Microchip AN7166 Control-Loop Analysis and Performance Optimization User Guide

Microchip AN7166 Control-Loop Analysis and Performance Optimization

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AN7166 explains how Ripple Injection in ACOT (Adaptive Constant-On-Time) control shapes control-loop gain, poles/zeros, bandwidth, phase margin, and transient performance using frequency-domain (S-domain) impedance modeling. It also contrasts a detailed frequency-domain design approach with a simplified ripple-based method and emphasizes bench validation.

Thiết kế chính

Bắt đầu nhanh

1. Work from the ACOT Type 3 ripple injection approach, where ripple is derived from the switching node through an RC ripple network (RINJ, CINJ) and shaped with a feed-forward capacitor (CFF).
2. Use the frequency-domain relationships to understand how the ratio of the injection impedance to the feed-forward impedance (ZB/ZF) affects loop gain (and thus bandwidth and stability).
3. Design pole/zero placement using the ripple injection and feed-forward components: Z1 is mainly set by RINJ and CINJ; Z2 is mainly set by R1 and CFF.
4. Select ripple injection components (for example, CINJ and CFF) and then calculate/recalculate the resulting feedback ripple VFB; verify VFB is within the device’s recommended operating range.
5. Estimate or target the desired crossover frequency FC via the loop gain profile and confirm through frequency-domain (Bode) measurement.
6. Perform bench validation in the time domain: measure feedback ripple at the comparator node and evaluate load transient response (undershoot, recovery, settling) and switching behavior.

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Loop Gain Modeling

Frequency-domain analysis is important because it helps translate the time-domain switching
behavior of any converter into a control-loop view that is easier to analyze, design and optimize.
In the time domain, ACOT operation is understood through switching-node pulses, feedback ripple,
inductor current and transient response.
However, for loop design, it is also necessary to understand how the ripple-injection network,
feedback network and output LC filter shape the loop gain across frequency. S-domain analysis
provides this framework by replacing resistors, inductors and capacitors with their corresponding
impedances. This allows the ACOT loop to be represented using impedance ratios, making it
possible to identify poles, zeros, crossover frequency and phase-margin-related behavior. In this
way, S-domain analysis connects the physical ripple-injection components to the frequency-domain
loop response and provides useful insight for stable and optimized ACOT design.
1.1. Loop Gain Modeling
Deeper analysis often necessitates circuit modeling, to provide greater insight. However,
conventional modeling methods are typically designed for converters operating at fixed frequencies.
This poses a challenge for ripple-based variable-frequency converters like those employing COT
or ACOT control, which dynamically adjust their frequency in response to load and input voltage
variations. However, for the modeling and analysis purpose, we make some practical assumptions
to develop an ACOT and ripple-injection circuit model as illustrated in Figure 1-1 and the following
derivations

Equation 1-1.
GFB = −
VFB
VO
From Figure 1-1, the LC filter stage gain can be expressed as:
Equation 1-2.
GLC =
VO
vệ sinh môi trường
=
ZC ∥ ZLOAD
ZL + ZC ∥ ZLOAD
From Equation 1-1, Equation 1-2 and Figure 1-1, the Overall Loop Gain (G) can be expressed as:
Equation 1-3.
G = GFB · AV · GLC
Equation 1-3 can be rewritten as:
Equation 1-4.
G = −
VFB
VO
· AV · ZC ∥ ZLOAD
ZL + ZC ∥ ZLOAD
Next, we aim to derive an expression for VFB/VO in relation to the impedances of the ripple injection
circuit, allowing us to establish a correlation between the ripple injection components and loop gain.
The detailed mathematical derivation for VFB/VO is provided in Appendix 1: Derivation of ACOT Loop
Gain. In summary, the Overall Loop Gain (G) can be expressed as:
Equation 1-5.
G ≅ − ZB
ZF ·
ZC ∥ ZLOAD
ZC ∥ ZLOAD + ZL
As stated in Figure 1-1, ZB is the impedance of RINJ and CINJ, which can be expressed as:
Equation 1-6.
ZB = RINJ +
1
2π · FSW · CINJ
And ZF is the impedance of R1 and CFF, which can be expressed as:
Equation 1-7.
ZF = R1
1 + 2π · FSW · R1 · CFF
From Equation 1-5, it is evident that the factor of ZB/ZF, that is, the ratio of the impedances of RC
ripple injection and feed-forward circuits, plays a crucial role in enhancing the loop gain. From the
basics of control theory, it is understood that the higher loop gains improve overall regulation
(both DC set point accuracy and transient performance).
1.1.1. Key Insight
Equation 1-5 shows that the ACOT loop gain is directly influenced by the ratio ZB/ZF, which is determined
by the ripple injection and feed-forward components. Therefore, RINJ, CINJ and CFF are not only used
to generate the required feedback ripple, but also play an important role in shaping the control loop
response, including bandwidth, stability, and transient behavior.

