Unit 7: Three-Phase Bridge Rectifiers: Commutation Overlap and Harmonic Profiles

Unit 7 — Small-signal dynamics and feedback

The engineering argument

A controlled converter is not just a static voltage ratio. Stored energy, load impedance, control delay, sensors and saturation shape its dynamic response. A compensator designed from an inappropriate low-frequency approximation may oscillate when the real converter has additional poles or zeros.

We linearize a buck stage near a nonzero operating point and compare open-loop plant dynamics with a closed-loop response. The objective is to understand what a feedback controller can and cannot fix using measured states.

Boost conversion contains a nonminimum-phase effect in continuous voltage-mode control: raising duty initially reduces the fraction of cycle delivering current to the output even though it can eventually raise the voltage. This limits achievable voltage-loop speed.

Dynamic models, regulation and delay. A static conversion equation is only an equilibrium condition. For feedback design, the inductor current and capacitor voltage obey differential equations; a control input changes their rates rather than changing the output instantaneously. Averaging yields useful low-frequency dynamics while hiding individual switching edges.

The reference command, measured output, actuator duty and applied duty should appear separately in every feedback validation plot. If the requested duty saturates while a numerical integrator keeps accumulating error, an apparently mysterious recovery overshoot has a specific mechanism: windup.

An additional delay in sampling, computation, filtering or PWM update shifts phase. It may be negligible relative to a very slow loop but decisive for a loop pushed toward switching frequency. The proper comparison is delay expressed as a fraction of the loop period.

A boost stage exhibits a particularly important limitation when commanded duty produces an initial output response in the direction opposite the eventual steady change. That nonminimum-phase effect requires a deliberately conservative loop design rather than an optimistic retuning of proportional gain.

Sequence and learning objectives

In the first lesson, identify the model, boundary and relevant physical state. In the second, compute the dominant currents, voltages or losses under a specified operating point. In the third, investigate where the ideal result ceases to hold. In the fourth, convert the model into a design review with independently falsifiable tests. This sequence is intended to prevent equation memorization without comprehension. The central question is not whether a formula can be remembered but whether the learner can recognize when it should not be used.

Unit reading

• 1. [Averaged buck state equations](lessons/U07-L01-averaged-buck-state-equations.html)
• 2. [Load regulation, PI compensation and windup](lessons/U07-L02-load-regulation-pi-compensation-and-windup.html)
• 3. [Boost nonminimum-phase response](lessons/U07-L03-boost-nonminimum-phase-response.html)
• 4. [Control bandwidth, sensing and digital delay](lessons/U07-L04-control-bandwidth-sensing-and-digital-delay.html)

Unit model: [interactive simulation](simulations/model-07.html). Unit project: [design study](projects/unit-07.html). Assessment: [review problems](assessments/unit-07.html).

Lesson 7.1 — Averaged buck state equations

Unit: Small-signal dynamics and feedback. Focus: energy balance, switching-state modeling, numerical prediction, design limitations, verification. Prerequisites: basic circuit analysis, the definition of average and RMS values, and the relevant preceding lessons. Safety: This is mathematical and simulated content; it does not license direct connection to mains or high-energy battery systems.

What the model is trying to explain

In the ideal continuous-conduction buck model, L di_L/dt = d Vin − v_C and C dv_C/dt = i_L − v_C/R. Equilibrium requires v_C=D Vin and i_L=v_C/R. These differential equations reveal transients absent from the static voltage-ratio formula.

The starting point is an electrical system with a stated input, a desired output, and at least one physical storage or loss mechanism. Choose voltage and current arrows before writing any balance equation. This apparently clerical step determines whether a negative term represents energy returned to a source, dissipated heat, or a mismatch of sign conventions. For a switching system the state of the controlled semiconductor changes the governing differential equation; a mean-value formula follows only after the switching states and the relevant steady-state conditions have been established. A time-domain numerical result should therefore always be accompanied by the state assumptions under which it was computed.

Complete numerical walkthrough

With Vin=24 V, d=0.5 and R=6 ohms, equilibrium voltage is 12 V and inductor current is 2 A. For L=100 microhenrys and C=470 microfarads, the undamped LC natural frequency is 1/(2π√LC)≈734 Hz before including load damping and parasitics.

How to verify the arithmetic independently. First convert every milliohm, microsecond, microfarad, kilohertz and microhenry into consistent base units. Next substitute the stated values into the governing relation in one line before rounding. Keep a separate column for the measured or assumed quantity and the calculated quantity. After calculation, check the physical limiting case: at zero load there should be no load power, at zero switching interval there should not be energy transfer through an ideal PWM command, and the sign of a voltage cannot be changed by efficiency alone. These simple limiting checks often expose a wrong topology or an invalid formula more reliably than another decimal place.

