# Quantifying Incomplete Zero-Voltage Switching Losses Using Equivalent Hard-Switching Losses

Mike Zäch, Szymon Bęczkowski, Asger Bjørn Jørgensen, Stig Munk-Nielsen

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Mike Zäch, Szymon Bęczkowski, Asger Bjørn Jørgensen, Stig Munk-Nielsen. Quantifying Incomplete Zero-Voltage Switching Losses Using Equivalent Hard-Switching Losses. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0116. hal-05095973

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# Quantifying Incomplete Zero-Voltage Switching Losses Using Equivalent Hard-Switching Losses

Mike Zäch, Szymon Bęczkowski, Asger Bjørn Jørgensen, Stig Munk-Nielsen,
AAU Energy,
Aalborg University,
Aalborg, Denmark,
mrz@energy.aau.dk

**Index Terms**—Incomplete ZVS, loss modelling, switching losses, power modules, power MOSFET, zero-voltage switching

**Abstract**— With fast switching wide band-gap semiconductors conquering high-power application, zero-voltage switching (ZVS) topologies are crucial to increase converter efficiency by minimizing switching losses. However, ZVS criteria cannot always be fulfilled in the entire operating range of the converter. In this case, the converter enters incomplete ZVS (iZVS). The losses during iZVS cannot be quantified using conventional double-pulse tests, but require calorimetric measurement setups, and separation of turn-on, conduction, and turn-off losses. This can be obtained for discrete devices, but is not possible for power modules. In this paper, a method to estimate the switching losses during iZVS using equivalent hard-switching losses is presented and experimentally verified, allowing to quantify the losses in power modules.

## I. INTRODUCTION

With the rise of fast-switching wide band-gap semiconductors in high power applications [1], [2], [3], quantifying and minimizing the switching losses becomes crucial in obtaining a high converter efficiency. Zero-voltage switching (ZVS) topologies are an effective choice in reducing the switching losses. However, converters might face operating points outside the ZVS region. To overcome this, overdimensioning the converter parameters or restricting the operating range might be the consequence. For converters with a wide operating range, such as EV chargers, the incomplete ZVS (iZVS) regions cannot always be avoided, and the transistors turn on with a remaining voltage across them, causing additional turn-on losses. Figure 1 shows the drain-source voltage transients during turn-on, where the load-dependent transition from hard-switching at a DC voltage of 600 V to complete ZVS is presented. The iZVS is

<figure><img src="image_2.png" /><figcaption>Fig. 1. Example waveforms of ZVS, iZVS, and hard-switching voltage transients [4].</figcaption></figure>

occurring in the intermediate region, with remaining turn-on voltages of 500 V, 250 V, and 100 V respectively.

The state-of-the-art of quantifying switching losses is the double-pulse test (DPT), which poses limitations on the turn-on current, as energy needs to be built up in the parallel inductor before the DUT is turned on under load [5]. Modifications can be applied to achieve a negative drain current required for iZVS [6], but still pose the challenge of compensating the propagation delays of the measurement equipment and de-skewing the time signals [7], as well as separating the turn-on and turn-off energies correctly [8]. Therefore, quantifying the losses between the ZVS region and the hard-switching region typically requires custom calorimetric setups, leading to converter specific models and cumbersome calibration procedures [9], [10], [11]. Furthermore, calorimetric setups need to be able to separate turn-
<figure><img src="image_3.png" /><figcaption>Fig. 2. Example waveforms of a) iZVS turn-on transient at V<sub>DC</sub> with negative I<sub>on</sub> and remaining turn-on voltage indicated, and b) hard-switching turn-on transient with positive I<sub>on</sub> and V<sub>rem</sub>.</figcaption></figure>

on, conduction, and turn-off losses. This is achieved by using a compensated power sensor to measure the transistor losses accurately on a full-bridge converter. Using a modulation scheme allowing full control of the amplitude of the turn-on and turn-off currents, the switching losses for both positive and negative drain currents can be measured [12]. For discrete devices, this measurement method can be reliably used to quantify the iZVS switching losses, but for power modules, the shared heat dissipation path for high-side and low-side dies make this approach unsuitable.

