A comparative study of digital low dropout regulators

Fig. 3 presents the basic block diagrams of the ALDO and the DLDO. The ALDO compares the output voltage (V_OUT) with the reference voltage (V_REF) through an error amplifier (EA). The EA's output feeds a power transistor that operates in the saturation region in most of the cases, although it sometimes operates in the linear region as well. To maintain good output accuracy, the ALDO should have an EA with sufficient loop gain. To circumvent the stability issue of a multistage system, it would be necessary to have a one- or two-stage EA with cascode devices. Yet, more cascode devices stacked imply a larger voltage headroom, which limits the minimum supply voltage of an energy-efficient SoC. Moreover, the analog EA has a weak process scalability, and requires re-design in process migration.

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The DLDO replaces the analog EA by a V_OUT sensor and a controller (integrator) implemented with digital circuits. The digital integrator manages to provide a very high DC gain that is irrelavent to its supply voltage, breaking the trade-off between loop gain and supply voltage in the analog design. As such, DLDO works well with a low supply voltage, and scales naturally with an advanced process. Meanwhile, the switch operation of the power transistors conducts more current with a full turn-on voltage (V_DD), saving silicon area.

In recent years, the inverter-based LDO^[42–44] stems from the ring amplifier, for low-voltage operation. They achieve a fast response with the inverters working in an analog fashion. However, to reduce shoot-through currents in the inverters, offset voltages should exist for proper biasing. Obviously, during process migration, the offset voltages require a dedicated calibration circuit, otherwise, re-design is necessary.

The widely adopted figure of merit (FoM) of speed proposed in Ref. [45] explains the power-speed trade-off of LDO:

$$\rm{FOM} = {T_{\rm{R}}}\frac{{{I_{\rm{Q}}}}}{{{I_{\rm{MAX}}}}} = \frac{{{C_{\rm{OUT}}} \cdot \Delta {V_{\rm{OUT}}}}}{{{I_{\rm{MAX}}}}} \cdot \frac{{{I_{\rm{Q}}}}}{{{I_{\rm{MAX}}}}}, \tag{ 1 }$$

where T_R is the transient response time, I_Q is the quiescent current, I_MAX is the maximum load current, C_OUT is output capacitor, and ΔV_OUT is the output voltage spike with the I_MAX load step.

For an ALDO, T_R is mainly determined by the bandwidth of the LDO loop and slew rate, which should be inversely proportional to the I_Q. Multiple-loop schemes could be a good way to reduce T_R without significantly increasing I_Q ^{[4, 46]}.

The synchronous DLDO should have a longer T_R and thus a slower transient response under the I_Q constraint, due to the conventionally discrete V_OUT sampling and synchronized control word changing, as illustrated in Fig. 4. When a load current (I_LOAD) step comes at t₁, DLDO fails to respond instantly. The output capacitor compensates the difference between I_LOAD and LDO output current I_LDO, leading to a V_OUT drop, until the next clock (CLK) rising edge (t₂) when the synchronized comparator can respond. To prevent metastability, the controller, lagging one-cycle behind t₂ (t₃), processes the comparator's output (CMP_OUT). After that, the control word n begins to increase, providing additional current I_LDO. We define Δt_D = t₃ − t₁ as the loop delay, and V_OUT the undershoot caused by Δt_D is ΔV_DLY. Subsequently, V_OUT does not stop decreasing until I_LDO = I_LOAD, depending on how fast the controller can find the desired control word within the processing time (Δt_P). This adds an additional voltage drop ΔV_REACT to the overall undershoot ΔV_MAX = ΔV_DLY + ΔV_REACT. Consequently, the T_R of the DLDO consists of Δt_D and Δt_P, where only Δt_P contributes to the T_R in analog counterparts.

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Due to the discrete sampling, the maximum Δt_D is two-clock cycle. Hence under the power consumption constraint, ΔV_DLY of the DLDO would be larger than that of the ALDO. To reduce Δt_D, continuous or asynchronous sampling techniques emerged, presented next in Section 3.

To reduce ΔV_REACT, many recent DLDO techniques tried to find out the desired control word within a few cycles, taking the advantage of digital operations (discussed in Section 4). This imposes the DLDO to have a comparable or even smaller ΔV_REACT than the ALDO.