For a useful second calculation, change one design variable while holding the others fixed. If a capacitor is changed, do not simultaneously alter switching frequency and load current, because the effect becomes difficult to attribute. Record which quantities are genuinely controlled and which are consequences of the change. Where the model is an approximation, report its domain of validity beside the number; a precise numerical answer derived outside the model's domain can be more misleading than a rough answer with honest uncertainty.

Why a technically plausible answer may be wrong

Averaging filters out switching ripple; it cannot predict MOSFET commutation spikes or subharmonic instability of a specific peak-current-mode controller without extra states.

In particular, distinguish a device rating, a nominal operating point, a predicted steady-state waveform and a measured transient peak. The four are not interchangeable. An ideal average voltage says little about a nanosecond-level overshoot; a calculated RMS current does not establish that a junction temperature is acceptable; a controller response without saturation says nothing about a hard overcurrent event. The reader should identify the first assumption likely to fail at an extreme input, a very light load, a startup event, or a component temperature different from the nominal value.

Engineering decision and defensible tradeoff

Write the model in matrix form for states [i_L,v_C]. Find the equilibrium and compare it to one numerical integration of the averaged differential equations.

State the requirement as a measurable inequality or a bounded interval instead of using only adjectives such as fast, small, efficient or robust. For example, a requirement for output voltage has to name the input-voltage range, load range, transient duration and maximum allowed deviation. A requirement for low loss has to identify the ambient-temperature condition and whether the gate driver or controller is included in the loss accounting. A comparison among device options is not reproducible unless all alternatives are evaluated under the same service conditions.

Also note what is sacrificed when the selected variable is improved. Raising frequency can decrease filter size but increase switching heat; lowering channel resistance may increase gate-charge cost; larger capacitance can improve low-frequency hold-up while increasing inrush; stronger voltage-loop action can improve a small load step but erode phase margin at a different point. Engineering is the management of coupled constraints rather than the maximization of one attractive metric.

Laboratory thinking and falsifiable observations

Investigative question: What information is missing from the averaged model when selecting a voltage probe bandwidth?

Develop a small simulated experiment with two deliberately different operating conditions and one held-constant reference. The output of the experiment should be a table with physical units, a plotted response and an explicit criterion that would disprove the initial interpretation. Include a counterexample that changes the topology or its conduction mode. If your model predicts an impossible output at a duty limit, do not hide that point; explain which neglected physical mechanism becomes dominant and replace the formula with an appropriate higher-fidelity model before continuing.

To compare simulation with future measurements, define where voltage is measured, how current direction is assigned, the required bandwidth, and the averaging interval. If the result is efficiency, synchronize input and output data. If the result is a switching peak, use appropriately rated equipment and identify probe-induced artifacts. If the result is a protection threshold, include comparator tolerance and intervention latency. This is a review workflow, not an instruction to energize unvalidated hardware.

Practice set — show a numerical answer and an engineering explanation

1. Reconstruct the example: Repeat the supplied computation in SI units, state every approximated parameter, and make a dimensional check. If any input is missing, state that a numerical result is indeterminate rather than inventing one.

2. Alternative operating point: Change the stated input or load by 20%. Explain whether the original formula remains applicable and calculate the new value if sufficient inputs are known.

3. Failure-mode challenge: Identify a capacitor, inductor, semiconductor, driver or sensing failure that could invalidate the optimistic ideal result. Explain which independent signal or check reveals the failure.

4. Specification writing: Rewrite the design decision as a testable requirement with a numerical tolerance, measurement point and operating range. Distinguish thermal limits from electrical limits.

5. Model comparison: Predict which conclusion would survive in an ideal switched simulation, an averaged simulation and a real physical implementation. Name at least one effect visible in one model and invisible in another.

6. Research extension: Find the parameter in the appropriate component datasheet that has the largest uncertainty at elevated temperature. Describe a conservative sensitivity study using that parameter instead of extrapolating one room-temperature curve.

Worked-answer guidance

For question 1, the numerical walkthrough gives the target relation and input values; correct work reports the substituted quantity with its units, not just the final number. In question 2, choose only a single independent change; be explicit about whether the converter remains in continuous conduction or the same thermal state. For question 3, a plausible defect must have an observable consequence and not merely a descriptive label. For question 4, a valid specification tells another engineer exactly what to measure and what constitutes passing. For question 5, an ideal averaged stage cannot resolve switching-edge electromagnetic interference, and a switched circuit model alone cannot establish manufactured-hardware safety. For question 6, a meaningful answer reports the temperature and bias conditions under which the parameter is specified.

