Unit 3: Resonant Converters: LLC Resonant Tanks, Zero-Voltage and Zero-Current Switching

Unit 3: Motor-drive converters and regenerative operation

Engineering situation

A motor drive coordinates electromagnetic torque, mechanical inertia, power-stage limits and a DC-link energy balance. Instantaneous electrical power can reverse during braking, but the DC source must have an actual path for the returned energy. A commanded negative torque is not evidence that a non-regenerative supply can absorb power.

Shared numerical scenario

We compare a simple scalar volts-per-hertz model with a field-oriented control block diagram. The examples assume idealized motor parameters for teaching; a real commissioning exercise requires motor identification, current sensors, thermal ratings, fault management and guarded rotating equipment.

Intended competence

Learners will calculate synchronous speed and slip, separate torque-producing and flux-producing current concepts, audit regenerative energy and determine why bus overvoltage requires a protective strategy independent of the speed controller.

Assumptions and model boundary

This unit uses specific equations at specific operating points; a model must not be extended merely because it produces a number outside the stated range. In particular, the switching network may be averaged over a PWM period for a slow state calculation, while magnetic reset, terminal-voltage stress and semiconductor switching energy demand individual switching intervals. Keep the mechanical, magnetic and electrical energy variables separate where appropriate. Every diagram should identify a reference node, current direction and whether the axes show peak, average, or RMS quantities. Where an external network or motor is discussed, the waveform is software-generated for learning and does not authorize connection to an energized system.

Unit reading route

Read Lessons 3.1 through 3.4 in sequence, then use the [unit simulation](../simulations/model-03.html). Complete the unit project before the assessment. Preserve the original scenario as a baseline so parameter changes are attributable; do not change source, loading and control law simultaneously. When citing a numerical result, write the values, units and limit cases, then label it as ideal analytical, simplified dynamic model, computed simulation or actual measurement. No measurement claims are made here.

Technical review checkpoint

The useful control output in a drive is mechanical torque or speed, but the actuator is an electrical converter. Energy travels through magnetic fields and rotor inertia. During braking the direction reverses, exposing any assumption that a DC source is a two-way energy sink.

Use separate vertical axes for q-axis current, speed, electromagnetic torque and DC-link voltage. State the assumptions that link them, including machine orientation, inertia, torque constant and converter loss. A single attractive speed trace does not validate the energy balance.

A good explanation specifies the assumptions under which a relation is true and names an experiment or simulation capable of detecting when it fails. A numerical waveform can agree with its own programmed equations and still be incapable of predicting a real apparatus. Use the following assessment to distinguish those two forms of evidence.

Lesson 3.1: Scalar volts-per-hertz control

Position: Unit 3, Motor-drive converters and regenerative operation. Reading level: undergraduate circuit analysis, switching converters and elementary calculus. Learning claim: distinguish the mathematical approximation from the physical device-level consequences. Status: unreviewed teaching chapter.

Physical picture and mechanism

An induction-machine scalar drive seeks to maintain approximately constant air-gap flux by raising stator voltage with commanded electrical frequency over part of the speed range. At low speed the stator resistance drop becomes significant, while at high speed the DC-link limit prevents indefinite voltage increase.

The practical boundary of the system matters. Draw where energy enters, where useful energy leaves, what storage changes within the process and where heat or field energy is likely to be lost. Define all voltage polarities before changing reference frames; a reported sign is physically meaningful only relative to a declared node or direction. The natural interval for averaging must also be stated: a switching-period equation, a line-frequency RMS result and a slowly varying thermal estimate describe different aspects of the same apparatus. One cannot simply average all three over an arbitrary interval and call the result a design guarantee.

Governing mathematical relation

For an induction machine with p poles, synchronous speed in rpm is ns=120f/p. The per-phase sinusoidal induced EMF approximately scales with frequency times flux, E≈k f Φ, within the chosen magnetic model.

Write the baseline relation symbolically first, then test it against a dimensional check and an elementary limiting case. For a variable expressed as a rate, verify what time base is implicit: per switching period, per electrical cycle, per unit mechanical rotation or per second. If the equation assumes continuous conduction, a linear magnetic inductance, an ideal waveform or a balanced three-phase input, say so before substituting numerical values. For a measured value, specify sensor position and measurement bandwidth; for a computed value, identify the exact source equation.

