My motivation. I study transformerless three-phase solar inverter systems because they offer small volume, low weight, high efficiency, and reduced cost compared with isolated solar inverter systems. However, because a transformerless solar inverter has no galvanic isolation between the photovoltaic array and the ac grid, a common-mode loop is formed through the photovoltaic parasitic capacitance and the grid. This loop produces leakage current, which can distort the grid current, create electromagnetic interference, and threaten personal safety. I therefore focus on modulation techniques for two-level and three-level three-phase four-leg solar inverter topologies. My goal is to suppress leakage current while preserving high dc voltage utilization, low switching loss, and reliable neutral-point voltage behavior.
Why leakage current appears in a solar inverter. In a practical photovoltaic array, the metal frame is grounded for safety. The photovoltaic terminals nevertheless have a parasitic capacitance to ground. The parasitic capacitance depends on array area, module construction, temperature, humidity, and mounting. For crystalline silicon modules, a typical value is about 50 nF/kW to 150 nF/kW. For thin-film modules, the value can be about 1 μF/kW. When the solar inverter switches, the common-mode voltage changes at high frequency, and the parasitic capacitance is charged and discharged repeatedly. The resulting leakage current can be written as
$$i_k(t)=-C_{\mathrm{PV}}\frac{d v_{\mathrm{PV}}(t)}{dt}.$$
Equation above shows that leakage current is proportional to the parasitic capacitance and to the time derivative of the voltage across the parasitic capacitance. Since the parasitic capacitance is difficult to eliminate, I must control the voltage across it. In a transformerless solar inverter, the common-mode voltage is the main excitation source. If the common-mode voltage is constant, the high-frequency component of the parasitic-capacitance voltage is greatly reduced, and leakage current is strongly suppressed.
Common-mode definitions for a four-leg solar inverter. For a three-phase four-leg solar inverter, I define the common-mode voltage as the average of the four leg-to-neutral voltages:
$$V_{\mathrm{CM}}=\frac{V_{AN}+V_{BN}+V_{CN}+V_{DN}}{4}.$$
When the four filter inductors are equal, the differential-mode terms cancel in the simplified common-mode model. The voltage across the photovoltaic parasitic capacitance can be expressed as a function of the common-mode voltage and the fourth-leg filter-capacitor voltage. In general form, I write
$$V_{\mathrm{PV}}(s)=a(s)V_{\mathrm{CM}}(s)+bV_{Cf4}(s),$$
where the coefficients depend on the filter inductance, the filter capacitance, and the parasitic capacitance. The fourth-leg capacitor voltage is also driven by the common-mode voltage. Therefore, the central requirement for leakage-current suppression in a solar inverter is to keep the common-mode voltage constant or to reduce its high-frequency variation as much as possible.
International limits. The leakage current of a solar inverter must satisfy safety standards. A commonly referenced requirement disconnects the solar inverter from the grid when the leakage current exceeds certain limits. The protection behavior can be summarized as follows.
| Leakage current condition | Maximum disconnection time |
|---|---|
| $$i_k > 300\,\mathrm{mA}$$ | 0.3 s |
| $$\Delta i_k > 30\,\mathrm{mA}$$ | 0.3 s |
| $$\Delta i_k > 60\,\mathrm{mA}$$ | 0.15 s |
| $$\Delta i_k > 150\,\mathrm{mA}$$ | 0.04 s |
For this reason, I evaluate my proposed modulation methods not only by waveform quality but also by leakage current peak and rms value. A solar inverter modulation strategy that produces a constant common-mode voltage is highly desirable.
