$$

(15)

$$\beta =C_1/C_{eq2}$$

(16)

$$L_X={\omega }_{}^{ 2}{C}_{IN}^{ }{L}_1{L}_t-L_1-L_t$$

(17)

$$C_X=C_{eq2}+C_{IN}{}_{}-1/\left({\omega }_{}^{ 2}{L}_b\right)$$

(18)

where C_eq2 is given by (2). β, L_X, and C_X, defined by (16), (17), and (18), respectively, were used to calculate T_SQ1-3. Figure 9 shows A_CG as a function of each design parameter. The tendencies of A_CG vs. L₁ (a) and C₁ (b) are matched with those for ESCO [16]. As expected, the smaller the C_IN, the closer to unity for A_CG in (c). (This tendency validated increasing the drain voltage amplitude with a smaller C_IN in the following sub-section, as expected in Section 1). There were substantial discrepancies in A_CG between the model and SPICE results for C_IN > 2 nF. A physical background is required here, but its investigation will be conducted in future work. Figure 9d suggests that Lt and Lb can be minor contributors to A_CG, resulting in no significant impact on the drain voltage amplitude, as discussed in the following subsection. In summary, the model calculation results were in good agreement with the SPICE results, except for C_IN > 2 nF.

Figure 9. A_CG as a function of each design parameter. (a) L₁ vs. A_CG, (b) C₁ vs. A_CG, (c) C_IN vs. A_CG, (d) L_t(=L_b) vs. A_CG.

3.3. Drain Voltage Amplitude v_da

In this subsection, the drain voltage amplitude v_da is modeled. In reality, there was a distortion in V_d and V_s due to the nonlinear behavior of the transconductance and drain conductance of M1, but it was assumed for simplicity in this study that the AC components of V_d and V_s are modeled with sinusoidal waveforms whose amplitudes are v_da and v_sa, respectively, as shown in Figure 10.

Figure 10. Waveform of V_d and V_s and operation modes in one cycle.

V_d has an offset voltage of V_IN. Thus, V_d and V_s can be written as (19) and (20), respectively.

$$V_d=V_{IN}+v_{da}\cos \theta$$

(19)

$$V_s=v_{sa}\cos \theta =\frac{v_{da}}{A_{CG}}\cos \theta$$

(20)

When v_sa is large enough, M1 operates in three different modes, i.e., in cut-off, saturation, and linear operation modes. θ₁ is the boundary between cut-off and saturation, resulting in (21), where V_TH is the threshold voltage of M1. θ₂ is the boundary between saturation and linear, resulting in (22).

$${\theta }_1{=\cos }^{-1}\frac{V_{IN}-V_{TH}}{v_{sa}}$$

(21)

$${\theta }_2{=\cos }^{-1}\frac{-V_{TH}}{v_{da}}$$

(22)

Let us assume a simple Shockley model for M1, as described in (23) and (24), where Ids_sat and Ids_lin are the drain current in saturation and liner modes, respectively, and k is a proportional coefficient of the drain current.

$$I_{ds\_sat}=\frac{1}{2}k{\left(V_{gs}-V_{th}\right)}^2$$

(23)

$$I_{ds\_lin}=k\left\{\left(V_{gs}-V_{th}\right)V_{ds}-\frac{1}{2}{V}_{ds}{}^2\right\}$$

(24)

The transconductance is expressed by (25) and (26) in saturation and liner modes, respectively.

$$g_{m_{sat}}=\frac{\partial I_{d{s}_{sat}}}{\partial V_{gs}}=k\left(V_{gs}-V_{th}\right)$$

(25)

$$g_{m\_lin}=\frac{\partial I_{ds\_lin}}{\partial V_{gs}}=k{V}_{ds}$$

(26)

Let us assume (27) holds when mESCO runs in steady state, where the peak drain voltage does not change by cycle, because when the left side value of (27) is positive, the amplitude increases by a cycle; whereas when it is negative, the amplitude decreases by a cycle.

$$\underset{0}{\overset{\pi }{{\displaystyle \int }}}{g}_m\left(\theta \right)d\theta =0$$

(27)

Using (21), (22), (25) and (26), (27) resulted in (28).