Worked numerical example with interpretation

For a four-pole machine commanded at f=40 Hz, synchronous speed is 120×40/4=1200 rpm. With measured shaft speed 1140 rpm, slip fraction is (1200−1140)/1200=0.05. The stated speed is mechanically plausible but does not establish motor torque or efficiency.

A credible numerical answer includes both a displayed number and its physical meaning. If a computed voltage is a peak, it cannot be directly used as an RMS rating. If a frequency is a natural resonance, it cannot be presented as a measured closed-loop bandwidth. If a result is an ideal output voltage, it cannot be equated with the real part's dielectric withstand requirement. Use at least three significant digits in the intermediate steps and round the final display to the precision actually supported by input parameters. A second independent check, such as energy conservation, a complementary voltage, or the expected response as one variable approaches zero, should be written beside the arithmetic.

Derivation workflow

Start from a physical definition and then introduce any simplification. Before calculating, sketch or describe the states that make the claimed equation meaningful. A derived relation is usually an operating-point statement, not a guarantee across startup, overload and faulted conditions. Distinguish independent inputs from outputs: a duty ratio may be commanded by the regulator, but a peak current is determined jointly by source voltage, load, storage and event timing.

Check dimensions at every substitution. In particular, microseconds and nanoseconds differ by a factor of one thousand; degrees and radians require explicit conversion before inserting a frequency; and a quantity in electrical radians per second is not a shaft speed in revolutions per minute. The direction of energy flow should be consistent with the current-arrow convention shown beside the circuit or graph.

Important failure modes and nonideal boundaries

The method is not direct torque control and can have weak low-speed dynamics. A measured shaft speed should not be assumed equal to the imposed stator frequency in rpm; pole count and slip both matter.

These limitations are not a minor footnote: they are the conditions under which this chapter's conclusion could reverse. Separate the design intent from the semiconductor's transient stress, and the requested controller action from the physical switching state. A controller display can be numerically smooth while its switching node is overshooting; a motor speed command can be well behaved while the DC bus is absorbing excessive regenerative energy. A waveform generated by a hand-written program is only as complete as the modeled parasitics and fault behavior.

Engineering judgment

Draft a V/f profile with a voltage ceiling and explain the reasons for low-frequency compensation and high-frequency field weakening. Compare this scalar strategy to a current-regulated drive for a fast torque transient.

An engineering comparison should use a common reference case and explicitly report one metric that improves and one that could worsen. For example, increasing switching frequency may reduce some storage ripple yet increase switching heat and radiated emissions; adding capacitance may smooth a voltage while raising surge energy and discharge requirements; increasing control bandwidth may improve tracking while reducing delay tolerance. The tradeoff depends on this lesson's topology and assumptions. Where the text omits a datasheet value, test configuration or certification criterion, the correct output is 'not established' rather than an invented margin.

Focused exercise set

Exercise A — reconstruct. Rewrite the governing relation above using named inputs and outputs. Draw or describe the source, load, storage or reference frame that makes it valid. Mark whether the calculated result is a switching-period average, fundamental value, transient peak, energy, current, power, frequency or loss.

Exercise B — parameter sensitivity. Reproduce the numerical case but change the most influential stated input by +10% while keeping all other inputs fixed. Show the substituted equation and the new number. Explain whether the response is expected to be linear, reciprocal, quadratic or nonlinear, and whether the modified point remains inside the same assumed operating regime.

Exercise C — challenge the assumption. Identify one concrete event, such as a load step, missing feedback signal, frequency detuning, angle-estimation error, current sign reversal or temperature change, that makes the basic calculation incomplete. State which extra measurement or model parameter would be necessary to proceed.

Exercise D — compare choices. Evaluate a second solution concept, not simply a different numerical parameter. Compare source-current stress, controllability, complexity and validation burden. Give a reason the choice could change at light load or at a different source voltage.

Exercise E — audit someone else's answer. A classmate reports a precise 'safe and efficient' result after using only the simplified equation in this lesson. Write a six-line technical review identifying one valid quantitative conclusion, one ignored uncertainty, one protection concern, one proposed simulation, one missing empirical test and the exact boundary of the claim.

Answer guidance and grading rubric

For Exercise A, the key is a consistent sign convention and correct categorization of the calculated quantity. In Exercise B the arithmetic must start from the same stated units, and a legitimate answer must note when a changed parameter would require a new operating regime. In Exercise C the missing variable must materially affect the behavior, rather than being a vague appeal to 'more data'. Exercise D earns full credit only when the alternate architecture changes an engineering tradeoff; merely changing a plotted color is not a design option. Exercise E must not describe software results as measurements or claim an external safety approval. Partial credit is appropriate for an otherwise correct derivation with an explicit, honest limitation; no credit for physically inconsistent units or an unstated phase-voltage reference.