Cross-link to the unit design study

This calculation contributes to Control-loop model integrity review. Put the result into the unit's requirements matrix and identify whether it changes a semiconductor voltage rating, magnetic current rating, controller operating range, expected efficiency or verification plan. A unit project is not complete when a formula produces one plausible number; it is complete when the predicted outcome, the assumptions and the unresolved risks are transparent to a second reader.

Source-to-load audit

Draw the current path through every conducting component in the operating state most relevant to this lesson. For a resistive component account for I²R heat; for a forward-biased diode account for the current-dependent drop; for an energy-storage element account for the sign of its energy derivative. Explain which path does not carry current during the complementary state. This diagram is a prerequisite for accurate peak, RMS and average claims. If the chosen model is averaged, mark the high-frequency phenomena that will be absent from its output.

Develop a three-column assumptions ledger: the property assumed ideal, the real physical effect omitted, and the likely sign of the error in estimated heat or voltage. Do not automatically assume every neglected effect reduces output voltage; ringing can raise instantaneous device stress while losses lower mean output. Use this ledger to decide which missing information is worth acquiring first. A model is a tool for ranking uncertainty, not a license to choose optimistic component ratings.

Lesson 7.2 — Load regulation, PI compensation and windup

Unit: Small-signal dynamics and feedback. Focus: energy balance, switching-state modeling, numerical prediction, design limitations, verification. Prerequisites: basic circuit analysis, the definition of average and RMS values, and the relevant preceding lessons. Safety: This is mathematical and simulated content; it does not license direct connection to mains or high-energy battery systems.

What the model is trying to explain

A voltage-loop PI controller increases duty when the output falls below reference. Integral action can remove a steady error in an ideal stable loop, but actuator clipping causes integral windup unless handled explicitly. Loop gains must be evaluated with switching frequency, output dynamics and sensor filtering in mind.

The starting point is an electrical system with a stated input, a desired output, and at least one physical storage or loss mechanism. Choose voltage and current arrows before writing any balance equation. This apparently clerical step determines whether a negative term represents energy returned to a source, dissipated heat, or a mismatch of sign conventions. For a switching system the state of the controlled semiconductor changes the governing differential equation; a mean-value formula follows only after the switching states and the relevant steady-state conditions have been established. A time-domain numerical result should therefore always be accompanied by the state assumptions under which it was computed.

Complete numerical walkthrough

In a toy sampled controller, e=Vref−Vmeas and u=Kp e+Ki∫e dt. If duty is clamped to 0.95 while the integral continues to increase during an overload, recovery after removing the overload may overshoot despite a nominally stable linear design.

How to verify the arithmetic independently. First convert every milliohm, microsecond, microfarad, kilohertz and microhenry into consistent base units. Next substitute the stated values into the governing relation in one line before rounding. Keep a separate column for the measured or assumed quantity and the calculated quantity. After calculation, check the physical limiting case: at zero load there should be no load power, at zero switching interval there should not be energy transfer through an ideal PWM command, and the sign of a voltage cannot be changed by efficiency alone. These simple limiting checks often expose a wrong topology or an invalid formula more reliably than another decimal place.

For a useful second calculation, change one design variable while holding the others fixed. If a capacitor is changed, do not simultaneously alter switching frequency and load current, because the effect becomes difficult to attribute. Record which quantities are genuinely controlled and which are consequences of the change. Where the model is an approximation, report its domain of validity beside the number; a precise numerical answer derived outside the model's domain can be more misleading than a rough answer with honest uncertainty.

Why a technically plausible answer may be wrong

There is no universally valid PI gain pair from switching frequency alone. The plant, sensor gain, modulation scaling, delay and operating point must be specified.

In particular, distinguish a device rating, a nominal operating point, a predicted steady-state waveform and a measured transient peak. The four are not interchangeable. An ideal average voltage says little about a nanosecond-level overshoot; a calculated RMS current does not establish that a junction temperature is acceptable; a controller response without saturation says nothing about a hard overcurrent event. The reader should identify the first assumption likely to fail at an extreme input, a very light load, a startup event, or a component temperature different from the nominal value.

Engineering decision and defensible tradeoff

Use a bounded integral or back-calculation antiwindup, and log requested duty versus applied duty. Compare a load-step response with and without clamping in a safe numerical model.

State the requirement as a measurable inequality or a bounded interval instead of using only adjectives such as fast, small, efficient or robust. For example, a requirement for output voltage has to name the input-voltage range, load range, transient duration and maximum allowed deviation. A requirement for low loss has to identify the ambient-temperature condition and whether the gate driver or controller is included in the loss accounting. A comparison among device options is not reproducible unless all alternatives are evaluated under the same service conditions.