This paper proposes a method to predict the losses during iZVS turn-on events as shown in Figure 2a using information extracted from hard-switching losses occurring at the remaining turn-on voltage and positive drain current of equal magnitude, as shown in Figure 2b. This allows to quantify the iZVS losses in power modules by using readily available hard-switching loss measurements and models. The obtained model can then be used to optimize power modules for wide-operating range converters.

## II. HARD SWITCHING LOSS EQUIVALENCE MODEL

In ZVS converters, the resonance occurring between a load inductor and the output capacitances of the high-side and low-side transistors in a half-bridge leg is used to fully discharge the output capacitance of the device under test (DUT) within the dead-time period. If the charge provided by the load inductor is not sufficient for a complete discharge, there will still be a remaining voltage present at the turn-on event, and iZVS will occur. The remaining voltage depends on the circuit parameters as well as the initial conditions at the beginning of the dead-time period, as described in [4]. During iZVS, the DUT will be turned on with a remaining voltage V<sub>rem</sub> across it and starting to

<figure><img src="image_4.png" /><figcaption>Fig. 3. Conceptual illustration of how the iZVS losses at I<sub>on</sub> and V<sub>DC</sub> can be quantified using the equivalent hard-switching losses at V<sub>rem</sub> and |I<sub>on</sub>|.</figcaption></figure>

conduct a turn-on current I<sub>on</sub>. The hypothesis of this work is that the iZVS losses can be predicted using the hard-switching losses dissipated at a DC voltage of V<sub>rem</sub> and a turn-on current |I<sub>on</sub>|, as illustrated in Figure 3. This can be expressed as:

$$E_{\text{on}}(V_{\text{DC}}, I_{\text{on}}) \approx E_{\text{on}}(V_{\text{rem}}, |I_{\text{on}}|) \quad (1)$$

Analyzing the turn-on process during iZVS and hard-switching, it becomes apparent that there are fundamental differences when turning on with a positive or negative drain current, affecting the energy dissipation. These differences must be identified and compensated for to calculate the equivalent turn-on losses.

### A. Influence of current direction on energy dissipation

To investigate the influence of the current direction on the energy dissipation during the turn-on process, the switching cell is simplified to a constant resistor denoting the channel resistance, and a variable capacitor, denoting the output capacitance of the transistor. A DC current source represents the drain current during the turn-on transient. The equivalent circuits are shown in as shown in Figure 4. The energy dissipated in a parallel RC circuit with a current source can be derived by solving the first-order ordinary differential equation (ODE). According to Kirchhoff's current law, the current in the output capacitor is given by:

$$i_{\text{coss}} = I_{\text{on}} - i_{\text{ch}} \quad (2)$$

<figure><img src="image_5.png" /><figcaption>Fig. 4. Simplified schematic of DUT during turn-on with negative (iZVS) and positive (hard-switching) turn-on current.</figcaption></figure>
Which can be rewritten as:

$$C(v) \frac{dv(t)}{dt} = I_{\text{on}} - \frac{v(t)}{R} \quad (3)$$

Rearranging, the following ODE is obtained:

$$\frac{dv(t)}{dt} = \frac{I_{\text{on}}}{C(v)} - \frac{v(t)}{RC(v)} \quad (4)$$

The general solution for the ODE can be expressed as:

$$v(t) = (v_0 - I_{\text{on}}R)e^{-\frac{t}{RC(v)}} + I_{\text{on}}R \quad (5)$$

Note that the direction of the turn-on current is positive during hard-switching, and negative during iZVS. As the output capacitance is non-linear and dependent on the voltage across it, the solution to this problem becomes complex and requires numerical solutions. To simplify the expression and the subsequent solution, the output capacitance is linearized using the charge equivalent capacitance [13].