Two-level three-phase four-leg solar inverter. I begin with the two-level three-phase four-leg solar inverter. It contains eight switches, four legs, four filter inductors, and four filter capacitors. Each leg has two states. State 1 means the upper switch is on, and the leg output is \(V_{dc}\). State 0 means the lower switch is on, and the leg output is 0. Therefore, the solar inverter has sixteen switching states. The common-mode voltage levels for these states are shown below.
| Common-mode voltage | Switching states of the four legs |
|---|---|
| $$V_{dc}$$ | 1111 |
| $$3V_{dc}/4$$ | 1110, 1101, 1011, 0111 |
| $$V_{dc}/2$$ | 1100, 1001, 0011, 0101, 1010, 0110 |
| $$V_{dc}/4$$ | 0001, 0010, 0100, 1000 |
| $$0$$ | 0000 |
With conventional space-vector modulation, the reference voltage vector is synthesized by nearby active vectors and zero vectors. The common-mode voltage then jumps among five levels, namely \(V_{dc}\), \(3V_{dc}/4\), \(V_{dc}/2\), \(V_{dc}/4\), and 0. This high-frequency common-mode variation causes a large leakage current in the solar inverter. I can express the volt-second balance for space-vector modulation as
$$V_{\mathrm{ref}}T_s=V_1T_1+V_2T_2+V_3T_3+V_0T_0.$$
The active-vector times are obtained from the reference vector components, and the zero-vector time is
$$T_0=T_s-T_1-T_2-T_3.$$
Although this method has high dc voltage utilization, its common-mode voltage is not constant. Therefore, I need improved modulation methods for the solar inverter.
Existing leakage-current suppression methods for the two-level solar inverter. I analyze three representative methods. The first is modified space-vector modulation. It replaces the zero vectors with a pair of complementary active vectors. This reduces the common-mode voltage range to \(V_{dc}/4\) through \(3V_{dc}/4\). The leakage current is reduced, but it is not eliminated. The second is remote-state space-vector modulation. It selects only vectors whose common-mode voltage is \(V_{dc}/2\). The common-mode voltage is constant, but the maximum modulation depth is limited to 1. The third is carrier modulation with a Boolean logic function. It uses logic operations to avoid zero states and to force the fourth leg state. The common-mode voltage is constant, but the switching count can be eight, twelve, or fourteen per carrier period, and the maximum modulation depth is also limited to 1.
| Method for two-level solar inverter | Common-mode voltage | Switching actions per period | Maximum modulation depth | Leakage current behavior |
|---|---|---|---|---|
| Conventional space-vector modulation | 0 to \(V_{dc}\), five levels | 8 | 1.15 | Large |
| Modified space-vector modulation | \(V_{dc}/4\) to \(3V_{dc}/4\) | 12 | 1.15 | Reduced but not eliminated |
| Remote-state space-vector modulation | Constant \(V_{dc}/2\) | 12 | 1 | Very small |
| Boolean-logic carrier modulation | Constant \(V_{dc}/2\) | 8, 12, or 14 | 1 | Very small |
My improved carrier modulation for the two-level solar inverter. I propose an improved carrier modulation method to combine constant common-mode voltage, high dc voltage utilization, and low switching loss. First, I inject a zero-sequence component into the three-phase sinusoidal modulation signals:
$$V_e=-\frac{V_{\max}+V_{\min}}{2}.$$
The final modulation signals become
$$V_{ma}^{*}=V_{ma}+V_e,\quad V_{mb}^{*}=V_{mb}+V_e,\quad V_{mc}^{*}=V_{mc}+V_e.$$
After the zero-sequence injection, the maximum and minimum phase modulation signals satisfy
$$V_{\max}^{*}=-V_{\min}^{*}.$$
I use this relationship to select different carrier forms. For the first three legs, the modulation signals are compared with falling sawtooth carriers, rising sawtooth carriers, and symmetric triangular carriers. The carrier form changes according to the sector. The fourth-leg upper-switch signal is generated by the exclusive-or function
$$S_7=S_1\oplus S_3\oplus S_5.$$
By choosing the carrier form properly, the zero switching states 000 and 111 are removed, and the solar inverter operates only in the six states whose common-mode voltage is \(V_{dc}/2\). These states are 1100, 1010, 1001, 0110, 0101, and 0011. Hence, the common-mode voltage of the solar inverter remains constant. The switching actions remain eight per carrier period, and the maximum modulation depth is 1.15. I show the carrier selection logic in the following tables.