$$\begin{array}{c}{{\displaystyle \int }}_0^{\pi }{g}_m\left(\theta \right)d\theta ={{\displaystyle \int }}_{{\theta }_1}^{{\theta }_2}{g}_{m\_sat}\left(\theta \right)d\theta +{{\displaystyle \int }}_{{\theta }_2}^{\pi }{g}_{m\_lin}\left(\theta \right)d\theta \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =V_{IN}(\pi -{\theta }_1)-V_{th}({\theta }_2-{\theta }_1)+v_{sa}\sin {\theta }_1-v_{da}\sin {\theta }_2=0\end{array}$$

(28)

Equation (28) shows that v_da is a function of A_CG, V_IN and V_TH. Therefore, the model was compared with the SPICE results in terms of v_da vs. A_CG, as shown in Figure 11a. Dots in colors show the SPICE results where the circuit parameters are varied as much as the ones in Figure 9. Because the SPICE model for NMOSFET used in this study has no breakdown at high drain voltages, all the simulated data were plotted. However, because there is a strict specification for the maximum voltages in reality, it can limit the design space for the circuit parameters. Even though the model result has an offset from the trend curve of the SPICE results, the model well-represents the tendency that v_da is determined by A_CG rather than individual circuit parameters. It is interesting to note that the data points of v_da and v_sa are plotted on a linear line with a slope of 1.00 for both the SPICE results and model calculated results with a slightly different offset of 1.05 and 1.25 V for the SPICE results and model calculated results, respectively, as shown in Figure 11b.

Figure 11. (a) A_CG vs. v_da and (b) v_sa vs. v_da.

Figure 12a,b show the waveforms for voltage, current, and gm in cases of A_CG = 1.2 and 1.8, respectively. The waveform for gm in Figure 12a suggests that gm in a linear operation can partially contribute to surplus for a very large voltage amplitude. For a relatively low voltage swing in Figure 12b, positive and negative gm appear in the saturation and linear regions, respectively.

Figure 12. Waveform in cases of A_CG = 1.2 (a) and 1.8 (b).

To compare the v_da of mESCO with that of ESCO, SPICE simulations and model calculations based on [16] were made with the same parameter conditions for L₁, C₁, and V_IN as mESCO. Figure 13 shows the comparison results. Even though the model accuracy of (28) for mESCO is not as sufficient as that of [16] for ESCO, the trend over circuit parameters represents the SPICE results. With the same values for L₁, L₂, C₁, and C₂ as those of ESCO, mESCO has a 2× or larger v_da than ESCO.

Figure 13. v_da of ESCO and mESCO as a function of L₁ (a), C₁ (b), and V_IN (c).

The data in Figure 13 are placed in a single v_da−A_CG graph as shown in Figure 14. mESCO, as well as ESCO, has a unified characteristic curve. It is essential to design mESCO and ESCO with A_CG as close to unity as possible to have a large v_da.

Figure 14. v_da of ESCO and mESCO as a function of A_CG.

4. Conclusions

We found that an additional L–C–L filter for ESCO, called mESCO, increased the peak drain voltage, which will contribute to increasing the boost ratio when it is used in DC–DC boost converter applications. In this study, the operation frequency, common gate voltage gain, and drain voltage amplitude were analyzed with simple proposed models for mESCO. Even though the frequency model potentially has three different frequencies as a solution, the lower two components were not allowed because an equivalent inductor and capacitor would have had negative inductance and capacitance, respectively. The highest frequency was matched with the SPICE results in wide circuit parameter ranges. The common gate voltage gain was also modeled with a simple circuit transformation. The model calculation results are in good agreement with the SPICE results within a 10% error, except for large C_IN. In addition, the drain voltage amplitude was modeled with an assumption that the average transconductance of the switching transistor over a half cycle is null. Even with a simple Shockley model, the drain voltage amplitude was successfully modeled so that it is a function of the common gate voltage gain rather than individual circuit parameters. This fact is valid for ESCO as well as mESCO. In wide ranges of L₁, C₁, and V_IN, the drain voltage amplitude of mESCO was 2× or larger than that of ESCO. ESCO and mESCO had the same trend in the drain voltage amplitude over the common gate voltage gain.

Further experiments to validate the model and SPICE results and an application of mESCO to DC–DC boost converters will be conducted in the future. When a rectifier is added to mESCO for the DC–DC boost converter, it would have another impedance at the drain of the switching transistor, potentially yielding a loss in the drain voltage amplitude and thereby a degradation of the output power of the DC–DC converter. One would need to analyze its impact to design a DC–DC converter with sufficient output power.