Bridge to the unit model

Energy through multiple physical domains: The useful control output in a drive is mechanical torque or speed, but the actuator is an electrical converter. Energy travels through magnetic fields and rotor inertia. During braking the direction reverses, exposing any assumption that a DC source is a two-way energy sink.

Use separate vertical axes for q-axis current, speed, electromagnetic torque and DC-link voltage. State the assumptions that link them, including machine orientation, inertia, torque constant and converter loss. A single attractive speed trace does not validate the energy balance.

The companion simulation uses deterministic, intentionally simplified equations. Preserve input settings and screenshot or export the computed parameters together with any graph. A successful interactive plot demonstrates software operation and internal mathematical consistency for the declared formulas; it does not prove real component compatibility. Return to the central question of this chapter: what is calculated, what is assumed, and what is still unknown?

Lesson 3.2: Field-oriented control reference frames

Position: Unit 3, Motor-drive converters and regenerative operation. Reading level: undergraduate circuit analysis, switching converters and elementary calculus. Learning claim: distinguish the mathematical approximation from the physical device-level consequences. Status: unreviewed teaching chapter.

Physical picture and mechanism

Field-oriented control expresses measured or estimated three-phase currents in a rotating d–q frame. Under suitable machine orientation, one component primarily controls flux and the other primarily controls torque. The transformation does not manufacture a rotor angle; that information must come from a sensor or an observer with limits.

The practical boundary of the system matters. Draw where energy enters, where useful energy leaves, what storage changes within the process and where heat or field energy is likely to be lost. Define all voltage polarities before changing reference frames; a reported sign is physically meaningful only relative to a declared node or direction. The natural interval for averaging must also be stated: a switching-period equation, a line-frequency RMS result and a slowly varying thermal estimate describe different aspects of the same apparatus. One cannot simply average all three over an arbitrary interval and call the result a design guarantee.

Governing mathematical relation

A common balanced Clarke transform gives iα=ia and iβ=(ia+2ib)/√3 when ia+ib+ic=0. A Park transform then gives id=iα cosθ+iβ sinθ and iq=−iα sinθ+iβ cosθ, for the stated sign convention.

Write the baseline relation symbolically first, then test it against a dimensional check and an elementary limiting case. For a variable expressed as a rate, verify what time base is implicit: per switching period, per electrical cycle, per unit mechanical rotation or per second. If the equation assumes continuous conduction, a linear magnetic inductance, an ideal waveform or a balanced three-phase input, say so before substituting numerical values. For a measured value, specify sensor position and measurement bandwidth; for a computed value, identify the exact source equation.

Worked numerical example with interpretation

For ia=2 A, ib=−1 A, ic=−1 A, iα=2 A and iβ=0. At θ=90°, the convention here gives id=0 A and iq=−2 A. A different dq sign or electrical angle orientation changes signs, so the transform definition must be displayed beside the result.

A credible numerical answer includes both a displayed number and its physical meaning. If a computed voltage is a peak, it cannot be directly used as an RMS rating. If a frequency is a natural resonance, it cannot be presented as a measured closed-loop bandwidth. If a result is an ideal output voltage, it cannot be equated with the real part's dielectric withstand requirement. Use at least three significant digits in the intermediate steps and round the final display to the precision actually supported by input parameters. A second independent check, such as energy conservation, a complementary voltage, or the expected response as one variable approaches zero, should be written beside the arithmetic.

Model comparison workflow

A second model is useful only if it changes an identifiable assumption. Compare an ideal lossless expression with a bounded nonideal version, or a fixed load with a changed load; do not modify several unrelated quantities and attribute the entire difference to a single mechanism. Make a table showing which equations survive the changed operating regime and which must be derived again.

Include the neglected process in plain language even when it cannot be computed: leakage inductance, dead time, load-angle error, winding temperature or non-sinusoidal current can change a conclusion. Giving an uncalibrated guessed correction factor to hide that omission is less useful than stating that the quantity remains unknown.

Important failure modes and nonideal boundaries

Noise, offset, dead-time distortion and angle estimation error can cause unintended torque. The dq current controller must also respect actual inverter voltage limits and current protection.