| Sector | Carrier for phase A | Carrier for phase B | Carrier for phase C |
|---|---|---|---|
| I | Falling sawtooth | Symmetric triangle | Rising sawtooth |
| II | Symmetric triangle | Falling sawtooth | Rising sawtooth |
| III | Rising sawtooth | Falling sawtooth | Symmetric triangle |
| IV | Rising sawtooth | Symmetric triangle | Falling sawtooth |
| V | Symmetric triangle | Rising sawtooth | Falling sawtooth |
| VI | Falling sawtooth | Rising sawtooth | Symmetric triangle |
| Method | Leakage current peak | Leakage current rms | Common-mode voltage | Switching actions | Modulation depth |
|---|---|---|---|---|---|
| Conventional space-vector modulation | 1080 mA | 671 mA | 0 to 120 V | 8 | 0 to 1.15 |
| Modified space-vector modulation | 850 mA | 472 mA | 30 to 90 V | 12 | 0 to 1.15 |
| Remote-state space-vector modulation | 23 mA | 10.2 mA | 60 V constant | 12 | 0 to 1 |
| Boolean-logic carrier modulation | 17 mA | 6.6 mA | 60 V constant | 8, 12, or 14 | 0 to 1 |
| My improved carrier modulation | 29 mA | 11.4 mA | 60 V constant | 8 | 0 to 1.15 |
Simulation and experimental findings for the two-level solar inverter. I simulated the two-level three-phase four-leg solar inverter with a dc input of 120 V, a switching frequency of 15 kHz, a modulation depth of 0.9, a parasitic capacitance of 300 nF, filter inductors of 5 mH, and filter capacitors of 10 μF. Conventional space-vector modulation produced a leakage-current rms of 671 mA. Modified space-vector modulation reduced it to 472 mA. Remote-state space-vector modulation and Boolean-logic carrier modulation reduced the leakage current to about 10 mA and 6.6 mA, respectively, but at the cost of modulation depth or additional switching actions. My improved carrier modulation achieved an rms leakage current of 11.4 mA while keeping eight switching actions and a modulation depth of 1.15. I then built a two-level solar inverter prototype. The experimental rms leakage currents were 433 mA for conventional space-vector modulation, 321 mA for modified space-vector modulation, 56.6 mA for remote-state space-vector modulation, and 26.5 mA for my improved carrier modulation. These results confirm that my method suppresses leakage current without increasing switching loss and while preserving high dc voltage utilization.
Three-level three-phase four-leg solar inverter. I also study a diode-clamped three-level three-phase four-leg solar inverter. Each leg has three states: state 2, state 1, and state 0. In state 2, the upper two switches are on and the leg output is \(V_{dc}\). In state 1, the middle switches are on and the leg output is \(V_{dc}/2\). In state 0, the lower two switches are on and the leg output is 0. The leg output voltage can be written as
$$V_{xN}=\frac{V_{dc}}{2}S_x,\quad x=A,B,C,D.$$
The common-mode voltage of the three-level solar inverter is
$$V_{\mathrm{CM}}=\frac{V_{dc}}{8}(S_A+S_B+S_C+S_D).$$
There are eighty-one switching states in total. To keep the common-mode voltage constant at \(V_{dc}/2\), the four leg states must satisfy
$$S_A+S_B+S_C+S_D=4.$$
The vectors satisfying this condition are shown below. These vectors include large vectors, medium vectors, small vectors, and the zero vector. Other states are invalid for leakage-current suppression because they produce a different common-mode voltage.
