Shuilin Tian

Small-Signal Analysis and Optimal Design of Constant Frequency

Recently, V2 control and its variety named ripple-based control has been gaining more and more popularity in academia research and commercial products. However, for constant frequency V2 control, design methodology is not clear due to insufficient knowledge about the small-signal model. This paper investigates the small-signal model and optimal design strategy for constant frequency V2 control. The factorized small-signal control-to-output voltage transfer function and output impedance are investigated. The stability criterion is obtained and design considerations are analyzed. Moreover, the small-signal model with ramp compensations is presented and optimal design guidelines from dynamic performance point of view are provided. For the first time, it is found the external ramp is good enough to get a well-damped performance when current feedback strength is strong (for example, when employing OSCON capacitors). However, the current ramp is necessary to achieve a good dynamic performance when the current feedback strength is weak (for example, when employing ceramic capacitors). As a result, a new control strategy with the hybrid ramp is proposed for ceramic capacitor applications. The small-signal model and proposed design guidelines are verified with Simplis simulation and experimental results.

V^{2}

Recently, constant on-time V2 control, and its variety named constant on-time control, or constant on-time ripple-based control is more and more popular in industry products due to features of high light-load efficiency, simple implementation, and fast transient response. However, subharmonic oscillation occurs when using multilayer ceramic caps due to the lagging phase of the capacitor voltage relative to the inductor current. External ramp compensation is one simple solution to solve the instability issue. However, the characteristics of constant on-time V2 control with external ramp are not fully understood and no explicit design guidelines for the external ramp are provided. This paper investigates the small-signal characteristics of constant on-time V2 control with external ramp compensation by providing a factorized, easy-to-use small-signal model. The external ramp is a critical parameter because it directly affects the position and damping of two pairs of double poles. Based on this fact, design guidelines of the external ramp for optimal dynamic performance are provided. Moreover, the effect of duty cycle is investigated. Finally, the small-signal experimental results and load transient performance are presented to verify the small-signal analysis and proposed design guideline.

Control

This paper presents a digital hybrid ripple-based constant on-time control scheme for voltage regulator modules (VRMs). Due to the sampling effects of the digital implementation, the stability issue becomes worse than the analog ripple-based control schemes, especially when low-ESR decoupling capacitors are used as the output filter. In order to stabilize the system and to fulfill the output impedance requirement of adaptive voltage positioning (AVP), a hybrid ramp compensation strategy, which includes the external ramp and the estimated current ramp, is proposed. The small-signal model of the proposed architecture is derived to provide the design guideline for the ramp compensation gains and the number of output and decoupling capacitors. Besides, only low sampling-rate Analog-to-Digital Converters (ADCs) are required to sample the input voltage, the output voltage, and the average current making the proposed architecture compatible with the cost/complexity constraints of VRM applications. Simulation and experimental results show that the ripple-based control can achieve high-bandwidth performance, and the proposed digital control architecture can fulfill the AVP design requirements of single-phase VRMs.

Small-Signal Analysis and Optimal Design of External Ramp for Constant On-Time V

V2 control has advantages of simple implementation and fast transient response and is widely used in industry for point-of-load applications. This control scheme is elegant when output capacitors with large RC time constant are employed, such as OSCON capacitors. However, in most cases using capacitors with small RC time constant, such as ceramic capacitors, instability problem will occur. Previous modeling methods including sampled-data modeling, discrete-time analysis, time-domain analysis, and describing function are all very mathematical and difficult to apply for practical engineers as little physical insight can be extracted. Up to now, no equivalent circuit model is proposed which is able to predict the instability issue and serve as a useful design tool for V2 control. This paper proposes a unified equivalent circuit model which is applicable to all types of capacitors by considering the effect of capacitor voltage ripple. The equivalent circuit provides the physical insight of V2 control as a nonideal voltage source, a dual concept of previous nonideal current source for current-mode control. The equivalent circuit model is a simple yet accurate complete model and is very helpful for design purpose. Optimal design guidelines for point-of-load applications are provided. The proposed equivalent circuit model is applicable to both variable frequency modulation and constant frequency modulation. The equivalent circuit model and design guidelines are verified with Simplis simulation and experimental results.

^{\bf 2}

Equivalent circuit models are useful design tools for control and have already well served their purposes in pulse width modulation dc-dc converters. However, no simple equivalent circuit model is available yet for resonant-type dc-dc converters. Up to now, the most successful equivalent circuit model of series resonant converter (SRC) is based on extended describing function concept, which was proposed by Yang et al. However, the equivalent circuit is a complicated fifth order with the cross-coupling effect and no analytical solution is provided for transfer functions. This paper proposes a simple third-order equivalent circuit model of SRC. The equivalent circuit model is derived by simplification of the original fifth-order equivalent circuit, based on the fact that the resonant capacitor behaves like an equivalent resonant inductor with respect to the modulation frequency. The equivalent circuit model can predict the dynamic behavior very well when the switching frequency is below, close to, or above the resonant frequency. Furthermore, for the first time, analytical expressions of all transfer functions, i.e., control-to-output, input-to-output, output impedance, and input impedance are provided. These analytical transfer functions will serve as a useful tool for the feedback design. The equivalent circuit model is verified by Simplis simulation and experimental results.

