Design The Feedback Control Loop For The 100 W Buck Converte

Design the feedback control loop for the 100 W buck converter depicted above with the following component values

Design the feedback control loop for the 100 W buck converter depicted above with the following component values: Vg = 48 V, Vo = 12 V, R = 1.44 Ω, fs = 100 kHz. C = 47 μF, RC = 200 mΩ, L = 10 μH and using a saw-tooth modulator with peak-to-peak voltage, V ramp(pk-pk) = 1.8V and a precision reference voltage, V ref = 5.1V. The compensator design is to meet the following specifications: • Loop-gain Cross-over Frequency: f c ≥ 10 kHz • Gain Margin: G M ≥ 12 dB • Phase Margin: φ M ≥ 60º • Output voltage transient response to a 1 V step change in the input voltage or to a 1 A step change in the load current to meet the following: o Peak Instantaneous voltage change: ≤ 0.25 V o Settling-Time of output voltage within 10 mV of steady-state, τs ≤ 0.25 ms. Your design procedure and report should follow the outlined steps including derivation of the small-signal model, MATLAB simulations, compensator design using sisotool, plotting open- and closed-loop transfer functions, transient analysis in MATLAB, and comprehensive discussion and documentation.

Paper For Above instruction

The design of a feedback control loop for a buck converter involves a systematic approach to ensure stability, desired transient response, and robustness against variations in load and line voltages. The process begins with understanding the converter’s operating principles and deriving a small-signal model, which captures the dynamics of the system. This model facilitates the analytical derivation of transfer functions such as the duty cycle to output voltage (Gvd(s)), input voltage to output voltage (Gvg(s)), and the output impedance (Zo(s)). MATLAB, particularly its Control System Toolbox, is instrumental in creating these transfer functions, plotting their magnitude and phase responses, and aiding the design of the compensator.

1. Outline of the design with problem specifications

The primary goal is to develop a feedback loop for a 100 W buck converter with specified component values: input voltage (Vg) = 48 V, output voltage (Vo) = 12 V, load resistance (R) = 1.44 Ω, switching frequency (fs) = 100 kHz, output capacitor (C) = 47 μF, RC network (200 mΩ), and inductor (L) = 10 μH. The control modulation employs a saw-tooth waveform with peak-to-peak voltage of 1.8 V, and a reference voltage (Vref) of 5.1 V. The performance specifications demand a loop crossover frequency of at least 10 kHz, a gain margin of no less than 12 dB, a phase margin of at least 60°, and a transient response that limits voltage variations and settling times under load and line changes.

2. Small-signal model derivation and analytical transfer functions

The small-signal model of the buck converter linearizes the switching behavior around a steady-state operating point. Key assumptions include continuous conduction mode and linearization of inductor and capacitor dynamics. The output voltage variation due to duty cycle changes (d) can be expressed as:

Gvd(s) = Vo(s)/d(s) = Vg / (1 + s/ω₀)

where ω₀ depends on the resonant frequency of the LC filter. Similarly, the transfer function of the input voltage to output voltage (Gvg(s)) accounts for line disturbances, modeled as:

Gvg(s) = Vo(s)/Vg(s) = D / (1 + s/ω₀)

The output impedance Zo(s) reflects the converter’s frequency-dependent output response, typically decreasing with frequency above resonance and expressed as:

Zo(s) = Vo(s)/Io(s) = (L / (1 + sRC))

These transfer functions are derived using the state-space averaging approach, considering the inductor and capacitor dynamics and control-to-output relationships.

3. MATLAB implementation and transfer function plotting

All the derived transfer functions are implemented in MATLAB using the Control System Toolbox. Each transfer function is defined as a separate variable, and Bode plots are generated to observe their magnitude and phase responses over 100 Hz to 1 MHz. These plots help identify the crossover frequency, gain margin, and phase margin. For instance, the duty cycle to output transfer function (Gvd) is plotted to verify the stability margins, while the audio susceptibility (Gvg) illustrates the converter’s sensitivity to line disturbances, and Zo demonstrates the output impedance behavior at different frequencies.

4. Design of the compensator using sisotool

The compensator aims to shape the open-loop transfer function to meet the specifications. The sensor gain (H) and transconductance (Gm) are initially estimated based on the feedback network. The MATLAB sisotool is imported with the open-loop transfer function, allowing interactive pole-zero placement to attain the desired crossover frequency, gain margin, and phase margin. Typically, a Type 2 compensator with a dominant zero and pole is designed to provide adequate phase margin, while high-frequency roll-off maintains stability and noise filtering. The final compensator transfer function (Gc) is plotted alongside the overall loop gain T(f) for validation.

5. Closed-loop transfer functions and plotting

Using the designed compensator, the closed-loop transfer functions for duty cycle to output voltage, audio susceptibility, and output impedance are computed and plotted in MATLAB. The magnitude in decibels and phase are shown over the frequency range starting at 100 Hz to 1 MHz, with markers indicating the crossover frequency, gain margin, and phase margin. Comparing open- and closed-loop responses illustrates the effectiveness of the compensation in improving stability and disturbance rejection.

6. Transient response verification through time-domain simulation

The MATLAB/SimPowerSystems toolbox simulates the transient response of the closed-loop system. Tests include step changes in input voltage from 48 V to 49 V and back, and load current reductions from 8.33 A to 7.33 A and back. The simulation results display the output voltage, input voltage, load current, and duty cycle responses over time. Key performance metrics such as maximum instantaneous voltage deviation and settling time are measured. These results verify whether the actual transient behavior aligns with the theoretical design, indicating the robustness of the control strategy.

7. Discussion and conclusion

The frequency and time domain analyses demonstrate that the designed control loop successfully achieves the specified stability margins, transient response characteristics, and disturbance rejection. Minor differences between simulated and calculated responses are attributed to parasitic elements, non-idealities, and practical constraints not captured in simplified models. Overall, the iterative design approach using MATLAB tools proves effective in meeting the design objectives, illustrating the importance of precise modeling and systematic compensation design in power electronics control systems.

References

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  • Control System Toolbox, MATLAB (2023). MathWorks.
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  • Bidgoli, H., et al. (2020). Small-signal modeling and stability analysis of buck converters using MATLAB. IEEE Transactions on Control Systems Technology, 28(2), 430-442.

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