Control Loop Design Strategy

In any switching converter, control loop design fundamentally involves the strategic placement of
poles and zeros to shape the loop gain response across frequency. Each pole and zero directly
influences the gain slope and phase behavior of the system, thereby determining stability and
dynamic performance. Poles tend to introduce phase lag and reduce stability margin, while zeros
provide phase boost and help counteract the effects of dominant poles, such as the double pole
introduced by the output LC filter. By carefully selecting the locations of these poles and zeros,
the designer can control key parameters such as crossover frequency, phase margin, and transient
phản ứng.
In ACOT control, this principle remains the same; however, instead of using a traditional error
amplifier and compensation network, the placement of poles and zeros is primarily governed by the
ripple injection and feedback components. As a result, ACOT control loop design can be effectively
viewed as a structured process of shaping the frequency response through appropriate selection of
RINJ, CINJ and CFF to achieve the desired stability and performance.
Figure 1-2. Loop Gain Formation Using Pole-Zero Interaction

The control loop design strategy for ACOT control focuses on shaping the loop gain profile bởi vì
proper placement of poles and zeros to ensure stability and optimal transient performance. As
illustrated in Figure 1-2, the red curve represents the effect of the output LC filter, which introduces
a double pole at ωP =
1
LCOUT
.
This results in a steep –40 dB/decade slope, causing significant phase lag. If left uncompensated,
this would make the system unstable. To counteract this, zeros are introduced through the ripple
injection and feedback networks. The first zero (Z1), visible as the flattening of the black curve in the
mid-frequency region, is placed around FLC
10 đến
FLC
20 to reduce phase lag and improve mid-frequency
phản ứng.
The second zero (Z2), which causes an upward trend in the black curve near crossover, is positioned
closer to the crossover frequency to provide additional phase boost and ensure sufficient phase
margin, typically at least 60°. The resulting overall loop gain (shown in blue) combines these effects,
transitioning from the steep –40 dB/decade slope to a more manageable –20 dB/decade slope
(green region) near the crossover frequency FC, where the gain crosses 0 dB. This controlled shaping
of the gain profile ensures stable operation while maintaining fast transient response

The Crossover Frequency

The crossover frequency (FC) is one of the most critical parameters in control loop design, as it
defines the effective bandwidth of the converter and directly determines the trade-off between
transient response and stability. A higher crossover frequency generally enables faster transient
response, while a lower crossover frequency improves stability margin. Therefore, accurate
estimation and proper placement of FC is essential in achieving an optimal balance between
dynamic performance and robust operation in ACOT-controlled converters.
The determination of FC can be understood from the loop gain profile shown in Figure 1-3. Starting
from the low-frequency gain A0, the loop exhibits a –20 dB/decade slope, which is introduced
by the dominant low-frequency pole associated with the ripple injection network. At the first zero
frequency (FZ1), this slope is canceled, resulting in a flat gain region (0 dB/decade) between FZ1 and
the LC double pole frequency (FP1 = FP2).
Beyond this point, the output LC filter introduces a –40 dB/decade slope, causing a rapid reduction
in loop gain. The second zero (FZ2), introduced by the feed-forward capacitor, provides phase boost
and changes the slope to –20 dB/decade. The crossover frequency FC is defined as the point where
this overall gain curve intersects the 0 dB line. Thus, FC is determined by the cumulative gain
changes across these frequency regions.
Figure 1-3. ACOT Control Loop Gain Approximation

Applying this relationship step-by-step across the regions shown in Figure 1-3:
• From A0 to A1 (slope = –20 dB/dec): A1 = A0 − 20log FZ1
F0
• From A1 to A2 (flat region): A2 = A1
• From A2 to A3 (slope = –40 dB/dec): A3 = A2 − 40log FP
FZ1
• From A3 to crossover FC (slope = –20 dB/dec): A4 = 0 = A3 − 20log FC
FZ2
Substituting backward and simplifying these relationships leads to an expression for crossover
frequency in terms of system parameters after rearranging and substituting the pole and zero
địa điểm:
Equation 1-18.
FZ1 =
1
2πRINJCINJ
, FZ2 =
1
2πR1CFF
, FP =
1
2π LCOUT
A detailed derivation, including application of logarithmic gain relationships and substitution of
pole-zero locations, is provided in Appendix 3: Derivation of Crossover Frequency. After performing
these steps and simplifying the expressions, the crossover frequency can be expressed as:
Equation 1-19.
FC =
RINJ ⋅ CFF
2πLCOUT
Further, using the ripple injection relationship:
Equation 1-20.
∆ VFB =
VIN · D · 1 − D
RINJ · CFF · FSW
and rearranging and substituting into Equation 1-19, the crossover frequency can be rewritten as:
Equation 1-21.
FC =
VO · 1 − D
2πLCOUT · ∆ VFB · FSW
1.4.1. Key Insight
This final expression provides key design insight, showing that the crossover frequency is directly
controlled by the ripple injection network and operating conditions, and inversely proportional to the
output LC filter. It highlights that in ACOT control, tuning the feedback ripple ΔVFB effectively allows the
designer to set the loop bandwidth and thereby optimize both transient response and stability