| Vector type | Example switching states | Common-mode voltage |
|---|---|---|
| Large vector | 0220, 2200, 2002, 2020, 0022, 0202 | $$V_{dc}/2$$ |
| Medium vector | 1201, 2101, 2011, 1021, 0121, 0211 | $$V_{dc}/2$$ |
| Small vector | 1210, 1102, 2110, 1012, 1120, 0112 | $$V_{dc}/2$$ |
| Zero vector | 1111 | $$V_{dc}/2$$ |
| Invalid vectors | 010X, 221X, 122X, 100X, 001X, 212X, 000X, 222X | Not constant |
Traditional modulation for the three-level solar inverter. In-phase disposition modulation uses two in-phase triangular carriers. Opposite-phase disposition modulation uses two carriers with a phase difference of 180 degrees. Both methods inject a third-harmonic zero-sequence component into the three-phase modulation signals. The zero-sequence component is
$$V_e=\frac{1}{6}\sin(3\omega t).$$
This injection allows the modulation depth to reach 1.15, which improves dc voltage utilization. However, the common-mode voltage still varies at high frequency. For in-phase disposition modulation, the common-mode voltage moves among five levels. For opposite-phase disposition modulation, the common-mode voltage moves among fewer levels, but it is still not constant. Therefore, both traditional methods produce leakage current in the solar inverter.
Existing leakage-current suppression for the three-level solar inverter. A known improved opposite-phase disposition method uses the fourth leg to force the sum of the four leg states to four:
$$S_D=4-S_A-S_B-S_C.$$
This method keeps the common-mode voltage constant and suppresses leakage current. However, the fourth leg switches six times per carrier period, so the total switching actions become twelve. This increases switching loss. I therefore propose a sawtooth combination modulation method for the three-level solar inverter.
My sawtooth combination modulation for the three-level solar inverter. The key idea is to use rising and falling sawtooth carriers for the upper and lower comparisons of the first three legs. The modulation signals are
$$V_{ma}^{*}=V_{ma}+V_e,\quad V_{mb}^{*}=V_{mb}+V_e,\quad V_{mc}^{*}=V_{mc}+V_e,$$
where the injected zero-sequence component is the same third-harmonic term:
$$V_e=\frac{1}{6}\sin(3\omega t).$$
I compare each modulation signal with an upper sawtooth carrier and a lower sawtooth carrier. The sum of the two comparison results gives the leg state \(S_x\) for \(x=A,B,C\). Then I determine the fourth leg state from
$$S_D=4-S_A-S_B-S_C.$$
Because each leg state is limited to 0, 1, or 2, this relation automatically removes the invalid vectors. The common-mode voltage becomes constant at \(V_{dc}/2\). In one carrier period, the first three legs switch twice each, and the fourth leg switches four times. Thus, the total switching actions are ten, which is lower than the twelve actions of the existing improved opposite-phase disposition method. The maximum modulation depth remains 1.15. I compare the methods below.
| Method for three-level solar inverter | Common-mode voltage | Switching actions per period | Modulation depth | Leakage current behavior |
|---|---|---|---|---|
| In-phase disposition modulation | Five levels | 8 | 1.15 | Large |
| Opposite-phase disposition modulation | Several levels | 8 | 1.15 | Reduced but not eliminated |
| Improved opposite-phase disposition modulation | Constant \(V_{dc}/2\) | 12 | 1.15 | Small |
| My sawtooth combination modulation | Constant \(V_{dc}/2\) | 10 | 1.15 | Small |
Neutral-point voltage in the three-level solar inverter. A diode-clamped three-level solar inverter has two dc-link capacitors. The midpoint voltage can deviate because of capacitor parameter mismatch, switching delay, or unequal charging and discharging by the midpoint current. I define the capacitor voltages as \(V_{C1}\) and \(V_{C2}\), and the midpoint voltage as \(V_O\). The relations are
$$V_{C1}=V_{dc}-V_O,\quad V_{C2}=V_O.$$
The midpoint current \(i_o\) is the difference between the currents entering the two capacitors. The midpoint voltage dynamics can be written as
$$\frac{dV_O}{dt}=-\frac{i_o}{2C_{dc}}.$$
This equation shows that any nonzero average midpoint current causes the midpoint voltage to drift or fluctuate. The fluctuation is larger when the dc-link capacitance is smaller. Neutral-point imbalance distorts the output voltage, increases harmonic content, raises device voltage stress, and can reduce capacitor life. In a solar inverter, it also affects leakage current because some switching states connect a leg to the midpoint. When a leg is in state 1, its output voltage is the midpoint voltage rather than an ideal \(V_{dc}/2\). If the midpoint voltage fluctuates, the common-mode voltage is no longer perfectly constant, and the leakage-current suppression performance is degraded.
