Turn-on energy loss of a power transistor with linear switching In the shown linear turn-on, voltage falls linearly from V to 0 while current rises linearly from 0 to I over time T. What is the switching energy lost during this interval?

Introduction / Context:
Switching losses in power electronic devices occur during finite transition times when a device simultaneously supports nonzero voltage and current. Estimating these losses is key to heat-sinking, efficiency calculations, and safe operating area selection. A common textbook approximation assumes linear voltage and current transitions for quick, closed-form estimates.

Given Data / Assumptions:

• Turn-on interval = T.
• Voltage across device v(t) decreases linearly from V to 0.
• Current through device i(t) increases linearly from 0 to I.
• Parasitics and nonidealities neglected beyond the linear approximation.

Concept / Approach:
Instantaneous switching power is p(t) = v(t) * i(t). With linear ramps, we can write simple expressions for v(t) and i(t), multiply them, and integrate over the transition time to obtain the turn-on energy.

Step-by-Step Solution:
Let v(t) = V * (1 − t/T), for 0 ≤ t ≤ T.Let i(t) = I * (t/T), for 0 ≤ t ≤ T.Instantaneous power: p(t) = v(t) * i(t) = V * I * (t/T) * (1 − t/T).Energy during turn-on: E_on = ∫_0^T p(t) dt = V * I * ∫_0^T (t/T − t^2/T^2) dt.Compute integrals: ∫_0^T t/T dt = (T/2), and ∫_0^T t^2/T^2 dt = (T/3).Therefore, E_on = V * I * (T/2 − T/3) = V * I * (T/6).

Verification / Alternative check:
Geometric method: the p(t) curve is a parabola peaking mid-interval. The average power over T equals the integral divided by T = (V * I)/6, giving the same energy E_on = (V * I)/6 * T.

Why Other Options Are Wrong:

• V * I * T: ignores overlap shape; far too large.
• (1/2) VIT or (1/3) VIT: overestimates compared to the correct parabolic overlap.
• (1/12) VIT: underestimates by a factor of two.

Common Pitfalls:

• Using a rectangular or triangular approximation without matching the specified linear rise/fall on both waveforms.
• Confusing turn-on and turn-off; formulas differ if the ramps differ.

Final Answer:
(1/6) * V * I * T