Also note what is sacrificed when the selected variable is improved. Raising frequency can decrease filter size but increase switching heat; lowering channel resistance may increase gate-charge cost; larger capacitance can improve low-frequency hold-up while increasing inrush; stronger voltage-loop action can improve a small load step but erode phase margin at a different point. Engineering is the management of coupled constraints rather than the maximization of one attractive metric.

Laboratory thinking and falsifiable observations

Investigative question: How could a controller appear stable at a small load step but recover poorly after a prolonged overload?

Develop a small simulated experiment with two deliberately different operating conditions and one held-constant reference. The output of the experiment should be a table with physical units, a plotted response and an explicit criterion that would disprove the initial interpretation. Include a counterexample that changes the topology or its conduction mode. If your model predicts an impossible output at a duty limit, do not hide that point; explain which neglected physical mechanism becomes dominant and replace the formula with an appropriate higher-fidelity model before continuing.

To compare simulation with future measurements, define where voltage is measured, how current direction is assigned, the required bandwidth, and the averaging interval. If the result is efficiency, synchronize input and output data. If the result is a switching peak, use appropriately rated equipment and identify probe-induced artifacts. If the result is a protection threshold, include comparator tolerance and intervention latency. This is a review workflow, not an instruction to energize unvalidated hardware.

Practice set — show a numerical answer and an engineering explanation

1. Reconstruct the example: Repeat the supplied computation in SI units, state every approximated parameter, and make a dimensional check. If any input is missing, state that a numerical result is indeterminate rather than inventing one.

2. Alternative operating point: Change the stated input or load by 20%. Explain whether the original formula remains applicable and calculate the new value if sufficient inputs are known.

3. Failure-mode challenge: Identify a capacitor, inductor, semiconductor, driver or sensing failure that could invalidate the optimistic ideal result. Explain which independent signal or check reveals the failure.

4. Specification writing: Rewrite the design decision as a testable requirement with a numerical tolerance, measurement point and operating range. Distinguish thermal limits from electrical limits.

5. Model comparison: Predict which conclusion would survive in an ideal switched simulation, an averaged simulation and a real physical implementation. Name at least one effect visible in one model and invisible in another.

6. Research extension: Find the parameter in the appropriate component datasheet that has the largest uncertainty at elevated temperature. Describe a conservative sensitivity study using that parameter instead of extrapolating one room-temperature curve.

Worked-answer guidance

For question 1, the numerical walkthrough gives the target relation and input values; correct work reports the substituted quantity with its units, not just the final number. In question 2, choose only a single independent change; be explicit about whether the converter remains in continuous conduction or the same thermal state. For question 3, a plausible defect must have an observable consequence and not merely a descriptive label. For question 4, a valid specification tells another engineer exactly what to measure and what constitutes passing. For question 5, an ideal averaged stage cannot resolve switching-edge electromagnetic interference, and a switched circuit model alone cannot establish manufactured-hardware safety. For question 6, a meaningful answer reports the temperature and bias conditions under which the parameter is specified.

Cross-link to the unit design study

This calculation contributes to Control-loop model integrity review. Put the result into the unit's requirements matrix and identify whether it changes a semiconductor voltage rating, magnetic current rating, controller operating range, expected efficiency or verification plan. A unit project is not complete when a formula produces one plausible number; it is complete when the predicted outcome, the assumptions and the unresolved risks are transparent to a second reader.

Parameter-sensitivity notebook

Construct two more operating points around the worked example by reducing and increasing the selected key parameter by 10%. Repeat the arithmetic and compare the percent response change to the percent parameter change. If the response is nonlinear, explain why its sensitivity grows or shrinks. Examples include the inverse factor 1/(1−D) at high duty, the quadratic I²R dependence of channel heat, and the inverse dependence of ripple on L, C or switching frequency.

For the same change, decide whether the numerical model has moved closer to a known mode boundary, a component maximum rating, a closed-loop stability restriction or a measurement limitation. A derivative taken inside a valid smooth model is useful; it is not sufficient to predict behavior across a discontinuous-conduction boundary or a fault-protection trip. Identify which conclusion is local and which remains valid over the full stated range.

Lesson 7.3 — Boost nonminimum-phase response

Unit: Small-signal dynamics and feedback. Focus: energy balance, switching-state modeling, numerical prediction, design limitations, verification. Prerequisites: basic circuit analysis, the definition of average and RMS values, and the relevant preceding lessons. Safety: This is mathematical and simulated content; it does not license direct connection to mains or high-energy battery systems.