Connection between carrier modulation and space-vector modulation. I use this connection to design neutral-point balance control. In space-vector modulation, redundant states can be distributed with a factor \(k\). In carrier modulation, the same effect is obtained by injecting a zero-sequence component. For example, in one sector, the modulation signals can be written as
$$V_{ma}^{*}=V_{ma}+V_e+V_0,\quad V_{mb}^{*}=V_{mb}+V_e+V_0,\quad V_{mc}^{*}=V_{mc}+V_e+V_0,\quad V_{md}^{*}=V_e+V_0.$$
The zero-sequence component \(V_0\) changes the duty ratios of the midpoint-connected states. Therefore, I can adjust \(V_0\) to control the average midpoint current. This is the basis of my neutral-point balance control for the three-level solar inverter.
My neutral-point balance control. I first estimate the required midpoint current from the capacitor voltage difference. Let the capacitor voltage difference be
$$\Delta U_C=U_{C1}-U_{C2}.$$
In one carrier period \(T_s\), the charge variation that should be compensated is
$$i_{np0}=\frac{C_{dc}\Delta U_C}{T_s}.$$
The average midpoint current produced by the solar inverter is
$$i_{np}=d_{ao}i_a+d_{bo}i_b+d_{co}i_c+d_{do}i_d,$$
where \(d_{xo}\) is the duty ratio of the midpoint-connected state in leg \(x\). I set the average midpoint current to the opposite of \(i_{np0}\), so that
$$i_{np}=-i_{np0}.$$
Solving this condition gives the zero-sequence component \(V_0\). For example, in one sector, the expression has the form
$$V_0=\frac{V_{\max}^{*}i_{\max}(i_x+i_{\max})+V_{\min}^{*}i_{\min}(i_{\max}-i_{\min})-V_{\mathrm{mid}}^{*}i_{\mathrm{mid}}(i_{\max}+i_{\min})}{2(i_{\max}+i_{\min})}.$$
More generally, the zero-sequence component must satisfy the modulation limit
$$-1-V_{\min}\leq V_0\leq 1-V_{\max}.$$
If the calculated \(V_0\) exceeds this range, I saturate it. Then the modified modulation signals are used by my sawtooth combination modulation method. This control keeps the average midpoint current near zero and maintains the dc-link capacitor voltages balanced. Because the midpoint voltage is stabilized, the common-mode voltage of the solar inverter is kept closer to a constant value, and the leakage current is further reduced.
| Condition | Effect on three-level solar inverter |
|---|---|
| No neutral-point control | Midpoint voltage ripple is large; common-mode voltage fluctuates; leakage current increases. |
| With neutral-point control | Midpoint voltage is balanced; common-mode voltage is more stable; leakage current is lower. |
| Small dc-link capacitance | Midpoint ripple is larger without control; the benefit of neutral-point control is more obvious. |
| High modulation depth | Zero-sequence injection range is narrower; neutral-point control authority is reduced. |
Simulation results for the three-level solar inverter. I simulated the three-level solar inverter with a dc input of 120 V, dc-link capacitance of 100 μF or 68 μF, parasitic capacitance of 300 nF, filter inductance of 5 mH, filter capacitance of 10 μF, switching frequency of 15 kHz, and modulation depth of 0.9. In-phase disposition modulation produced a leakage-current rms of 328 mA. Opposite-phase disposition modulation reduced it to 150 mA. The existing improved opposite-phase disposition method produced 21 mA. My sawtooth combination modulation produced 17 mA. When I added neutral-point balance control, the midpoint voltage ripple was reduced from about 10 V to about 0.6 V for a 100 μF dc link. The leakage-current rms fell from 17.53 mA to 6.76 mA. For a 68 μF dc link, the midpoint ripple fell from about 16 V to about 0.8 V, and the leakage-current rms fell from 24.42 mA to 6.69 mA. These results show that neutral-point balance control improves the leakage-current performance of the solar inverter.