These limitations are not a minor footnote: they are the conditions under which this chapter's conclusion could reverse. Separate the design intent from the semiconductor's transient stress, and the requested controller action from the physical switching state. A controller display can be numerically smooth while its switching node is overshooting; a motor speed command can be well behaved while the DC bus is absorbing excessive regenerative energy. A waveform generated by a hand-written program is only as complete as the modeled parasitics and fault behavior.

Engineering judgment

Draw the complete feedback loop from ADC sampling through coordinate transforms, current regulators, inverse transforms, PWM and the motor. Mark which parts need encoder calibration or parameter identification.

An engineering comparison should use a common reference case and explicitly report one metric that improves and one that could worsen. For example, increasing switching frequency may reduce some storage ripple yet increase switching heat and radiated emissions; adding capacitance may smooth a voltage while raising surge energy and discharge requirements; increasing control bandwidth may improve tracking while reducing delay tolerance. The tradeoff depends on this lesson's topology and assumptions. Where the text omits a datasheet value, test configuration or certification criterion, the correct output is 'not established' rather than an invented margin.

Focused exercise set

Exercise A — reconstruct. Rewrite the governing relation above using named inputs and outputs. Draw or describe the source, load, storage or reference frame that makes it valid. Mark whether the calculated result is a switching-period average, fundamental value, transient peak, energy, current, power, frequency or loss.

Exercise B — parameter sensitivity. Reproduce the numerical case but change the most influential stated input by +10% while keeping all other inputs fixed. Show the substituted equation and the new number. Explain whether the response is expected to be linear, reciprocal, quadratic or nonlinear, and whether the modified point remains inside the same assumed operating regime.

Exercise C — challenge the assumption. Identify one concrete event, such as a load step, missing feedback signal, frequency detuning, angle-estimation error, current sign reversal or temperature change, that makes the basic calculation incomplete. State which extra measurement or model parameter would be necessary to proceed.

Exercise D — compare choices. Evaluate a second solution concept, not simply a different numerical parameter. Compare source-current stress, controllability, complexity and validation burden. Give a reason the choice could change at light load or at a different source voltage.

Exercise E — audit someone else's answer. A classmate reports a precise 'safe and efficient' result after using only the simplified equation in this lesson. Write a six-line technical review identifying one valid quantitative conclusion, one ignored uncertainty, one protection concern, one proposed simulation, one missing empirical test and the exact boundary of the claim.

Answer guidance and grading rubric

For Exercise A, the key is a consistent sign convention and correct categorization of the calculated quantity. In Exercise B the arithmetic must start from the same stated units, and a legitimate answer must note when a changed parameter would require a new operating regime. In Exercise C the missing variable must materially affect the behavior, rather than being a vague appeal to 'more data'. Exercise D earns full credit only when the alternate architecture changes an engineering tradeoff; merely changing a plotted color is not a design option. Exercise E must not describe software results as measurements or claim an external safety approval. Partial credit is appropriate for an otherwise correct derivation with an explicit, honest limitation; no credit for physically inconsistent units or an unstated phase-voltage reference.

Bridge to the unit model

Energy through multiple physical domains: The useful control output in a drive is mechanical torque or speed, but the actuator is an electrical converter. Energy travels through magnetic fields and rotor inertia. During braking the direction reverses, exposing any assumption that a DC source is a two-way energy sink.

Use separate vertical axes for q-axis current, speed, electromagnetic torque and DC-link voltage. State the assumptions that link them, including machine orientation, inertia, torque constant and converter loss. A single attractive speed trace does not validate the energy balance.

The companion simulation uses deterministic, intentionally simplified equations. Preserve input settings and screenshot or export the computed parameters together with any graph. A successful interactive plot demonstrates software operation and internal mathematical consistency for the declared formulas; it does not prove real component compatibility. Return to the central question of this chapter: what is calculated, what is assumed, and what is still unknown?

Lesson 3.3: Regenerative braking and DC-link energy

Position: Unit 3, Motor-drive converters and regenerative operation. Reading level: undergraduate circuit analysis, switching converters and elementary calculus. Learning claim: distinguish the mathematical approximation from the physical device-level consequences. Status: unreviewed teaching chapter.

Physical picture and mechanism

A traction-style motor can act as an electrical generator during mechanical deceleration. Mechanical kinetic energy may return to the DC link, but without an absorber, battery charging path or qualified braking circuit the link capacitor rises in voltage. Energy conservation provides a useful first check on the scale of this event.