What the model is trying to explain

In continuous-conduction boost voltage-mode control, increasing duty lengthens the interval when the inductor does not deliver current to the output. The output capacitor can initially discharge faster, even as the eventual steady voltage rises. This inverse response is associated with a right-half-plane zero in the duty-to-output transfer function.

The starting point is an electrical system with a stated input, a desired output, and at least one physical storage or loss mechanism. Choose voltage and current arrows before writing any balance equation. This apparently clerical step determines whether a negative term represents energy returned to a source, dissipated heat, or a mismatch of sign conventions. For a switching system the state of the controlled semiconductor changes the governing differential equation; a mean-value formula follows only after the switching states and the relevant steady-state conditions have been established. A time-domain numerical result should therefore always be accompanied by the state assumptions under which it was computed.

Complete numerical walkthrough

For the commonly used ideal CCM model, omega_RHPZ≈R(1−D)²/L. At R=30 ohms, D=0.6 and L=150 microhenrys, omega≈32,000 rad/s or about 5093 Hz. This is a model-based estimate, not a recommended loop crossover frequency.

How to verify the arithmetic independently. First convert every milliohm, microsecond, microfarad, kilohertz and microhenry into consistent base units. Next substitute the stated values into the governing relation in one line before rounding. Keep a separate column for the measured or assumed quantity and the calculated quantity. After calculation, check the physical limiting case: at zero load there should be no load power, at zero switching interval there should not be energy transfer through an ideal PWM command, and the sign of a voltage cannot be changed by efficiency alone. These simple limiting checks often expose a wrong topology or an invalid formula more reliably than another decimal place.

For a useful second calculation, change one design variable while holding the others fixed. If a capacitor is changed, do not simultaneously alter switching frequency and load current, because the effect becomes difficult to attribute. Record which quantities are genuinely controlled and which are consequences of the change. Where the model is an approximation, report its domain of validity beside the number; a precise numerical answer derived outside the model's domain can be more misleading than a rough answer with honest uncertainty.

Why a technically plausible answer may be wrong

A controller attempting to exceed this dynamic limit can amplify inverse response and erode phase margin. The expression changes with nonidealities and the selected model.

In particular, distinguish a device rating, a nominal operating point, a predicted steady-state waveform and a measured transient peak. The four are not interchangeable. An ideal average voltage says little about a nanosecond-level overshoot; a calculated RMS current does not establish that a junction temperature is acceptable; a controller response without saturation says nothing about a hard overcurrent event. The reader should identify the first assumption likely to fail at an extreme input, a very light load, a startup event, or a component temperature different from the nominal value.

Engineering decision and defensible tradeoff

Plot RHP-zero frequency as D changes and design a conservative bandwidth below it. Show why low input voltage and high duty can make voltage-loop control more difficult.

State the requirement as a measurable inequality or a bounded interval instead of using only adjectives such as fast, small, efficient or robust. For example, a requirement for output voltage has to name the input-voltage range, load range, transient duration and maximum allowed deviation. A requirement for low loss has to identify the ambient-temperature condition and whether the gate driver or controller is included in the loss accounting. A comparison among device options is not reproducible unless all alternatives are evaluated under the same service conditions.

Also note what is sacrificed when the selected variable is improved. Raising frequency can decrease filter size but increase switching heat; lowering channel resistance may increase gate-charge cost; larger capacitance can improve low-frequency hold-up while increasing inrush; stronger voltage-loop action can improve a small load step but erode phase margin at a different point. Engineering is the management of coupled constraints rather than the maximization of one attractive metric.

Laboratory thinking and falsifiable observations

Investigative question: Why is a right-half-plane zero a problem even though it is not itself an unstable pole?

Develop a small simulated experiment with two deliberately different operating conditions and one held-constant reference. The output of the experiment should be a table with physical units, a plotted response and an explicit criterion that would disprove the initial interpretation. Include a counterexample that changes the topology or its conduction mode. If your model predicts an impossible output at a duty limit, do not hide that point; explain which neglected physical mechanism becomes dominant and replace the formula with an appropriate higher-fidelity model before continuing.

To compare simulation with future measurements, define where voltage is measured, how current direction is assigned, the required bandwidth, and the averaging interval. If the result is efficiency, synchronize input and output data. If the result is a switching peak, use appropriately rated equipment and identify probe-induced artifacts. If the result is a protection threshold, include comparator tolerance and intervention latency. This is a review workflow, not an instruction to energize unvalidated hardware.

Practice set — show a numerical answer and an engineering explanation

1. Reconstruct the example: Repeat the supplied computation in SI units, state every approximated parameter, and make a dimensional check. If any input is missing, state that a numerical result is indeterminate rather than inventing one.