| Three-level modulation method | Leakage-current rms in simulation | Common-mode voltage | Switching actions |
|---|---|---|---|
| In-phase disposition modulation | 328 mA | Five levels | 8 |
| Opposite-phase disposition modulation | 150 mA | Several levels | 8 |
| Improved opposite-phase disposition modulation | 21 mA | Constant \(V_{dc}/2\) | 12 |
| My sawtooth combination modulation | 17 mA | Constant \(V_{dc}/2\) | 10 |
| DC-link capacitance | Midpoint ripple without control | Midpoint ripple with control | Leakage rms without control | Leakage rms with control |
|---|---|---|---|---|
| 100 μF | About 10 V | About 0.6 V | 17.53 mA | 6.76 mA |
| 68 μF | About 16 V | About 0.8 V | 24.42 mA | 6.69 mA |
Experimental results for the three-level solar inverter. I built a three-level three-phase four-leg solar inverter prototype. The experimental leakage-current rms values were 560 mA for in-phase disposition modulation, 87.7 mA for opposite-phase disposition modulation, 26.6 mA for the existing improved opposite-phase disposition method, and 27.7 mA for my sawtooth combination modulation. The switching actions per period were 8, 8, 12, and 10, respectively. Therefore, my method achieves leakage-current suppression comparable to the existing method while using fewer switching actions. For neutral-point balance, with a 100 μF dc link, the midpoint ripple was about 10 V without control and became nearly constant with control. The leakage-current rms changed from 27.4 mA to 25.1 mA. With a 68 μF dc link, the midpoint ripple was about 22 V without control and became nearly constant with control. The leakage-current rms changed from 33 mA to 25.6 mA. These experimental results confirm that my neutral-point balance control works with the sawtooth combination modulation and further improves leakage-current suppression in the solar inverter.
| Three-level method | Experimental leakage rms | Switching actions | Main advantage |
|---|---|---|---|
| In-phase disposition modulation | 560 mA | 8 | Simple carrier implementation |
| Opposite-phase disposition modulation | 87.7 mA | 8 | Lower common-mode variation |
| Improved opposite-phase disposition modulation | 26.6 mA | 12 | Constant common-mode voltage |
| My sawtooth combination modulation | 27.7 mA | 10 | Constant common-mode voltage with lower switching loss |
| DC-link capacitance | Midpoint behavior without control | Midpoint behavior with control | Leakage rms without control | Leakage rms with control |
|---|---|---|---|---|
| 100 μF | About 10 V ripple | Nearly constant | 27.4 mA | 25.1 mA |
| 68 μF | About 22 V ripple | Nearly constant | 33 mA | 25.6 mA |
My conclusions. I have studied leakage-current suppression in transformerless three-phase solar inverter systems. For the two-level three-phase four-leg solar inverter, I proposed an improved carrier modulation method. It selects falling sawtooth, rising sawtooth, and symmetric triangular carriers according to the sector and uses an exclusive-or function for the fourth leg. It keeps the common-mode voltage constant at \(V_{dc}/2\), suppresses leakage current, uses only eight switching actions per carrier period, and allows a maximum modulation depth of 1.15. For the three-level three-phase four-leg solar inverter, I proposed a sawtooth combination modulation method. It uses rising and falling sawtooth carriers, determines the fourth leg state from the sum of the first three leg states, and keeps the common-mode voltage constant. It suppresses leakage current with ten switching actions per carrier period and retains high dc voltage utilization. I also proposed a neutral-point balance control for the three-level solar inverter. By injecting a suitable zero-sequence component, the average midpoint current is driven toward zero. This balances the dc-link capacitor voltages, reduces midpoint ripple, stabilizes the common-mode voltage, and further lowers leakage current. Simulation and experimental results support my analysis. In future work, I will extend the carrier-selection concept to parallel solar inverter systems, study topology-based neutral-point balancing, and validate the proposed modulation methods under grid-connected operation.