The practical boundary of the system matters. Draw where energy enters, where useful energy leaves, what storage changes within the process and where heat or field energy is likely to be lost. Define all voltage polarities before changing reference frames; a reported sign is physically meaningful only relative to a declared node or direction. The natural interval for averaging must also be stated: a switching-period equation, a line-frequency RMS result and a slowly varying thermal estimate describe different aspects of the same apparatus. One cannot simply average all three over an arbitrary interval and call the result a design guarantee.

Governing mathematical relation

Rotational kinetic energy is Ek=½Jω². For an ideal capacitor, the energy change from V1 to V2 is ΔEc=½C(V2²−V1²). If an idealized fraction η of mechanical energy is recovered, ηΔEk must be accounted for across the DC link and energy sink.

Write the baseline relation symbolically first, then test it against a dimensional check and an elementary limiting case. For a variable expressed as a rate, verify what time base is implicit: per switching period, per electrical cycle, per unit mechanical rotation or per second. If the equation assumes continuous conduction, a linear magnetic inductance, an ideal waveform or a balanced three-phase input, say so before substituting numerical values. For a measured value, specify sensor position and measurement bandwidth; for a computed value, identify the exact source equation.

Worked numerical example with interpretation

Let inertia J=0.05 kg·m², speed fall from 100 to 60 rad/s and ideal recovery fraction 0.8. The kinetic energy decrease is 0.5×0.05×(10000−3600)=160 J and returned energy 128 J. A 2 mF link starting at 300 V would require V2=√(300²+2×128/0.002)≈467 V if there were no sink—an unacceptable rise for many systems.

A credible numerical answer includes both a displayed number and its physical meaning. If a computed voltage is a peak, it cannot be directly used as an RMS rating. If a frequency is a natural resonance, it cannot be presented as a measured closed-loop bandwidth. If a result is an ideal output voltage, it cannot be equated with the real part's dielectric withstand requirement. Use at least three significant digits in the intermediate steps and round the final display to the precision actually supported by input parameters. A second independent check, such as energy conservation, a complementary voltage, or the expected response as one variable approaches zero, should be written beside the arithmetic.

Measurement and inference workflow

Define what an instrument would actually observe, even though this exercise is software only. A spectrum analyzer reports a frequency-domain estimate, a current sensor reports a bandwidth-limited sample, and a thermal measurement reports temperature at a location and time. None directly measures every term in an analytical state equation. Design an independent cross-check using conservation of energy or symmetry before trusting a displayed result.

Record the intended sample rate, window, triggering event and reference node. For transient switching analyses distinguish a 10 ns peak from a 1 ms average and identify whether the probe itself could perturb the observed circuit. If a claim needs actual bench data, mark it explicitly as not yet established.

Important failure modes and nonideal boundaries

This capacitor calculation is not a proposed hardware braking strategy. It omits battery behavior, power conversion efficiency variation, link protection and switching dynamics. Mechanical friction may also absorb some energy.

These limitations are not a minor footnote: they are the conditions under which this chapter's conclusion could reverse. Separate the design intent from the semiconductor's transient stress, and the requested controller action from the physical switching state. A controller display can be numerically smooth while its switching node is overshooting; a motor speed command can be well behaved while the DC bus is absorbing excessive regenerative energy. A waveform generated by a hand-written program is only as complete as the modeled parasitics and fault behavior.

Engineering judgment

Specify the desired destination of braking energy. Then describe a high-voltage cutoff independent of the speed controller, and enumerate evidence required before claiming safe regenerative operation.

An engineering comparison should use a common reference case and explicitly report one metric that improves and one that could worsen. For example, increasing switching frequency may reduce some storage ripple yet increase switching heat and radiated emissions; adding capacitance may smooth a voltage while raising surge energy and discharge requirements; increasing control bandwidth may improve tracking while reducing delay tolerance. The tradeoff depends on this lesson's topology and assumptions. Where the text omits a datasheet value, test configuration or certification criterion, the correct output is 'not established' rather than an invented margin.

Focused exercise set

Exercise A — reconstruct. Rewrite the governing relation above using named inputs and outputs. Draw or describe the source, load, storage or reference frame that makes it valid. Mark whether the calculated result is a switching-period average, fundamental value, transient peak, energy, current, power, frequency or loss.

Exercise B — parameter sensitivity. Reproduce the numerical case but change the most influential stated input by +10% while keeping all other inputs fixed. Show the substituted equation and the new number. Explain whether the response is expected to be linear, reciprocal, quadratic or nonlinear, and whether the modified point remains inside the same assumed operating regime.