2. Alternative operating point: Change the stated input or load by 20%. Explain whether the original formula remains applicable and calculate the new value if sufficient inputs are known.

3. Failure-mode challenge: Identify a capacitor, inductor, semiconductor, driver or sensing failure that could invalidate the optimistic ideal result. Explain which independent signal or check reveals the failure.

4. Specification writing: Rewrite the design decision as a testable requirement with a numerical tolerance, measurement point and operating range. Distinguish thermal limits from electrical limits.

5. Model comparison: Predict which conclusion would survive in an ideal switched simulation, an averaged simulation and a real physical implementation. Name at least one effect visible in one model and invisible in another.

6. Research extension: Find the parameter in the appropriate component datasheet that has the largest uncertainty at elevated temperature. Describe a conservative sensitivity study using that parameter instead of extrapolating one room-temperature curve.

Worked-answer guidance

For question 1, the numerical walkthrough gives the target relation and input values; correct work reports the substituted quantity with its units, not just the final number. In question 2, choose only a single independent change; be explicit about whether the converter remains in continuous conduction or the same thermal state. For question 3, a plausible defect must have an observable consequence and not merely a descriptive label. For question 4, a valid specification tells another engineer exactly what to measure and what constitutes passing. For question 5, an ideal averaged stage cannot resolve switching-edge electromagnetic interference, and a switched circuit model alone cannot establish manufactured-hardware safety. For question 6, a meaningful answer reports the temperature and bias conditions under which the parameter is specified.

Cross-link to the unit design study

This calculation contributes to Control-loop model integrity review. Put the result into the unit's requirements matrix and identify whether it changes a semiconductor voltage rating, magnetic current rating, controller operating range, expected efficiency or verification plan. A unit project is not complete when a formula produces one plausible number; it is complete when the predicted outcome, the assumptions and the unresolved risks are transparent to a second reader.

Competing design alternatives

Compare two configurations with a fixed source, load and service requirement. One should prioritize lower conduction loss or larger energy storage, and the other should prioritize faster switching, smaller size or more economical control. Build a scorecard with heat, component stress, signal noise, cost and verification effort. Give each criterion a unit or a precise definition rather than assigning arbitrary star ratings.

A defensible conclusion should change when the service envelope changes. For example, a low-resistance transistor can be attractive under heavy current while a lower-gate-charge device can be preferable at light load and high frequency. A large bulk capacitor can improve hold-up but worsen startup inrush. Demonstrate at least one such reversal without changing unrelated assumptions at the same time. The objective is to learn why there is rarely one universally best component.

Lesson 7.4 — Control bandwidth, sensing and digital delay

Unit: Small-signal dynamics and feedback. Focus: energy balance, switching-state modeling, numerical prediction, design limitations, verification. Prerequisites: basic circuit analysis, the definition of average and RMS values, and the relevant preceding lessons. Safety: This is mathematical and simulated content; it does not license direct connection to mains or high-energy battery systems.

What the model is trying to explain

A closed-loop power stage samples a filtered output and applies duty updates after computation and PWM timing delays. Delay contributes phase lag, while sensor noise grows at high bandwidth. A sound loop design trades fast disturbance rejection against switching ripple, noise and stability margin.

The starting point is an electrical system with a stated input, a desired output, and at least one physical storage or loss mechanism. Choose voltage and current arrows before writing any balance equation. This apparently clerical step determines whether a negative term represents energy returned to a source, dissipated heat, or a mismatch of sign conventions. For a switching system the state of the controlled semiconductor changes the governing differential equation; a mean-value formula follows only after the switching states and the relevant steady-state conditions have been established. A time-domain numerical result should therefore always be accompanied by the state assumptions under which it was computed.

Complete numerical walkthrough

At 100 kHz switching, a 10 microsecond delay contributes phase lag of 2πfτ radians. At a 2 kHz loop frequency this is about 7.2°, whereas at 10 kHz it is 36°. These are delay contributions only; they do not include plant phase.

How to verify the arithmetic independently. First convert every milliohm, microsecond, microfarad, kilohertz and microhenry into consistent base units. Next substitute the stated values into the governing relation in one line before rounding. Keep a separate column for the measured or assumed quantity and the calculated quantity. After calculation, check the physical limiting case: at zero load there should be no load power, at zero switching interval there should not be energy transfer through an ideal PWM command, and the sign of a voltage cannot be changed by efficiency alone. These simple limiting checks often expose a wrong topology or an invalid formula more reliably than another decimal place.