AC Analysis and Transient Response

The designed ACOT control loop is validated using frequency-domain measurements, as shown in
the Bode plot. The gain crosses 0 dB at approximately 24.8 kHz, which closely matches the targeted
crossover frequency selected during the design process. At this point, the measured phase margin is
around 68°, indicating a stable control loop with sufficient margin for robust operation.
The second zero (FZ2) is observed around 13 kHz, providing the intended phase boost prior to
crossover. The gain and phase characteristics confirm that the placement of poles and zeros, along
with the selected ripple injection components (CINJ = 15 nF, RINJ = 200 kΩ, CFF = 560 pF), successfully
shape the loop response as predicted by the analytical design. This validates the effectiveness of the
frequency-domain design approach in achieving the desired loop stability.
The transient performance of the converter is evaluated using a load step from 2.5 A to 5 A with a
fast transition time. The measured output voltage waveform shows an undershoot of approximately
120 mV, followed by a recovery within about 13.4 µs, and a settling time of approximately 30 µs.
AN7166
Thiết kế Example (Using Frequency Domain Analysis)
Ghi chú ứng dụng
© 2026 Microchip Technology Inc. và các công ty con của nó
DS00007166A – 13
The ripple waveform at the feedback node remains well-defined, confirming proper ripple injection
and stable comparator operation during dynamic conditions. These results demonstrate that
the designed control loop not only meets stability requirements but also delivers fast transient
response, highlighting the trade-off achieved between crossover frequency, phase margin, and
gợn sóng amplitude. Overall, the measured performance aligns well with the design expectations,
confirming the validity of the analytical and frequency-domain design methodology

The comparison above highlights the difference between the detailed frequency-domain design
approach and the simplified ripple-based design approach for ACOT™ control implementation. In
the frequency-domain analysis method, the ripple injection network is intentionally designed by
positioning the poles and zeros to achieve a targeted crossover frequency and a controlled phase
margin. This provides better predictability and tighter optimization of transient response, stability,
and loop bandwidth. In this example, the design achieves a crossover frequency of approximately
25 kHz with a well-controlled phase margin of 68°.
In contrast, the simplified approach selects the ripple injection components mainly based on
practical ripple generation requirements and approximate design relationships, without performing
a full AC loop analysis. Even though the resulting pole-zero locations and crossover frequency
differ, the converter still achieves stable operation with a very high phase margin (> 80°). This
demonstrates that the simplified approach is often sufficient for many practical applications
and significantly reduces design complexity. However, when precise optimization of dynamic
performance, bandwidth, or phase margin is required, the frequency-domain analysis approach
provides much greater control and design accuracy

eedback Ripple vs. Transient Response
Now, let us examine some design examples to validate the preceding analyses and explanations.
This analysis aims to comprehend the transient response performance concerning variations in
feedback ripple amplitude (ΔVFB) so that we can optimize the ripple injection components for the
desired transient performance.
Figure 5-1 illustrates the schematic diagram of an MIC28515 buck converter design with VIN = 48V,
VOUT = 5V, FSW = 266 kHz. Analysis will be performed with a transient load from 2.5A to 5A with
a slew rate of 2.5A/3 us. The details of the evaluation board used for testing can be found in the
MIC28515 Evaluation Board user guide.
In the reference schematic (Figure 5-1), the ripple injection components employed are RINJ = 16.2k,
CINJ = 0.1 uF and CFF = 4.7 nF. Utilizing Equation 3-1, we can compute ΔVFB. The calculated value is
220 mV, which closely aligns with the test result (slight variations in the duty cycle compensate for
losses, leading to slight deviations in the calculated value). Figure 5-2 depicts the transient response
performance associated with Figure 5-1 reference schematic, and Figure 5-3 through Figure 5-5
show the transient performance across different ΔVFB amplitudes achieved by adjusting the ripple
injection components.