Exercise C — challenge the assumption. Identify one concrete event, such as a load step, missing feedback signal, frequency detuning, angle-estimation error, current sign reversal or temperature change, that makes the basic calculation incomplete. State which extra measurement or model parameter would be necessary to proceed.

Exercise D — compare choices. Evaluate a second solution concept, not simply a different numerical parameter. Compare source-current stress, controllability, complexity and validation burden. Give a reason the choice could change at light load or at a different source voltage.

Exercise E — audit someone else's answer. A classmate reports a precise 'safe and efficient' result after using only the simplified equation in this lesson. Write a six-line technical review identifying one valid quantitative conclusion, one ignored uncertainty, one protection concern, one proposed simulation, one missing empirical test and the exact boundary of the claim.

Answer guidance and grading rubric

For Exercise A, the key is a consistent sign convention and correct categorization of the calculated quantity. In Exercise B the arithmetic must start from the same stated units, and a legitimate answer must note when a changed parameter would require a new operating regime. In Exercise C the missing variable must materially affect the behavior, rather than being a vague appeal to 'more data'. Exercise D earns full credit only when the alternate architecture changes an engineering tradeoff; merely changing a plotted color is not a design option. Exercise E must not describe software results as measurements or claim an external safety approval. Partial credit is appropriate for an otherwise correct derivation with an explicit, honest limitation; no credit for physically inconsistent units or an unstated phase-voltage reference.

Bridge to the unit model

Energy through multiple physical domains: The useful control output in a drive is mechanical torque or speed, but the actuator is an electrical converter. Energy travels through magnetic fields and rotor inertia. During braking the direction reverses, exposing any assumption that a DC source is a two-way energy sink.

Use separate vertical axes for q-axis current, speed, electromagnetic torque and DC-link voltage. State the assumptions that link them, including machine orientation, inertia, torque constant and converter loss. A single attractive speed trace does not validate the energy balance.

The companion simulation uses deterministic, intentionally simplified equations. Preserve input settings and screenshot or export the computed parameters together with any graph. A successful interactive plot demonstrates software operation and internal mathematical consistency for the declared formulas; it does not prove real component compatibility. Return to the central question of this chapter: what is calculated, what is assumed, and what is still unknown?

Lesson 3.4: Torque transients and protection coordination

Position: Unit 3, Motor-drive converters and regenerative operation. Reading level: undergraduate circuit analysis, switching converters and elementary calculus. Learning claim: distinguish the mathematical approximation from the physical device-level consequences. Status: unreviewed teaching chapter.

Physical picture and mechanism

Fast torque requests interact with current-loop bandwidth, semiconductor current ratings, magnetic saturation, mechanical resonances and temperature. A current command saturating at its software limit does not replace independent hardware overcurrent protection or rotor overspeed detection.

The practical boundary of the system matters. Draw where energy enters, where useful energy leaves, what storage changes within the process and where heat or field energy is likely to be lost. Define all voltage polarities before changing reference frames; a reported sign is physically meaningful only relative to a declared node or direction. The natural interval for averaging must also be stated: a switching-period equation, a line-frequency RMS result and a slowly varying thermal estimate describe different aspects of the same apparatus. One cannot simply average all three over an arbitrary interval and call the result a design guarantee.

Governing mathematical relation

For a simplified motor with torque constant kt in N·m/A and q-axis current iq, T≈kt iq under fixed flux and a defined q-axis convention. Mechanical angular acceleration follows J dω/dt=Tmotor−Tload−bω for inertia J and viscous coefficient b.

Write the baseline relation symbolically first, then test it against a dimensional check and an elementary limiting case. For a variable expressed as a rate, verify what time base is implicit: per switching period, per electrical cycle, per unit mechanical rotation or per second. If the equation assumes continuous conduction, a linear magnetic inductance, an ideal waveform or a balanced three-phase input, say so before substituting numerical values. For a measured value, specify sensor position and measurement bandwidth; for a computed value, identify the exact source equation.

Worked numerical example with interpretation

If kt=0.6 N·m/A, iq=5 A, J=0.04 kg·m² and negligible load/friction, torque is 3 N·m and initial acceleration 75 rad/s². Over 0.2 seconds the linear approximation predicts a speed change of about 15 rad/s, but voltage and current constraints may invalidate constant torque.

A credible numerical answer includes both a displayed number and its physical meaning. If a computed voltage is a peak, it cannot be directly used as an RMS rating. If a frequency is a natural resonance, it cannot be presented as a measured closed-loop bandwidth. If a result is an ideal output voltage, it cannot be equated with the real part's dielectric withstand requirement. Use at least three significant digits in the intermediate steps and round the final display to the precision actually supported by input parameters. A second independent check, such as energy conservation, a complementary voltage, or the expected response as one variable approaches zero, should be written beside the arithmetic.

Design-release workflow

A mathematical design is complete only relative to its declared requirements. Produce a decision matrix connecting efficiency, voltage stress, current stress, temperature, dynamics, isolation where applicable, EMC and fault response. Each column has a different pass criterion; a high numerical efficiency cannot compensate for a failed voltage rating or missing fault isolation.

Audit the design under the worst credible input, the largest and smallest loads, startup, shutdown and a sensor fault. State which cases have been derived, which have only been simulated, and which demand independent qualified testing. A student solution can be rigorous while acknowledging that specific manufacturers' datasheets and safety standards have not been checked.

Important failure modes and nonideal boundaries

Do not use a first-order mechanical calculation to claim a motor can supply constant commanded current at any speed. Back EMF consumes voltage headroom, sensors saturate, and torque estimates depend on temperature and machine parameters.

These limitations are not a minor footnote: they are the conditions under which this chapter's conclusion could reverse. Separate the design intent from the semiconductor's transient stress, and the requested controller action from the physical switching state. A controller display can be numerically smooth while its switching node is overshooting; a motor speed command can be well behaved while the DC bus is absorbing excessive regenerative energy. A waveform generated by a hand-written program is only as complete as the modeled parasitics and fault behavior.

Engineering judgment

Build a mode/state table for normal torque, current limiting, emergency disable, speed-sensor loss and regenerative overvoltage. Give each fault a physical output rather than relying on a dashboard alarm.

An engineering comparison should use a common reference case and explicitly report one metric that improves and one that could worsen. For example, increasing switching frequency may reduce some storage ripple yet increase switching heat and radiated emissions; adding capacitance may smooth a voltage while raising surge energy and discharge requirements; increasing control bandwidth may improve tracking while reducing delay tolerance. The tradeoff depends on this lesson's topology and assumptions. Where the text omits a datasheet value, test configuration or certification criterion, the correct output is 'not established' rather than an invented margin.

Focused exercise set

Exercise A — reconstruct. Rewrite the governing relation above using named inputs and outputs. Draw or describe the source, load, storage or reference frame that makes it valid. Mark whether the calculated result is a switching-period average, fundamental value, transient peak, energy, current, power, frequency or loss.

Exercise B — parameter sensitivity. Reproduce the numerical case but change the most influential stated input by +10% while keeping all other inputs fixed. Show the substituted equation and the new number. Explain whether the response is expected to be linear, reciprocal, quadratic or nonlinear, and whether the modified point remains inside the same assumed operating regime.

Exercise C — challenge the assumption. Identify one concrete event, such as a load step, missing feedback signal, frequency detuning, angle-estimation error, current sign reversal or temperature change, that makes the basic calculation incomplete. State which extra measurement or model parameter would be necessary to proceed.

Exercise D — compare choices. Evaluate a second solution concept, not simply a different numerical parameter. Compare source-current stress, controllability, complexity and validation burden. Give a reason the choice could change at light load or at a different source voltage.

Exercise E — audit someone else's answer. A classmate reports a precise 'safe and efficient' result after using only the simplified equation in this lesson. Write a six-line technical review identifying one valid quantitative conclusion, one ignored uncertainty, one protection concern, one proposed simulation, one missing empirical test and the exact boundary of the claim.

Answer guidance and grading rubric

For Exercise A, the key is a consistent sign convention and correct categorization of the calculated quantity. In Exercise B the arithmetic must start from the same stated units, and a legitimate answer must note when a changed parameter would require a new operating regime. In Exercise C the missing variable must materially affect the behavior, rather than being a vague appeal to 'more data'. Exercise D earns full credit only when the alternate architecture changes an engineering tradeoff; merely changing a plotted color is not a design option. Exercise E must not describe software results as measurements or claim an external safety approval. Partial credit is appropriate for an otherwise correct derivation with an explicit, honest limitation; no credit for physically inconsistent units or an unstated phase-voltage reference.

Bridge to the unit model

Energy through multiple physical domains: The useful control output in a drive is mechanical torque or speed, but the actuator is an electrical converter. Energy travels through magnetic fields and rotor inertia. During braking the direction reverses, exposing any assumption that a DC source is a two-way energy sink.

Use separate vertical axes for q-axis current, speed, electromagnetic torque and DC-link voltage. State the assumptions that link them, including machine orientation, inertia, torque constant and converter loss. A single attractive speed trace does not validate the energy balance.

The companion simulation uses deterministic, intentionally simplified equations. Preserve input settings and screenshot or export the computed parameters together with any graph. A successful interactive plot demonstrates software operation and internal mathematical consistency for the declared formulas; it does not prove real component compatibility. Return to the central question of this chapter: what is calculated, what is assumed, and what is still unknown?

Unit 3 project: Drive speed and braking energy journal

Scope: software-only engineering study and analytical notebook. No mains, traction battery, energized utility interface, or uncontrolled rotating machine is authorized. Deliverables: calculation sheet, documented assumptions, parameter sweep, self-check plot, uncertainty register and short technical decision record.

Project brief

Analyze a four-pole induction drive with commanded electrical frequency 40 Hz and rotor mechanical speed 1140 rpm. Estimate synchronous speed and slip. Then model the energy change of a separate motor inertia J=0.05 kg m² decelerating from 100 to 60 rad/s.

Begin with a clear control volume and state where power enters and exits. Draw the useful storage variables alongside the PWM switching states. Specify both time scales that matter: a switching or sampled control step, and the slower physical response. Fix a baseline and use identical physical units in every table. Give each graph explicit axis units and distinguish the dependent variable from the manipulated independent parameter.

Experiment 1 — reproduce the baseline

Calculate all baseline quantities by hand or a reproducible script using the equations in Lessons 3.1–3.4. Retain three significant figures before rounding. Use a dimensional cross-check and, where possible, an energy-conservation check. If the model has only a steady-state answer, state why it cannot predict transient overshoot or a semiconductor edge event. Record a short comment explaining whether the computed output is peak, RMS, mean or instantaneous.

Experiment 2 — a controlled perturbation

Repeat energy-accounting predictions for link capacitances 1, 2 and 4 mF with initial link voltage 300 V and recovery energy 128 J. Graph the predicted ideal final capacitor voltage versus capacitance, and discuss realistic sink options conceptually.

An informative parameter sweep changes one physical variable at a time and makes the boundary of the model visible. Where the response is proportional to a square, plot both the direct input and its square so a linear relationship in the transformed variable is evident. Where frequency is varied, avoid mixing hertz and radians per second. Add a small error budget that separates rounding, parameter tolerance and explicitly unmodeled physics. Never present the neatness of a plotted line as independent validation of a physical circuit.

Deliverable and evidence standard

Include a power-flow diagram during motoring and braking, a minimum current/frequency/speed sensor set, a state transition table for an overvoltage event, and a clear note that brake-energy calculations are neither battery charge acceptance data nor hardware design.

Include a table with columns: claim; governing equation; assumed parameter; predicted number; sensitivity to a changed parameter; required physical measurement; status. At least three rows must be marked 'analytical estimate', one must be marked 'simulation only', and two must be marked 'unverified without real data'. If a control function is said to protect a system, describe the software state that changes; do not imply that a high-energy circuit is safely disconnected by an untested algorithm.

Critical decision memo

Write 300–500 words comparing the baseline design against one alternative, showing what improves, what becomes more difficult and what would stop a release. Quantify at least one tradeoff. A technically credible answer may decline to rank options when necessary thermal or component properties were not specified. In that case identify the least-cost experiment or authoritative datasheet information that would resolve the ambiguity. Conclude with an operating-region sketch rather than an unsupported single optimum.

Instructor solution and boundary conditions

The indicated speed case gives ns=1200 rpm and s=0.05. For the energy case, the increase in squared bus voltage equals 2E/C; doubling capacitor size reduces this squared-voltage increment but does not magically absorb the recovered energy indefinitely.

An acceptable submission reproduces the numerical baseline independently; makes the units and sign conventions unambiguous; describes what extra physical or sampled-time behavior could invalidate the results; and includes an explicit exclusion of hardware safety certification. An incorrect submission treats a nominal fundamental voltage as total broadband RMS, mistakes a mathematical natural frequency for an empirical stability limit, or declares a component safe without its applicable manufacturer ratings. Project success demonstrates reasoning and traceable calculation, not a built converter.