## Equivalent Circuit Model for Current Mode Controls and Its Extensions

##### Abstract

Current-mode control architectures have been an indispensable technique in many applications, such as Voltage Regulator, Point-of-load converters, power factor correction, battery charger and LED driver. Since the inductor current ramp is used in the modulator in current-mode control without any low pass filter, high order harmonics play important role in the feedback control. This is the reason for the difficulty in obtaining the small-signal model for current-mode control in the frequency domain. A continuous time domain model was recently proposed as a successful model for current-mode control architectures. However, the model was derived by describing function method, which is very arithmatically complicated, not to mention time consuming.

For the analysis and design of non-linear system, equivalent circuit model, which is user friendly and intuitive, is an effective tool. In this dissertation, the primary objective is to develop a unified three-terminal switch model for current-mode controls using the results of describing function derivation, which characterizes the small signal property of the common subcircuit of current mode controlled PWM converters. Its application is extended to average current mode control, V2 control and other proposed novel current mode control schemes.

First, the existing model for current mode control is reviewed. The limitations of exsiting model for current-mode control are identified. Based on the universal small signal relationship between terminal currents and the results of describing function derivation, a unified three-terminal switch model for current mode control is proposed. A three-terminal equivalent circuit is developed to represent the small signal behavior of this common sub-circuit. The proposed model is applicable in both constant frequency and variable frequency modulation.

After that, the modeling of digital predictive current mode control is presented. Predictive current mode control is one of the promising digital current mode control method featuring fast dynamic response and low sample rate requirement. Many implementations were presented in past ten years. To understand the benefit and the limitation of each implementation, help the engineer to choose the modulation scheme and design the control loop, a small signal Laplace-domain model for digital predictive current mode controls is proposed. The model is extended to the multi-sampled implementation. The modeling result is summarize as the small signal equivalent circuit mode, whose form is consistant with that of analog current mode controls. Based on S-domain model, digital predictive current mode controls are compared with analog implementation to demonstrate the advantages and limitation. Implementation selection guideline and compensation is discussed based on the modeling results.

Then, using the proposed unified model is used in the analysis of average current mode control. Under proper design, the inductor current ripple passes through the current compensator and appears in PWM comparator. It significantly influence the high frequency small signal property of the converter. In chapter 3, the proportional feedback is separated from integral feedback so that the sideband frequency feedback effect can be taken into consideration. It extends the results obtained in peak-current model control to average current mode control. The proposed small signal model is accurate up to half switching frequency, predicting the sub-harmonic instability. Based on the proposed model, a new feedback design guideline is proposed. By designing the external ramp following the proposed design guideline, quality factor of the double poles at half of switching frequency in control-to-output transfer function can be precisely controlled. This helps the feedback design to achieve widest control bandwidth and proper damping.

V2 control is a popular control scheme in Point-of-load converters due to the unique fast transient response. As the output voltage ripple is used as PWM modulation ramp, V2 control has close relationship with current mode control but this relationship was not addressed in the exsiting model. Chapter 4 utilizes the three-terminal switch model to build the equivalent circuit model for V2 control, which clearly shows that V2 control is a particular implementation of current mode control, with proportional capacitor voltage feedback and load current feedback embeded.

The analysis presented in Chapter 3 provides a clear physical understanding of average current mode control. With constant frequency modulation, the control bandwidth is usually limited by the double pole at half of swithcing frequency, especially in the converters with wide duty cycle range. Chapter 5 proposed a novel I2 current mode control to improve the dynamic performance of average current mode control. In particular, constant on-time I2 control eliminates the need of external ramp while the current loop is inherently stable. Moreover, constant on-time modulation improves the light load efficiency.

As a conclusion, this dissertation proposed a unified three-terminal switch model for current mode controls. The application of this equivalent circuit model is extended to average current mode control, V2 control and the novel I2 current mode control. The Laplace-domain model of predictive current mode control is also presented. All the modeling results are verified through simulation and experiments.

For the analysis and design of non-linear system, equivalent circuit model, which is user friendly and intuitive, is an effective tool. In this dissertation, the primary objective is to develop a unified three-terminal switch model for current-mode controls using the results of describing function derivation, which characterizes the small signal property of the common subcircuit of current mode controlled PWM converters. Its application is extended to average current mode control, V2 control and other proposed novel current mode control schemes.

First, the existing model for current mode control is reviewed. The limitations of exsiting model for current-mode control are identified. Based on the universal small signal relationship between terminal currents and the results of describing function derivation, a unified three-terminal switch model for current mode control is proposed. A three-terminal equivalent circuit is developed to represent the small signal behavior of this common sub-circuit. The proposed model is applicable in both constant frequency and variable frequency modulation.

After that, the modeling of digital predictive current mode control is presented. Predictive current mode control is one of the promising digital current mode control method featuring fast dynamic response and low sample rate requirement. Many implementations were presented in past ten years. To understand the benefit and the limitation of each implementation, help the engineer to choose the modulation scheme and design the control loop, a small signal Laplace-domain model for digital predictive current mode controls is proposed. The model is extended to the multi-sampled implementation. The modeling result is summarize as the small signal equivalent circuit mode, whose form is consistant with that of analog current mode controls. Based on S-domain model, digital predictive current mode controls are compared with analog implementation to demonstrate the advantages and limitation. Implementation selection guideline and compensation is discussed based on the modeling results.

Then, using the proposed unified model is used in the analysis of average current mode control. Under proper design, the inductor current ripple passes through the current compensator and appears in PWM comparator. It significantly influence the high frequency small signal property of the converter. In chapter 3, the proportional feedback is separated from integral feedback so that the sideband frequency feedback effect can be taken into consideration. It extends the results obtained in peak-current model control to average current mode control. The proposed small signal model is accurate up to half switching frequency, predicting the sub-harmonic instability. Based on the proposed model, a new feedback design guideline is proposed. By designing the external ramp following the proposed design guideline, quality factor of the double poles at half of switching frequency in control-to-output transfer function can be precisely controlled. This helps the feedback design to achieve widest control bandwidth and proper damping.

V2 control is a popular control scheme in Point-of-load converters due to the unique fast transient response. As the output voltage ripple is used as PWM modulation ramp, V2 control has close relationship with current mode control but this relationship was not addressed in the exsiting model. Chapter 4 utilizes the three-terminal switch model to build the equivalent circuit model for V2 control, which clearly shows that V2 control is a particular implementation of current mode control, with proportional capacitor voltage feedback and load current feedback embeded.

The analysis presented in Chapter 3 provides a clear physical understanding of average current mode control. With constant frequency modulation, the control bandwidth is usually limited by the double pole at half of swithcing frequency, especially in the converters with wide duty cycle range. Chapter 5 proposed a novel I2 current mode control to improve the dynamic performance of average current mode control. In particular, constant on-time I2 control eliminates the need of external ramp while the current loop is inherently stable. Moreover, constant on-time modulation improves the light load efficiency.

As a conclusion, this dissertation proposed a unified three-terminal switch model for current mode controls. The application of this equivalent circuit model is extended to average current mode control, V2 control and the novel I2 current mode control. The Laplace-domain model of predictive current mode control is also presented. All the modeling results are verified through simulation and experiments.

##### Collections

- Doctoral Dissertations [11316]