For conventional ALDO designs, an ALDO will become unstable or even oscillate once its phase margin is not enough. But regarding the DLDO, which typically has a DC pole from the digital integrator, large signal unstable behaviour is less likely to happen. However, DLDOs suffer from another oscillation phenomenon inhexistent in ALDOs. Because of the digitized output current with limited resolution, there will always be a quantization error in the DLDO, inciting a limit cycle oscillation (LCO).

The quantization level determines the quantization error. Fig. 5(a) displays a measured V_OUT LCO waveform of a two-level quantized DLDO^[11], with a period of integer clock cycles and an amplitude of several milli-volt. Refs. [12, 13] explain the difference between an LCO and oscillation in analog circuits, with the small-signal model of a two-level quantized DLDO in Fig. 5(b). With a DC pole and an output pole, Fig. 5(c) plots the root locus of this system. The LCO appears when the root locus crosses the unit circle (a "healthy" oscillation). By contrast, the analog loop instability basically occurs outside the unit circle. Ref. [11] further investigated a method to predict the period and amplitude of the LCO.

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Generally speaking, a smaller least significant bit (LSB) current reduces the quantization error and subsequently the LCO. Nonetheless, the small LSB exponentially increases the control bits required for the targeted output current. A pulse-width-modulated (PWM) LSB scheme proposed in Ref. [14] reduces the effective LSB. Additionally, Ref. [11] added an auxiliary path to reduce the LCO mode to 1 across a wide load current range.

An ALDO can achieve a good PSR with a properly designed EA and a dominant pole allocation^{[4, 9]}, or by adding a feedforward path to cancel the supply input ripple^[7]. But the low supply and dropout voltages in an energy-efficient SoC inevitably undermine the achievable PSR.

Fig. 6 illustrates the power supply ripple rejection process of a DLDO, where discontinuous sampling and control word changing will excite significant glitches with a supply voltage ripple. We define the optimum control word n (n_opt) that imposes V_OUT = V_REF in the steady-state, with the change of n synchronized with the clock rising edge. In an ideal case, the controller manages to find out n_opt (n₁ = n_opt) at each sampling point (t₀, t₁, t₂, t₃), with V_OUT pulled-back to V_REF. However, during the sampling intervals, e.g., from t₀ to t₁, with n fixed, the instant PSR becomes:

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$$\rm{PSR} = \frac{1}{{1 + \dfrac{{{R_{\rm{LOAD}}}}}{{{r_{\rm{ds}}}}}}}, \tag{ 2 }$$

where r_ds represents the instance resistance of the power transistor (constant value during the intervals), R_LOAD is the load resistance. This indicates that the DLDO is vulnerable to the supply ripple during the sampling intervals.

For the non-ideal cases when n₁ ≠ n_opt, e.g., at t₁, it causes an error V_e = V_OUT − V_REF. Then before the next sampling, V_OUT starts to increase with V_IN from V_e, leading to a larger V_OUT ripple. Therefore, the only way for a conventional DLDO to improve the PSR is to minimize the sampling interval using a faster CLK. Unfortunately, this contradicts the power-efficient prerequisite. Section 5 introduces several techniques to incorporate a continuous analog path to enhance the PSR.

A quantizer will approximate the voltage error between V_OUT and V_REF for further processing. Conventional DLDOs use a two-level, synchronized, latch-based comparator^{[2, 10, 13, 15–18]}, as shown in Fig. 7, quantizing V_OUT to a binary output. The complementary clock-controlled switches in the comparator eliminate the steady quiescent current. However, the clock limits the quantization speed, and the discontinuous sampling causes a certain delay Δt_D. Meanwhile, the unbalanced parasitic capacitance and transistor mismatches will cause input offset^[19].

