 #### Time domain analysis of capacitor voltage self-balance mechanism Part 2

This article analyzes the characteristics of the current ix(t) and its average value Ix(avg) from the midpoint of the capacitor arm when Vd=0 and Vd≠0.

(1) Vd=0

Vd=0 means that the capacitor voltage is balanced. At this time, iinv(t) is denoted as iinv_b(t), and its expression is:

Among them, φZmn is the impedance angle of Zeq, and φZmn∈[-π/2,π/2], m=0,1,2,3,…,n=±1,±3,±5…

It can be seen from equation (1.1) that iinv_b(t) only contains odd harmonics, while Figure 1 shows that Sd contains only DC components and even harmonics. According to formula (1.2) ix=-iinvSd, the average value Ix(aveg)_b of the current ix(t) out of the midpoint of the capacitor bridge arm is:

From equations (1.3) and (1.4), it can be seen that the average current of capacitors Cdc1 and Cdc2: I1 (avg) = I2 (avg) = 0, which shows that when the capacitor voltage of 3L-NPC is balanced, there are:

① The average value of the current flowing out of the midpoint of the capacitor bridge arm is 0:

② There is no DC component in the capacitor current, and the average value of the capacitor voltage will stabilize at Udc/2;

③Factors such as load nature and capacitance deviation do not affect the equilibrium state of capacitor voltage.

(2) Vd≠0.

Vd≠0 means that the capacitor voltage is not balanced. At this time, iinv(t) is recorded as iinv_b(t), and its expression is rewritten as formula (1.5):

Then the current iinv-nb(t) consists of two parts, iinv-b(t) and iinv-d(t), iinv-b(t) is shown in formula (1.1), iinv-d(t) is:

It can be seen from equation (1.6) that iinv-d(t) contains DC components and even harmonics. It also shows that when the capacitor voltage is unbalanced, the current iinv-nb(t) before filtering contains a DC component, which is recorded as iinv-nb(avg), and its magnitude is:

Therefore, when the capacitor voltage is unbalanced, the average value Ix(avg)_nb of the current ix(t) flowing out of the midpoint of the capacitor bridge arm is:

Combining formula (1.3) with formula (1.6) and the property that the product of different trigonometric functions is zero in one cycle, we can get:

Among them, φZmn is the impedance angle of Zeq, and φZmn∈[-π/2, π/2], m=0, 1, 2, 3., n=0, ±2, ±4….

From the formula (1.9), it can be seen that the magnitude of Ix(avg)_nb is derived from the contribution of the two properties of the bridge arm output current iinv(t), which are the DC component and even harmonic components of iinv(t) respectively, where Kdc and Kharm are respectively:

According to formula (1.9) and formula (1.4), if (Kdc+Kharm)>0, then Ix(avg)_nb is inversely proportional to Vd. This shows that when Vd>0, that is, U1>U2, Ix(avg)_nb<0, then IC1(avg)<0, lC2(avg)>0, the capacitor Cdc1 will discharge and the capacitor Cdc2 will charge until the voltages of the two are equalized; when Vd<0, the same is true. This use of topology and SPWM modulation characteristics to achieve capacitor voltage balance is called self-balance characteristic, of course, this self-balance characteristic is established under the condition of (Kdc+Kharm)>0.

Therefore, it is necessary to discuss the situation of (Kdc+Kharm)>0, that is, what factors determine (Kdc+Kharm)>0. Figure 2 shows the self-balancing characteristics of 3L-NPC and the source of self-balancing current when Zeq is a load of different nature. In Figure 2, iinv(t) is the output current of the inverter bridge arm, which can be seen in combination with equation (1.9). out:

①Except for Zeq which is purely capacitive, the 3L-NPC topology has the characteristics of capacitive voltage self-balancing, and the pure capacitive load condition does not exist, because the actual circuit has a certain parasitic impedance;

②According to the relationship between |Zeq(jω)| and ω according to different load properties, combined with the frequency spectrum of Sd in Figure 1, Sd_dc and Sd_10 (at fs) are larger, then the magnitude of Ix(avg)_nb mainly comes from the DC component of iinv(t) and even harmonic components near the switching frequency;

③Usually, the filter has a greater effect on the attenuation of the switching frequency sub-harmonics, but has no effect on the DC component, so the DC component of iinv(t) is also reflected in the load current io(t), then: