Exercise set

Inverter Drive, Filter, Protection, and Thermal Reliability Exercises

Worked inverter-drive exercises for output current, PWM error, LCL filters, braking, fast protection, EMI leakage, thermal limits and lifetime.

Branch: Electrical Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises practise inverter and converter-system release as a waveform, filter, protection, thermal and lifetime problem. They cover output current, modulation headroom, dead-time voltage error, LCL filter resonance, braking energy, fast trip timing, EMI leakage, semiconductor loss, switching-frequency trade-offs, cooling loss, power cycling, efficiency acceptance, gate-driver response, regenerative overvoltage, harmonic current and release gates.

The focus is narrower than DC-link component sizing. An inverter-drive page should answer whether the converter can switch, protect, cool, filter and survive the intended operating envelope.

How to Use These Exercises

For each calculation, define:

converter topology, voltage, modulation method, switching frequency and load type;
current, voltage, filter, thermal or protection limit being tested;
measured or calculated margin, including uncertainty where relevant;
firmware, gate-driver, interlock and protection timing boundary;
commissioning evidence required before release.

The common mistake is proving rated power while leaving protection timing, dead time, filter resonance, EMI leakage, cooling loss or thermal cycling outside the release package.

Release Evidence Notes

Inverter evidence should include waveforms. RMS current, phase voltage, DC-link voltage, switching frequency, dead time, filter current and measured temperatures should be tied to the same operating point.

Protection evidence should separate hardware and firmware layers. Hardware trips limit destructive energy; firmware limits manage overloads and derating. One should not mask the other.

Thermal evidence should cover degraded cooling and cycling. A converter that passes steady-state loss at rated load may still fail under fan loss, coolant loss or repeated power cycles.

EMC and leakage evidence should be measured in the installed configuration. Cable length, shield termination, filter topology and grounding path can change common-mode behaviour.

Engineering Boundary Notes

These exercises are simplified engineering screens. Real inverter and converter-system release requires device datasheets, gate-driver tests, EMC testing, thermal instrumentation, firmware review, functional safety review, grid-code or motor-drive requirements, measurement uncertainty and qualified approval.

A system-level inverter pass does not remove the need for DC-link precharge, discharge, capacitor ripple and stored-energy checks.

Scenario Map

Scenario	Exercises	Primary calculation	Engineering decision
Waveform and filter behaviour	1-5, 15	output current, modulation, dead time, LCL resonance and harmonic current	Decide whether the inverter waveform and filter are acceptable.
Protection and braking	6-9, 14	braking power, fast trip timing, gate-driver delay and regenerative overvoltage	Decide whether abnormal energy is controlled.
Thermal and reliability release	10-13, 16-18	leakage, junction temperature, switching trade-off, cooling loss, lifetime damage and release scoring	Decide whether the converter can operate reliably.

Exercise 1: Inverter Output Current for a Motor Drive

A three-phase inverter delivers:

P=75\ \text{kW}

to a motor at:

V_{LL}=400\ \text{V},\qquad \text{pf}=0.88,\qquad \eta=0.94

Estimate line current.

Solution

Electrical input to motor terminals:

P_e=\dfrac{75{,}000}{0.94}=79{,}787\ \text{W}

Line current:

I=\dfrac{P_e}{\sqrt{3}V_{LL}\text{pf}}

I=\dfrac{79{,}787}{\sqrt{3}(400)(0.88)}=131\ \text{A}

Engineering Comment

Drive current rating should include overload duty, ambient temperature, carrier frequency and enclosure derating, not only rated mechanical power.

Plausibility Check

A $75$ kW low-voltage drive typically draws current in the hundred-ampere range.

Exercise 2: Modulation Index Headroom

A two-level inverter has DC-link voltage:

V_{DC}=700\ \text{V}

The requested fundamental line-line RMS voltage is:

V_{LL}=400\ \text{V}

Use the simplified linear PWM relation:

V_{LL}\approx 0.612mV_{DC}

Calculate modulation index.

Solution

Modulation index:

m=\dfrac{400}{0.612(700)}=0.934

Headroom to $m=1$ :

H=1-0.934=0.066

Engineering Comment

The inverter is near the linear modulation limit. Voltage drop, DC-link sag and dead-time error can remove the remaining headroom.

Plausibility Check

A $700$ V DC bus is a common basis for $400$ V output, but the margin is not large.

Exercise 3: PWM Dead-Time Voltage Error

An inverter leg has dead time:

t_d=2.0\ \mu\text{s}

Switching frequency is:

f_s=10\ \text{kHz}

DC-link voltage is:

V_{DC}=650\ \text{V}

Use the simple per-leg average voltage error:

\Delta V=t_df_sV_{DC}

Solution

Voltage error:

\Delta V=(2.0\times10^{-6})(10{,}000)(650)=13.0\ \text{V}

Engineering Comment

Dead-time error distorts low-speed motor voltage and current. Compensation should be validated with current polarity, device delays and minimum pulse constraints.

Plausibility Check

The dead-time fraction is $2\%$ of the switching period, and $2\%$ of $650$ V is $13$ V.

Exercise 4: LCL Filter Resonant Frequency

An inverter output filter has:

L_1=1.2\ \text{mH},\qquad L_2=0.8\ \text{mH},\qquad C=18\ \mu\text{F}

Use:

f_r=\dfrac{1}{2\pi}\sqrt{\dfrac{L_1+L_2}{L_1L_2C}}

Solution

Substitute SI units:

f_r=\dfrac{1}{2\pi}\sqrt{\dfrac{0.0012+0.0008}{(0.0012)(0.0008)(18\times10^{-6})}}

f_r=1711\ \text{Hz}

Engineering Comment

The resonance should be separated from fundamental frequency and switching ripple, and damping should be validated in the installed control loop.

Plausibility Check

Millihenry inductors and microfarad capacitors often resonate in the kilohertz range.

Exercise 5: Filter Resonance Separation from Switching Frequency

The filter resonant frequency is:

f_r=1711\ \text{Hz}

The switching frequency is:

f_s=10\ \text{kHz}

The design rule requires:

f_r<0.25f_s

Check separation.

Solution

Limit:

0.25f_s=2500\ \text{Hz}

Margin:

M=2500-1711=789\ \text{Hz}

The resonance passes this simplified separation rule.

Engineering Comment

Frequency separation is not sufficient by itself. Damping, sensor delay, digital control and grid impedance can still create oscillation.

Plausibility Check

$1711$ Hz is clearly below one quarter of $10$ kHz.

Exercise 6: Braking Energy and Chopper Resistor Power

A drive decelerates a load and removes kinetic energy:

E=420\ \text{kJ}

over:

t=18\ \text{s}

The braking chopper sends $82\%$ of that energy to the resistor. Calculate average resistor power during braking.

Solution

Resistor energy:

E_R=0.82(420)=344.4\ \text{kJ}

Average power:

P_R=\dfrac{344.4\ \text{kJ}}{18\ \text{s}}=19.1\ \text{kW}

Engineering Comment

Braking resistor release needs pulse energy, average power, enclosure temperature, thermal switch behaviour and fault response if the chopper fails open.

Plausibility Check

Hundreds of kilojoules over tens of seconds should produce tens of kilowatts.

Exercise 7: Fast Protection Timing

A converter hardware trip path has:

Segment	Time
desaturation detection	$3\ \mu\text{s}$
gate-driver propagation	$2\ \mu\text{s}$
turn-off transition	$5\ \mu\text{s}$
fault latch confirmation	$4\ \mu\text{s}$

The hardware trip limit is:

20\ \mu\text{s}

Check timing.

Solution

Total time:

t=3+2+5+4=14\ \mu\text{s}

Margin:

M=20-14=6\ \mu\text{s}

The hardware trip path passes.

Engineering Comment

Hardware trip timing should be measured with the real gate driver, device, DC-link voltage, fault current and layout parasitics.

Plausibility Check

Microsecond-scale segments add to a value below the $20\ \mu$ s limit.

Exercise 8: Firmware Current-Limit Latency

Firmware samples current every:

100\ \mu\text{s}

The filter and control computation add:

180\ \mu\text{s}

PWM update waits at most:

100\ \mu\text{s}

The overload response requirement is:

0.50\ \text{ms}

Check latency.

Solution

Worst-case latency:

t=100+180+100=380\ \mu\text{s}=0.380\ \text{ms}

Margin:

M=0.50-0.380=0.120\ \text{ms}

The firmware current-limit response passes the overload requirement.

Engineering Comment

Firmware current limiting is not a substitute for fast hardware protection. It manages overloads, not destructive short-circuit energy.

Plausibility Check

Several hundred microseconds is below half a millisecond, so the pass is plausible.

Exercise 9: EMI Filter Leakage Current

A line filter has phase-to-ground capacitance:

C_Y=4.7\ \text{nF}

per phase on a $230$ V, $50$ Hz system. Estimate total three-phase leakage current:

I=3(2\pi f C_Y V)

Solution

Per-phase leakage:

I_\phi=2\pi(50)(4.7\times10^{-9})(230)=0.340\ \text{mA}

Total leakage:

I=3(0.340)=1.02\ \text{mA}

Engineering Comment

Leakage current affects residual-current devices, touch current and EMC filter selection. Installed cable and parasitic capacitance can add to this value.

Plausibility Check

Nanofarad capacitors at line frequency produce milliampere-scale leakage.

Exercise 10: Semiconductor Loss and Junction Temperature

A power module dissipates:

P=420\ \text{W}

Thermal resistance from junction to coolant is:

R_{\theta}=0.075\ ^\circ\text{C/W}

Coolant temperature is:

T_c=54^\circ\text{C}

Calculate junction temperature.

Solution

Temperature rise:

\Delta T=PR_{\theta}=420(0.075)=31.5^\circ\text{C}

Junction temperature:

T_j=54+31.5=85.5^\circ\text{C}

Engineering Comment

Thermal release should include sensor location, transient load, thermal interface material, coolant flow, fouling and uncertainty in the loss model.

Plausibility Check

Hundreds of watts through a small thermal resistance gives a few tens of degrees Celsius rise.

Exercise 11: Switching Frequency Trade-Off

At:

f_s=8\ \text{kHz}

switching loss is:

P_{sw}=160\ \text{W}

Assume switching loss scales linearly with frequency. Estimate switching loss at:

12\ \text{kHz}

Solution

Switching loss:

P_{sw,2}=160\dfrac{12}{8}=240\ \text{W}

Increase:

\Delta P=240-160=80\ \text{W}

Engineering Comment

Higher switching frequency can improve current ripple and acoustic behaviour but increases loss, junction temperature and sometimes EMI stress.

Plausibility Check

Frequency increases by $50\%$ , so linearly scaled loss also increases by $50\%$ .

Exercise 12: Cooling-Loss Thermal Hold Time

During coolant loss, thermal capacitance of the module and cold plate is:

C_{th}=11{,}000\ \text{J}/^\circ\text{C}

Loss remains:

P=520\ \text{W}

Available temperature rise before derating is:

18^\circ\text{C}

Estimate hold time.

Solution

Thermal energy capacity:

E=C_{th}\Delta T=11{,}000(18)=198{,}000\ \text{J}

Hold time:

t=\dfrac{198{,}000}{520}=381\ \text{s}=6.35\ \text{min}

Engineering Comment

Thermal hold time should trigger derating or shutdown before junction temperature exceeds limit. Sensor delay and hot-spot gradients reduce usable margin.

Plausibility Check

Hundreds of watts into hundreds of kilojoules gives several minutes of hold time.

Exercise 13: Power-Cycling Lifetime Damage

A module is rated for:

N_f=80{,}000

cycles at a given temperature swing. A duty profile uses:

n=1200

such cycles per month. Use Miner damage:

D=\dfrac{n}{N_f}

Calculate monthly damage and months to $D=1$ .

Solution

Monthly damage:

D_m=\dfrac{1200}{80{,}000}=0.015

Months to unity damage:

T=\dfrac{1}{0.015}=66.7\ \text{months}

Engineering Comment

Power-cycling models depend strongly on temperature swing, mean temperature, dwell time and module technology. The duty profile must represent real operation.

Plausibility Check

At $1.5\%$ damage per month, unity damage should occur after about $67$ months.

Exercise 14: Regenerative DC-Link Overvoltage

A drive regenerates:

P=22\ \text{kW}

for:

t=0.40\ \text{s}

before the braking chopper engages. DC-link capacitance is:

C=6.0\ \text{mF}

Initial voltage is $680$ V. Estimate final voltage if all energy charges the capacitor.

Solution

Regenerated energy:

E=22{,}000(0.40)=8800\ \text{J}

Use:

V_f=\sqrt{V_i^2+\dfrac{2E}{C}}

V_f=\sqrt{680^2+\dfrac{2(8800)}{0.0060}}=1844\ \text{V}

Engineering Comment

This impossible voltage shows that the chopper or active front end must act much earlier. The calculation is useful because it exposes how little delay a DC link can tolerate during regeneration.

Plausibility Check

The regenerated energy is far larger than the capacitor energy margin, so an unrealistic overvoltage result is expected.

Exercise 15: Harmonic Current Filter Gate

A converter output has fundamental current:

I_1=120\ \text{A RMS}

The RMS harmonic current above the fundamental is:

I_h=8.4\ \text{A}

The project gate requires harmonic current below $8\%$ of the fundamental. Check the gate.

Solution

Harmonic ratio:

H=\dfrac{8.4}{120}=0.070=7.0\%

The filter gate passes.

Engineering Comment

Harmonic current should be measured over the operating envelope, including low load, high load, weak grid and switching-frequency changes.

Plausibility Check

Eight amps relative to $120$ A is a small single-digit percentage.

Exercise 16: Efficiency Acceptance with Uncertainty

Measured efficiency is:

\eta_m=97.1\%

Expanded measurement uncertainty is:

U=0.35\ \text{percentage points}

The acceptance requirement is:

\eta\ge 96.8\%

Use a guarded lower value:

\eta_g=\eta_m-U

Solution

Guarded efficiency:

\eta_g=97.1\%-0.35\%=96.75\%

This is below $96.8\%$ , so the guarded acceptance fails narrowly.

Engineering Comment

Efficiency release should include measurement uncertainty, sensor calibration, thermal state and load point. A nominal pass can become a guarded fail.

Plausibility Check

The measured value is only $0.3$ percentage points above the requirement, and uncertainty is larger than that margin.

Exercise 17: Fan-Loss Derating Gate

At rated load, the converter dissipates:

P=680\ \text{W}

With fans healthy, thermal resistance to ambient is:

0.055^\circ\text{C/W}

With one fan failed, it rises to:

0.090^\circ\text{C/W}

Ambient is $40^\circ$ C and maximum allowed case temperature is $95^\circ$ C. Check fan-failed operation.

Solution

Case temperature with one fan failed:

T_c=40+680(0.090)=101.2^\circ\text{C}

Margin:

M=95-101.2=-6.2^\circ\text{C}

Rated-load operation fails with one fan failed.

Engineering Comment

The release option is derating, alarmed shutdown, fan redundancy or a lower ambient envelope. Fan-loss cases should not be hidden behind normal cooling data.

Plausibility Check

Increasing thermal resistance by more than half should create a large temperature rise at hundreds of watts.

Exercise 18: Inverter Release Gate

A converter release review assigns five gates:

Gate	Weight	Result
waveform and modulation headroom	$0.20$	$0.94$
filter and harmonic evidence	$0.20$	$0.91$
hardware and firmware protection	$0.25$	$0.89$
thermal and cooling reliability	$0.20$	$0.93$
commissioning records	$0.15$	$0.96$

The release threshold is:

S\ge 0.92

and protection may not be below $0.90$ . Calculate the score and decision.

Solution

Weighted score:

\begin{aligned} S&=0.20(0.94)+0.20(0.91)+0.25(0.89)+0.20(0.93)+0.15(0.96)\\ &=0.188+0.182+0.2225+0.186+0.144\\ &=0.9225 \end{aligned}

The weighted score is:

92.25\%

The score passes, but protection evidence fails its floor:

0.89<0.90

Release is held.

Engineering Comment

Protection evidence is a critical floor. Rated waveform and thermal performance do not compensate for incomplete trip-chain validation.

Plausibility Check

The total score is just over threshold while one mandatory gate fails, so the hold decision follows the rule.

Validation Package Checklist

Rated current, modulation headroom, dead time and waveform quality are validated at the same DC-link and load condition.
LCL filter resonance, damping and harmonic current are checked across the operating envelope.
Hardware protection, firmware current limit, interlocks and rollback states are tested separately.
Braking and regeneration checks include DC-link voltage, chopper timing, resistor energy and fault response.
EMI leakage and grounding evidence use the installed cable, filter and shield configuration.
Thermal release includes normal cooling, degraded cooling, sensor uncertainty and power-cycling duty.
Commissioning records identify firmware, parameter file, gate-driver settings, measurements and release limits.

Common Release Mistakes

Proving output current while ignoring modulation headroom and dead-time distortion.
Checking an LCL filter frequency without validating damping and control-loop interaction.
Treating firmware current limit as a replacement for hardware desaturation or fast trip.
Sizing braking power from average production duty while missing short regenerative peaks.
Measuring efficiency without applying uncertainty to the acceptance margin.
Releasing at normal fan operation without a fan-loss or coolant-loss derating plan.
Counting a thermal steady-state pass as proof of power-cycling lifetime.

REF