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High-performance DLDO designs^[20–23] use continuous quantization, typically achieved with a multi-level quantization (functioning as an analog-to-digital converter, ADC). In Ref. [21], a transconductance stage continuously compares V_OUT and V_REF, and a current-to-code converter generates a multi-bit output (Fig. 8(a)). A code-dependent reference increases the dynamic range and resolution of the converter. Ref. [22] designed a continuous quantizer also based on inverters, as shown in Fig. 8(b). An auto-zero technique removes the input offset voltage of the aforementioned inverter-based analog LDO. A time-to-digital converter (TDC) is another candidate for continuous multi-level quantization^[23], but needs calibration on the delay time. Plus, the quantized multi-level signals processed by digital logic lead to a more coherent DLDO. But in general, the power consumption of the multi-level quantizer is much higher.

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A voltage-controlled oscillator (VCO) can also be a continuous quantizer^{[24, 25]}. Fig. 9 shows how with the two VCOs driven by V_OUT and V_REF, the DLDO system looks like a phase-locked loop (PLL). In the steady-state, the VCOs should oscillate at the same frequency, resulting in a fixed phase error subsequently converted to a control word by the phase detector (PD). When a load current step occurs, the V_OUT spike information turns into an incremental phase error, captured by the PD and processed by the controller. The main drawback of this topology is that the VCO is essentially an integrator (1/s), that needs enhancement of the transient speed by a proportional control, as presented next.

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As a brief summary, the features of the quantizers in DLDO design are included in Table 1.

One of the possible solutions to address the power-speed trade-off of the DLDO is to use asynchronous control logic^{[26, 27]}, but the asynchronous design may not be robust under process, voltage, temperature (PVT) variations.

Another straightforward way combines the proportional, integral (PI), and even derivative (PID) controls^[28] for better transient response. Fig. 10 displays the conceptual transient waveforms of these controls. For a conventional I-control that has a large DC gain, DLDO achieves an accurate steady-state output but low speed. By contrast, P-control can minimize V_OUT undershoot, but lacks the capability of reducing the steady-state error. PI-control can address both drawbacks, while PID-control further suppresses the recovery overshoot.

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3.2.1. Digital PID control

As discussed above, the number of the quantization levels should be proportional to the power consumption. For a low power design, the detection of whether V_OUT exceeds the V_REF window^{[2, 13]} can activate the P-control, namely a three-level quantization. Fig. 11 presents the comparison of V_OUT in two comparators, with V_REFH and V_REFL, respectively, forming a V_REFH −V_REFL window. When V_OUT exceeds this window, the sampling frequency increases with P-control, driving V_OUT back to the window quickly. When it is within the window, the comparison of V_OUT with V_REF generates the control word after the integrator, as an I-control.

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High-performance designs employ typically a multi-level quantization with an ADC^{[14, 20–22]}. With the V_OUT information converted to the digital domain, the control circuits can perform easily the PID control. However, to reduce the loop delay, a power-hungry ADC may be necessary.

Granular power management oversees not only the V_OUT overshoot/undershoot, but also the recovery time. For a linear increment control used in the conventional DLDO^[10], the control word can only change one count per clock cycle, resulting in a very long recovery time. The coarse-fine tuning scheme used in Refs. [2, 13, 29], changes the control word in multiple counts per clock cycle (coarse-tuning) once V_OUT exceeds the predefined window. A more aggressive scheme utilizes binary search with the control word changing with 2^{N [14]}, with D-control added to prevent a recovery overshoot. To reduce the overshoot without D-control, Ref. [30] used exponential control expressed as:

$${\left( {W/L} \right)_n} = \rm{Const}. \times {q^n} ,\quad {q \gt 1} , \tag{ 3 }$$

where the recovery time may be a bit longer than the binary search.

3.2.2. Analog-assisted loop

By using analog circuits to sense V_OUT ^{[15–18, 31]} as the fast loop can also be a candidate, due to the continuous nature of the analog circuits. Meanwhile, it is worthwhile to utilize passive devices, as active circuits may still have limitations in terms of their low supply voltage and small dropout.