Control With Multilayer Ceramic Caps

Multi-phase constant on-time current mode control based on pulse distribution structure is widely used in Voltage Regulator application for microprocessor. To minimize ripple cancellation effect, external ramp compensation is used in commercial products. However, external ramp will introduce dynamic to the system and AVP requirement will be violated without considering its effect. This paper first studies the effect of external ramp by deriving small-signal transfer function based on describing function method. It is found that external ramp brings additional dynamic, with time constant related with switching period. Then, a simple equivalent circuit model based on three-terminal switch concept is proposed, which considers the effect of external ramp by adding an additional R-L branch. The equivalent circuit model can be reduced to previous unified three-terminal switch model when external ramp is zero and can be reduced to model of constant on-time voltage mode control when external ramp is much larger than inductor current ramp. The proposed three-terminal switch model is a complete model which can be used to examine all transfer functions and is accurate up to half of switching frequency. The model is verified by Simplis simulation.

Digital Hybrid Ripple-Based Constant On-Time Control for Voltage Regulator Modules

Predictive current mode control is one of the promising digital current mode control method featuring fast dynamic response and low cost implementation. To understand the benefit and the limitation of each implementation, choose the modulation scheme and design the control loop, this paper proposes small signal Laplace-domain models for a family of digital predictive current mode controls, including peak current control, valley current control and average current control with single edge modulation and symmetric/asymmetric double edge modulation. The model is extended to the multi-sampled implementation. The proposed model considers the dynamic nonlinearity caused by the feedback of sideband frequency components of virtual inductor current, so that it is more precise than conventional Z-domain model based on averaging modeling concept and discretization. Implementation selection and sampling number selection guideline are also discussed. The theoretical analysis of this paper is verified by both simulations and experimental results.

Unified Equivalent Circuit Model and Optimal Design of

A simple third-order equivalent circuit model of series resonant converter (SRC) is proposed in this paper. Up to now, the most successful equivalent circuit model of SRC is based on extended describing function concept, which is proposed by Dr. E. Yang [30]. However, the equivalent circuit is a complicated fifth-order circuit with the cross-coupling effect and no analytical solution is provided for transfer functions. This paper proposes a methodology to simplify the fifth-order equivalent circuit to a third-order equivalent circuit. The equivalent circuit model can predict the dynamic behavior very well when switching frequency is below, close to or above resonant frequency. Furthermore, for the first time, analytical expressions of transfer functions are provided to serve as a useful tool for feedback design. The equivalent circuit model is verified by Simplis simulation and experimental results.

V^{2}

Constant-on-time V2 control is widely used in DC-DC buck converters. However, sub-harmonic oscillations occur in constant-on-time V2 control employing low-ESR caps. For digital constant-on-time V2 control, the instability issue is worse due to the sampling effect. In both cases, external ramp compensation is one simple solution to solve the instability issue, although the characteristic of constant-on-time V2 with external ramp is not fully understood. This paper provides design guidelines for external ramp by factorizing the previous small-signal model of analog constant-on-time V2 control. Then the design strategy of the external ramp is extended to digital constant-on-time V2 control. Moreover, the small-signal models between analog and digital constant-on-time V2 control are compared to reveal the sampling effect. The small-signal analysis is verified with simulation and experimental results.

Controlled Buck Converters

Multiphase constant on-time current-mode control based on pulse distribution structure is widely used in voltage regulator application for microprocessor. To minimize ripple cancellation effect, external ramp compensation is used in commercial products. However, external ramp will introduce dynamic to the system and stability margin will be suffered without considering its effect. This paper first studies the effect of external ramp by deriving small-signal transfer function based on describing function method. It is found that external ramp brings additional dynamic, with time constant related with switching period. Then, a simple equivalent circuit model based on three-terminal switch concept is proposed, which considers the effect of external ramp by adding an additional R-L branch. The equivalent circuit model can be reduced to previous unified three-terminal switch model when external ramp is zero and can be reduced to model of constant on-time voltage mode control when external ramp is much larger than inductor current ramp. The proposed three-terminal switch model is a complete model, which can be used to examine all transfer functions and is accurate up to half of switching frequency. The analytical transfer functions are provided for easy reference. The model is verified by SIMPLIS simulation and experimental measurement.