$$C_{\text{q,eq}}(v_{\text{ds}}) = \frac{Q_{\text{oss}}(v_{\text{ds}})}{v_{\text{ds}}} \quad (6)$$

The energy dissipation in the resistor can be written as the time integral of the dissipated power.

$$E_{\text{ch}} = \int_{0}^{T} \frac{v^2(t)}{R} dt \quad (7)$$

Inserting (5) into (7) and evaluating the integral, the dissipated energy can be written as:

$$E_{\text{ch}} = C \left( \frac{V_0^2}{2} - V_0 I_{\text{on}} + \frac{I_{\text{on}}^2 R^2}{2} \right) \left( 1 - e^{-2 \frac{T}{RC}} \right) \quad (8) \\ + 2CI_{\text{on}}(V_0 - I_{\text{on}}R) \left( 1 - e^{-\frac{T}{RC}} \right) + I_{\text{on}}^2 RT$$

If the current changes direction in the case of iZVS, the difference in energy dissipation caused by the opposite current can be evaluated by considering

$$\Delta E_{\text{ch}} = E_{\text{ch}}(I_{\text{on}}) - E_{\text{ch}}(-I_{\text{on}}) \quad (9)$$

Assuming $T \gg RC$, the exponential terms approach zero. Likewise, parts containing the square of $I_{\text{on}}$ will cancel out. Thus, the difference in energy dissipation as a result of a change in current direction can be expressed as

$$\Delta E_{\text{ch}} = 4V_{\text{DC}}C_{\text{q,eq}}(v_{\text{ds}})I_{\text{on}} \quad (10)$$

Where $V_{\text{DC}}$ is the DC bus voltage, $C_{\text{q,eq}}$ is the charge equivalent output capacitance described in (6), and $I_{\text{on}}$ is the positive turn-on current during hard-switching. Thus, the energy dissipation at negative turn-on current can be written as

$$E_{\text{neg}} = E_{\text{pos}} + \Delta E_{\text{ch}} \quad (11)$$

## B. Reverse recovery losses

During the dead-time in hard-switching, $Q_2$ starts conducting the inductor current. When the DUT turns on, the carriers in the junction need to be swept, causing a reverse recovery current. The current cannot be drawn from the inductor, as the direction cannot change immediately, but has to be drawn through the DUT, as shown in Figure 5, causing additional losses [14]. The inflicted reverse recovery losses in the DUT can be expressed as

$$E_{\text{rr,compl}} = \frac{1}{2} V_{\text{DC}} I_{\text{f}} \sqrt{\frac{2Q_{\text{a}}}{di/dt}} \quad (12)$$

where the $I_f$ is the forward current in the body diode, $Q_a$ is the reverse recovery charge during the fall period of the reverse recovery process, and $di/dt$ is the rate of change of the reverse recovery current. The reverse recovery charge $Q_a$ can be derived from the total reverse recovery charge $Q_{rr}$, the total reverse recovery time $t_{rr}$, and the peak reverse recovery current $I_{rrm}$, commonly found in the device data sheet.

$$Q_{\text{a}} = \frac{1}{2} I_{\text{rrm}} \cdot t_{\text{a}} \quad (13)$$

$$t_{\text{a}} = \frac{I_{\text{rrm}}}{di/dt} \quad (14)$$

With these differences in energy dissipation between positive and negative current turn-on identified, the initial hypothesis formulated in (1) has to be adjusted accordingly.

$$E_{\text{on,iZVS}}(V_{\text{DC}}, I_{\text{on}}) = E_{\text{on,HS}}(V_{\text{rem}}, |I_{\text{on}}|) + \Delta E_{\text{ch}} - E_{\text{rr,compl}} \quad (15)$$

<figure><img src="image_6.png" /><figcaption>Fig. 5. During the dead-time period in hard-switching, the body diode $Q_2$ is forward biased and conducting the load current. When $Q_1$ turns on, the reverse recovery current is drawn through $Q_1$.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Conceptual illustration of how the iZVS losses at $I_{on}$ and $V_{DC}$ can be quantified using the equivalent hard-switching losses at $V_{rem}$ and $|I_{on}|$ compensated for the additional losses caused by the reversed current direction.</figcaption></figure>

Figure 6 shows how the original hypothesis from Figure 3 has to be adjusted for the difference in energy dissipation caused by the opposite current direction during turn-on, as described in (15).