Fig. 12(a) presents an "analog-assisted" method proposed in Refs. [15, 16]. A passive high-pass network R_C and C_C couples to the ground of the driving inverters (V_SSB) the high-frequency components of V_OUT. R_C biases V_SSB to the ground in the steady-state, while drops to a negative value (almost equal to the V_OUT undershoot) at the load transient with a minor delay, allowing the power transistors to generate more instantaneous output current as a high-pass path. Ref. [17] further reduces the V_OUT undershoot by providing more instantaneous current, through coupling V_OUT to the body of the power transistors. The drawback of these schemes is that the analog circuit can only assist the turned-on power transistors, setting a limit on the minimum output current. Ref. [18] proposed an improved version (Fig. 12(b)), where analog paths include not only the high-pass network (P₂), but also NMOS pass devices (P₁). This advances the PMOS counterpart, by providing more instantaneous current from both turned-on and -off pass devices. Nevertheless, this benefit trades with the need for a gate drive voltage higher than the supply voltage. Ref. [31], although not implemented from a digitized output power transistor array, also utilizes a capacitor to couple the output spike to the gate of the power transistor as a high-pass path (Fig. 12(c)). A charge pump generates the gate voltage of the power transistor. Though it is analog in nature, it can be a good candidate for a low-power DLDO.

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In addition, the analog-assisted methods can be an effective solution to improve the DLDO transient response without using advanced process. Table 2 summarizes the process nodes and FoMs (from (1)) of state-of-the-art DLDO works, to evaluate how the process advancement enhances the DLDO performance. As can be seen, the FoM value decreases with process scaling down for the ADC-based DLDO designs^{[21, 22, 36, 39, 40]}. This stems from the fact that the ADC and digital processing has a higher speed and better power efficiency with advanced process. Yet, it is interesting to find out that the analog-assisted-based DLDOs^{[15, 18]} manage to achieve even better FoMs with a cost-effective process, which is facilitated by the continuous response of the analog circuits.

Table 2. FoM of speed versus process nodes.

Year	Process (nm)	Architecture	FoM (ps)
2020[39]	130	ADC + PID	63.9
2016[22]	65	ADC + Event driven	20
2016[21]	28	ADC + Coarse/fine	9.57
2020[40]	20	ADC + Computational	6.7
2020[36]	10	TDC + PID	5.2
2017[15]	65	Analog-assisted	0.23
2018[18]	28	Analog-assisted	0.026

As discussed in Section 2, the DLDO has inherently mediocre PSR even when the optimum control word can be found. Instead, analog circuits may be the only solution under the power consumption constraint. As illustrated in Fig. 13(a), by parallelizing an analog resistance r_ds,A with the digitized resistance r_ds,D, the overall resistance can change continuously. Additionally, feedback loops may adjust r_ds,A from the supply voltage noise, just like in the ALDO. Then, with the supply voltage V_IN-to-V_OUT response mainly determined by the analog circuits, they allow a longer digital sampling interval (t₄ to t₅), and thus higher power consumption. Fig. 13(b) exhibits a topology that combines analog and digitized resistance classified as an hybrid LDO^[32].

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Although the working principle of the PSR improving techniques in the digital/hybrid LDO are similar to those in the ALDO, there are two critical issues remaining. Firstly, we expect the current provided by the analog part to be small. Hence even with a continuously adjustable r_ds,A, this LDO fails to respond to a large supply ripple. Secondly, the limited supply voltage and dropout undermine the achievable PSR of the analog circuits.

Ref. [33] employed one ALDO array and one DLDO array, and proposed a feedforward PSR cancellation technique (Fig. 14(a)), implemented with a resistor and capacitor between the supply voltage and the gate node of the power transistor. It improves the light-load PSR, degraded at heavy load. To maintain the PSR performance, the architecture should comprise more ALDOs and less DLDOs under heavy load conditions.

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As presented in Fig. 14 (b), Refs. [34, 35] added an active replica loop to enhance the PSR. The main part of the ALDO constructed from a flipped voltage follower (FVF)^{[4, 5]} (M_PA, M₂, A₁), manages to work under a low supply voltage with a fast transient response. Fig. 15 plots the simulated PSR, showing a significant improvement above 10 kHz, when compared with a conventional DLDO. The addition of the replica loop, composed by M_Pr, M_2r, and A₂, increases the loop gain and then improves the PSR by 6 dB (5.3 in simulation). Additionally, the current provided by the ALDO dynamically increases with heavy load PSR. However, the utilization of many analog circuits obviously undermines the process scalability.

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