Compensators are electronic filters that enhance the speed and stability of control systems during dynamic operations. Traditionally, compensators are active circuits integrated with operational amplifiers (op amps), assumed to have perfect characteristics. However, when applied to low-bandwidth systems, such as modern converters with minimal output capacitance, achieving crossover frequencies above 100 kHz can lead to faster transient responses, limiting output voltage drops. In these cases, treating the op amp as ideal no longer holds true, causing significant distortions in both gain and phase.
By examining the open-loop gain and the influence of the low and high-frequency poles of the chosen op amp, the appropriate components can be selected to avoid compromising the necessary gain and phase characteristics required for the crossover. This discussion will initially focus on the effects of open-loop gain, omitting considerations of low and high-frequency poles. The second part will delve into these additional poles and demonstrate how improper selection can weaken the overall performance.
Different types of compensators exist, typically labeled as Type 1, Type 2, and Type 3 in switching converters. Each type includes a pole at the origin to maximize quasi-static gain, ensuring precise control over output variables. Type 1 compensators are simple integrators offering no phase boost. Type 2 builds upon Type 1, incorporating a pole-zero pair to provide a phase boost of up to 90 degrees. Type 3 further enhances phase boost by another pole-zero pair, reaching up to 180 degrees. Figure 1 illustrates the frequency responses (both amplitude and phase) and respective transfer functions for these three compensators.
[Insert Figure 1]
Type-2 compensators are common in current-mode power supplies, offering a substantial phase boost of up to 90 degrees for enhanced compensation. Figure 2 displays a typical implementation around an op amp. A resistor divider monitors the controlled variable (output voltage in this case), while passive components form the filter. To analyze the transfer function of the circuit, we'll first consider the open-loop gain (AOL) of the op amp and its influence on the final expression.
[Insert Figure 2]
To efficiently analyze such circuits, we employ Fast Analysis Circuit Techniques (FACTs), as detailed in references [2] and [3]. FACTs rely on determining circuit time constants under two conditions: when the excitation signal is removed (Vout drops to 0V) and when the response is cleared (VFB equals 0). This approach simplifies the process of deriving transfer functions, making it both quick and intuitive.
A transfer function for a first-order system with non-zero quasi-static gain can be expressed as:
[Insert Equation (1)]
Here, G0 represents the gain exhibited by the system when s=0, and N(s) governs the zero of the transfer function. The denominator D(s) contains the natural time constants of the circuit, derived by setting the excitation to zero and observing the resistance provided by the energy storage elements.
For instance, in Figure 3, we determine the time constant by setting the excitation to zero and observing the resistance at the capacitor terminals.
[Insert Figure 3]
This straightforward method allows us to calculate time constants and subsequently determine poles and zeros, simplifying the transfer function derivation.
[Insert Figures 4-6]
By applying FACTs to a Type-2 compensator, we can derive its transfer function step-by-step. Considering the second-order nature of the system and non-zero quasi-static gain, the general form of the transfer function is:
[Insert Equation (11)]
Further simplifications yield the denominator and numerator terms, allowing us to express the transfer function in a low-entropy format, easily identifying key parameters like gain, poles, and zeros.
Comparing the dynamic responses of an ideal op amp and one with finite open-loop gain reveals differences in gain and phase boost at lower frequencies. For example, selecting an op amp with a higher AOL minimizes these discrepancies, aligning the responses closely.
In conclusion, understanding the impact of open-loop gain in non-ideal op amps is crucial for designing effective compensators. Future discussions will address additional complexities introduced by low and high-frequency poles, essential for ensuring operational amplifier stability.
References:
1. C. Basso, “Designing Control Loops for Linear and Switching Converters – A Tutorial Guideâ€, Artech House 2012.
2. C. Basso, “Linear Circuit Transfer Functions – An Introduction to Fast Analytical Techniquesâ€, Wiley 2016.
3. V. Vorpérian, “Fast Analytical Techniques for Electrical and Electronic Circuitsâ€, Cambridge University Press 2002.
4. C. Basso, “Fast Analytical Techniques at Work with Small-Signal Modelingâ€, APEC Professional Seminar, Long Beach (CA), 2016.
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