For a useful second calculation, change one design variable while holding the others fixed. If a capacitor is changed, do not simultaneously alter switching frequency and load current, because the effect becomes difficult to attribute. Record which quantities are genuinely controlled and which are consequences of the change. Where the model is an approximation, report its domain of validity beside the number; a precise numerical answer derived outside the model's domain can be more misleading than a rough answer with honest uncertainty.

Why a technically plausible answer may be wrong

A phase-margin estimate that neglects current-limit operation or saturating duty is a local linear claim. Passing a nominal load-step simulation cannot establish stability across startup and input extremes.

In particular, distinguish a device rating, a nominal operating point, a predicted steady-state waveform and a measured transient peak. The four are not interchangeable. An ideal average voltage says little about a nanosecond-level overshoot; a calculated RMS current does not establish that a junction temperature is acceptable; a controller response without saturation says nothing about a hard overcurrent event. The reader should identify the first assumption likely to fail at an extreme input, a very light load, a startup event, or a component temperature different from the nominal value.

Engineering decision and defensible tradeoff

Select sampling instant, anti-aliasing, PWM update schedule and a finite set of operating points. Document both phase-margin estimates and worst-case time-domain disturbances.

State the requirement as a measurable inequality or a bounded interval instead of using only adjectives such as fast, small, efficient or robust. For example, a requirement for output voltage has to name the input-voltage range, load range, transient duration and maximum allowed deviation. A requirement for low loss has to identify the ambient-temperature condition and whether the gate driver or controller is included in the loss accounting. A comparison among device options is not reproducible unless all alternatives are evaluated under the same service conditions.

Also note what is sacrificed when the selected variable is improved. Raising frequency can decrease filter size but increase switching heat; lowering channel resistance may increase gate-charge cost; larger capacitance can improve low-frequency hold-up while increasing inrush; stronger voltage-loop action can improve a small load step but erode phase margin at a different point. Engineering is the management of coupled constraints rather than the maximization of one attractive metric.

Laboratory thinking and falsifiable observations

Investigative question: How would you tell from test logs that a delayed sensor reading, not a weak capacitor, caused oscillatory recovery?

Develop a small simulated experiment with two deliberately different operating conditions and one held-constant reference. The output of the experiment should be a table with physical units, a plotted response and an explicit criterion that would disprove the initial interpretation. Include a counterexample that changes the topology or its conduction mode. If your model predicts an impossible output at a duty limit, do not hide that point; explain which neglected physical mechanism becomes dominant and replace the formula with an appropriate higher-fidelity model before continuing.

To compare simulation with future measurements, define where voltage is measured, how current direction is assigned, the required bandwidth, and the averaging interval. If the result is efficiency, synchronize input and output data. If the result is a switching peak, use appropriately rated equipment and identify probe-induced artifacts. If the result is a protection threshold, include comparator tolerance and intervention latency. This is a review workflow, not an instruction to energize unvalidated hardware.

Practice set — show a numerical answer and an engineering explanation

1. Reconstruct the example: Repeat the supplied computation in SI units, state every approximated parameter, and make a dimensional check. If any input is missing, state that a numerical result is indeterminate rather than inventing one.

2. Alternative operating point: Change the stated input or load by 20%. Explain whether the original formula remains applicable and calculate the new value if sufficient inputs are known.

3. Failure-mode challenge: Identify a capacitor, inductor, semiconductor, driver or sensing failure that could invalidate the optimistic ideal result. Explain which independent signal or check reveals the failure.

4. Specification writing: Rewrite the design decision as a testable requirement with a numerical tolerance, measurement point and operating range. Distinguish thermal limits from electrical limits.

5. Model comparison: Predict which conclusion would survive in an ideal switched simulation, an averaged simulation and a real physical implementation. Name at least one effect visible in one model and invisible in another.

6. Research extension: Find the parameter in the appropriate component datasheet that has the largest uncertainty at elevated temperature. Describe a conservative sensitivity study using that parameter instead of extrapolating one room-temperature curve.

Worked-answer guidance

For question 1, the numerical walkthrough gives the target relation and input values; correct work reports the substituted quantity with its units, not just the final number. In question 2, choose only a single independent change; be explicit about whether the converter remains in continuous conduction or the same thermal state. For question 3, a plausible defect must have an observable consequence and not merely a descriptive label. For question 4, a valid specification tells another engineer exactly what to measure and what constitutes passing. For question 5, an ideal averaged stage cannot resolve switching-edge electromagnetic interference, and a switched circuit model alone cannot establish manufactured-hardware safety. For question 6, a meaningful answer reports the temperature and bias conditions under which the parameter is specified.