### III. METHODOLOGY

To verify the model, the switching losses of a MOSFET are measured in situ in a full-bridge converter at various DC voltages and load currents to generate loss maps, allowing to compare iZVS losses to the equivalent hard-switching losses. A non-invasive compensated power sensor presented in [15] is used to quantify the dissipated losses. The sensor operates using a thermal electric generator (TEG) on the symmetry axis of a copper fixture. On one side of the fixture, the DUT is located, operating continuously and dissipating power through the copper fixture. On the second side of the fixture, a transistor of the same type with characterized forward characteristics is located. This reference device is dissipating a known power through its body diode. The reference power is controlled and adjusted until the voltage of the TEG reaches zero. In this case, thermal equilibrium is obtained, and both transistors dissipate the same power. The working principle is shown in Figure 7.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Compensated power sensor presented in [15]; a) working principle; b) schematic of sensor; c) thermal equilibrium when same power is dissipated in reference device as in the DUT.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. a) Full-bridge test circuit with DUT and inductive load; b) Modulation of inductor current to vary turn-on and turn-off current [9].</figcaption></figure>

#### A. Separating transistor switching losses

As the DUT is operating continuously in a full-bridge configuration, the losses measured with the compensated power sensor include both the turn-on losses, the conduction losses, and the turn-off losses [12].

$$P_{\text{tot}} = P_{\text{on}} + P_{\text{cond}} + P_{\text{off}} \quad (16)$$

Therefore, a modulation method to separate the three parts presented in [9] is applied. With the proposed modulation method, the magnitude of the turn-on current and the turn-off current can be controlled in first and third quadrant operation, as shown in Figure 8b. The DUT can therefore be turned on in both ZVS, iZVS, and hard-switching. Two experiments are required to generate a full loss map. In an initial experiment, the losses are measured while the converter is operating with a fixed turn-on current beyond the ZVS boundary. In this case, the turn-on losses are negligible and only the conduction losses and the turn-off losses are measured. To ensure ZVS, the ZVS current defined in [4] is calculated.

$$I_{\text{ZVS}} = -\frac{\omega_0}{\sin(\omega_0 T_d)} \cdot 2Q_{\text{oss}}(V_{\text{DC}}) \quad (17)$$

Where $T_d$ is the dead-time, and $\omega_0$ denotes the LC resonance frequency between the output capacitances of $Q_1$ and $Q_2$ and the load inductor. The resonance frequency can be determined using

$$\omega_0 = \frac{1}{\sqrt{2L \frac{Q_{\text{oss}}(V_{\text{DC}})}{V_{\text{DC}}}}} \quad (18)$$

By subtracting the conduction losses from the losses measured during ZVS, the turn-off losses can be extracted. The conduction losses are estimated by measuring the inductor current $i_L(t)$, which is equal to the drain current $i_D(t)$ while the DUT is conducting, as shown in Figure 8b. The static characteristics of the reference device are characterized, and the conduction losses are determined using

$$P_{\text{cond}} = f_{\text{sw}} \int_{t_{\text{on}}}^{t_{\text{off}}} v_{\text{DS}}(i_{\text{D}}, T_j, v_{\text{gs}}) \cdot i_{\text{D}} \, dt \quad (19)$$
The experiment is repeated with a fixed turn-off current, while the turn-on current is swept through iZVS and hard-switching operating points. The losses measured losses during this experiment will include all three terms of (16). With the turn-off losses known from the first experiment, and the conduction losses calculated according to (19), the turn-on losses can be determined, separating the power dissipation of the DUT into turn-on, conduction, and turn-off losses for both iZVS and hard-switching. The extracted turn-on losses can then be inserted into (15) and compensated for the difference in energy dissipation due to the reverse current direction and the reverse recovery induced losses in the DUT to verify the proposed method of quantification of iZVS turn-on losses.