Cross-link to the unit design study

This calculation contributes to Control-loop model integrity review. Put the result into the unit's requirements matrix and identify whether it changes a semiconductor voltage rating, magnetic current rating, controller operating range, expected efficiency or verification plan. A unit project is not complete when a formula produces one plausible number; it is complete when the predicted outcome, the assumptions and the unresolved risks are transparent to a second reader.

Review at a nonnominal condition

Assume the environment becomes hotter, the supply reaches its minimum allowed voltage and the output experiences a short overload. Which number in the worked example is likely to become obsolete first? Rewrite the corresponding balance or stress relation with a conservative placeholder for the uncertain quantity. If no trusted bound is available, leave the conclusion unresolved. An engineering report should explicitly state what has not been established.

Prepare a two-stage verification plan: first check the symbolic result in a calculator or independent script; next check the behavior in a simulation using nonideal elements or a supervised low-voltage teaching bench. Name a stopping condition for the second stage. Independent verification means an alternative physical or computational route to the result, not simply repeating the same source calculation in a different font.

Unit 7 design study — Control-loop model integrity review

Objective. Show that the reader can connect switching equations to a bounded engineering requirement, rather than only repeat a nominal calculation.

Scenario and data

Build the ideal averaged buck equations for Vin=24 V, R=6 Ω, L=100 µH and C=470 µF. Solve equilibrium with D=0.5, then estimate undamped resonant frequency. Introduce a first-order sensor lag and a one-sample PWM update delay. Compare what each element changes in a Bode interpretation.

Explain why a PI gain chosen only from a 100 kHz switching rate is unsupported. Define an antiwindup action when applied duty is limited to [0.05,0.95]. Create a test that would detect whether the controller continues integrating during a 0.5-second overload.

Engineering notebook procedure

1. Requirements table. Write columns for demanded function, allowed input range, load range, maximum stress, accuracy tolerance, missing information and evidence. Reject adjectives without an associated test. Where values have not been provided, label them 'to be determined' and explain the measurement or datasheet that will resolve them.

2. Energy and switching state diagram. Give the current path in each interval and show inductor voltage or capacitor current with consistent sign conventions. State which approximation lets the calculation use cycle averages. If conduction mode can change, identify the condition at which the derivation stops being applicable.

3. Hand calculation. Carry through at least one numerical case from this unit's lessons, report intermediate values with units, check a limiting case, and recompute it with a 20% parameter variation. Separate assumed input values from computed results and ensure no conclusion depends on a hidden choice of waveform convention.

4. Comparison with a different model. Run the companion offline simulation at three operating points; record how its assumptions differ from the hand calculation. The model must not be described as validated physical hardware. If the simulation refuses an impossible operating point, that refusal is a finding rather than a reason to bypass the guardrail.

5. Failure analysis. Identify at least two independent failures: one electrical stress or thermal failure and one sensing, timing, protection or measurement failure. For each propose an observable symptom and a method that could falsify the simplified design claim.

6. Review and decision. Assemble a one-page design summary with chosen assumptions, calculated quantities, sensitivity, unresolved safety risks, measurement uncertainty and a justified next test. Do not specify live-mains experiments or unsupervised high-energy hardware as part of this project.

Evaluation rubric (20 points)

• Four points: correct switching-state or control-volume representation with reference directions.
• Four points: independently checkable arithmetic, correct SI units and a valid domain of application.
• Four points: design tradeoff and a nonnominal sensitivity study.
• Four points: an explicit falsification or failure test and appropriate safety bounds.
• Four points: clear interpretation of simulation limitations and remaining uncertainty.

Instructor checkpoint

A submission that uses the right numerical formula with the wrong conduction regime cannot receive full credit. Similarly, a convincing waveform plot does not compensate for missing component limits. The strongest submissions explain what would change their conclusion. Ask learners to present a one-minute defense of the least reliable assumption and to identify the next measurement or calculation that is most likely to overturn the result.

Optional deeper investigation

Vary temperature, parasitic resistance or sensor delay separately. Compare the relative importance of each effect at light load versus rated load. Use an uncertainty table rather than drawing more decimal places. Explain if the most important uncertainty is irreducible without physical measurement, or whether a manufacturer specification supplies a conservative bound. Finally, distinguish the verified result—perhaps an algebraic equality or a checked simulator calculation—from an unverified implication about real hardware.

Safety boundary: EDUCATIONAL MODEL ONLY. Do not connect mains voltage or high-energy batteries to a classroom prototype. A schematic, component value, or simulation in this volume is not a certified electrical-safety design. Use isolated, current-limited training hardware only under competent supervision and follow relevant safety rules, ratings, and test procedures.