## IV. EXPERIMENT

### A. Experimental setup

The loss map is generated using a full-bridge converter with an inductive load, and the compensated power sensor connected to the high-side transistor on the input side. The circuit parameters are listed in Table I and the experimental setup is shown in Figure 9.

<table><caption>TABLE I
CIRCUIT PARAMETERS USED IN EXPERIMENTAL SETUP</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Transistors</td><td>IPB65R125C7 Si CoolMos</td></tr><tr><td>Load inductor</td><td>170 μH</td></tr><tr><td>Driver voltage</td><td>-5 / + 20 V</td></tr><tr><td>Switching frequency</td><td>20 kHz</td></tr><tr><td>Dead-time</td><td>300 ns</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Experimental setup of the full-bridge converter with the compensated power loss sensor attached to the DUT, and the Fluke multimeter and precision SMU in the background.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Switching waveforms of turn-on and turn-off events for iZVS at 60 V and hard-switching at 32 V.</figcaption></figure>

During operation of the full-bridge converter, the initial drain current at the beginning of the dead-time period is set to control the switching mode, i.e. ZVS, iZVS, or hard-switching. The inductor current, drain-source voltage and the gate-source voltage of both the high and low-side device are measured using an oscilloscope. The voltage of the TEG of the compensated power sensor is measured using a FLUKE 8845A 6-digit precision multimeter. The reference device, responsible for compensating the dissipated power, is supplied by a Keysight B2902A precision source measure unit (SMU) using a four-wire configuration.

### B. Experimental results

The described experiment is repeated for multiple DC voltages to generate loss maps including both ZVS, iZVS, and hard-switching. Figure 10 shows a turn-on and turn-off transient for both iZVS and hard-switching. The initial drain current is -1 A during iZVS and 0.6 A during hard-switching. The turn-off current is 2.1 A in both cases. From these waveforms and the measured power dissipation during the two sweeps, the loss map is generated using (16) and (19). The obtained loss maps showing the turn-on and turn-off energy at the various voltages is shown in Figure 11.

The turn-on energy during iZVS from the 60 V experiment is compared to the turn-on losses during hard-switching at the positive current with the same amplitude and the corresponding remaining turn-on voltage. The remaining turn-on voltage during iZVS can be extracted from the electrical waveform of the drain-source voltage during the turn-on transient.
<figure><img src="image_12.png" /><figcaption>Fig. 11. Experimentally obtained loss maps for turn-on and turn-off losses for various DC voltages.</figcaption></figure>

Zooming in on the turn-on transient during iZVS in Figure 12, it is possible to extract the remaining turn-on voltage at $t_{on}$. The switching time marks the beginning of the dead-time period. During the dead-time period, the inductor current resonates between the load inductance and the output capacitances, as elaborated in [4], leading to a turn-on current different from the initial drain current.

The remaining turn-on voltage is extracted at different initial drain currents at 60 V, as seen in Figure 13 to map the hard-switching losses at the respective voltage level. The remaining turn-on voltage decreases as the initial drain current approaches the ZVS boundary. The shape of the curve reveals the non-linearity of the output capacitances, which are charged and discharged during the switching transition. It is to be noted that the turn-on losses at 0 A are not equal to zero, as the DUT dissipates the required energy $E_{oss}(V_{DC})$ to charge the output capacitance of the complementary device, which can be calculated by integrating the increase in voltage over the charge $Q_{oss}(v_{ds})$ as described in [13], [16].

<figure><img src="image_13.png" /><figcaption>Fig. 12. Switching transient during iZVS with the remaining turn-on voltage at 12 V with an initial drain current of –1 A and a turn-on current of –0.5 A.</figcaption></figure>

With the obtained data, it is possible to compare the measured iZVS turn-on losses at 60 V and the equivalent hard-switching losses. To compare the iZVS to the equivalent hard-switching losses, the following procedure is followed:

1. Measure iZVS losses at a DC voltage of 60 V

2. Extract remaining turn-on voltages

3. Measure hard-switching losses at corresponding remaining turn-on voltages and positive drain currents

4. Map hard-switching losses to corresponding iZVS drain currents and remaining voltages

5. Compensate for the opposite current direction using (10) & (12)

<figure><img src="image_14.png" /><figcaption>Fig. 13. Remaining turn-on voltage during iZVS and hard-switching with a DC voltage of 60 V.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 14. Measured turn-on energy during iZVS at a DC voltage of 60 V, and hard-switching turn-on energy at the corresponding turn-on current at the equivalent remaining turn-on voltage.</figcaption></figure>

Figure 14 shows the results of the experimental verification listen in Table II. The turn-on energy during the equivalent hard-switching is compared to the iZVS turn-on energy, and it can be observed that the energy dissipation deviates significantly without taking for the effects caused by the reversed current direction into account. When compensating for the reverse recovery current and power dissipation in the channel, the iZVS losses can be approximated accurately.

<table><caption>TABLE II
EXPERIMENTAL RESULTS</caption><thead><tr><th>I<sub>sw</sub><br>[A]</th><th>V<sub>rem</sub><br>[V]</th><th>I<sub>on</sub><br>[A]</th><th>E<sub>on,HS</sub><br>[µJ]</th><th>E<sub>on,comp</sub><br>[µJ]</th><th>E<sub>on,iZVS</sub><br>[µJ]</th><th>ΔE<sub>on</sub><br>[µJ]</th></tr></thead><tbody><tr><td>-0.89</td><td>14.00</td><td>-0.92</td><td>2.44</td><td>0.66</td><td>0.45</td><td>0.22</td></tr><tr><td>-0.72</td><td>23.48</td><td>-0.70</td><td>3.34</td><td>0.89</td><td>1.11</td><td>-0.22</td></tr><tr><td>-0.62</td><td>32.20</td><td>-0.59</td><td>4.18</td><td>1.15</td><td>1.53</td><td>-0.37</td></tr><tr><td>-0.53</td><td>40.00</td><td>-0.54</td><td>5.22</td><td>1.65</td><td>1.68</td><td>-0.03</td></tr><tr><td>-0.44</td><td>50.00</td><td>-0.39</td><td>4.88</td><td>1.59</td><td>1.96</td><td>-0.38</td></tr></tbody></table>

V. CONCLUSION

A method to estimate the turn-on losses occurring during incomplete zero-voltage switching using equivalent hard-switching losses was presented and verified using a full-bridge converter with full control of the turn-on and turn-off currents, and a compensated power sensor, allowing for separation of turn-on, conduction, and turn-off losses. By predicting the remaining turn-on voltage at a given initial drain current, the remaining voltage can be used to quantify the hard-switching losses at the positive drain current of the same magnitude. The hard-switching losses need to be compensated for the differences due to the opposite current direction to obtain an accurate

estimation of the incomplete zero-voltage switching turn-on losses. This makes it possible to calculate the incomplete zero-voltage switching losses without a calorimetric measurement method, but with methods suitable for hard-switching topologies, such as double-pulse tests. Furthermore, this makes it possible to estimate the losses during incomplete zero-voltage switching for power modules, where calorimetric measurement setups are not suitable to separate turn-on and turn-off losses due to the shared heat sink. While the proposed method presents an accurate model to estimate incomplete zero-voltage switching losses using equivalent hard-switching losses, more parameters need to be considered in the future, such as the influence of circuit parasitics, parallel devices, and different semiconductor materials, such as SiC transistors to reduce the impact of reverse recovery losses.

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