2.2. The Operational Principle of the Coupled Inductance
There are two main ways of performing coupled inductance, namely flux mutual and flux cancellation. As shown in
Figure 2, coil1 corresponds to inductance
and has
turns whilst coil2 corresponds to inductance
and has
turns.
When a current
flows through coil1 in
Figure 2a, the total self-induced magnetic flux linkage can be expressed as (
1), and the mutual magnetic flux linkage can be expressed as (
2), where
is the self-induced magnetic flux linkage,
is the mutual magnetic flux generated by cycle of coil1,
is the self-induction of coil1, and
is the mutual magnetic flux linkage generated by coil1 and affecting coil2,
is the mutual magnetic flux by cycle of coil2, while
M is the mutual inductance of coil1 and coil2.
Similarly, coil2 also generates a self-induced magnetic flux linkage and mutual magnetic flux linkage. Its self-induced magnetic flux linkage is denoted by and its mutual-induced magnetic flux linkage is denoted by , as is the self-induced magnetic flux linkage, is the self-induction of coil2, is the mutual magnetic flux linkage generated by coil2 and affecting coil1, is the mutual magnetic flux generated by coil2 and affecting coil1, and M is the mutual inductance of coil1 and coil2.
Under linear conditions,
=
=
M, and hereafter
M is used to denote mutual inductance. According to the right-handed helix rule, the self- and mutual-inductive flux of the two coils shown in
Figure 2a go in the same direction, which is defined as flux mutual, and the total magnetic flux linkage of coil1 and coil2 is denoted by (
5) and the port voltage is denoted by (
6).
The two coils shown in the corresponding
Figure 2b have their self-inductive and mutual-inductive fluxes in opposite directions, which is defined as flux cancellation, and the total magnetic flux linkage of coil1 and coil2 is denoted by (
7), and the port voltage can be denoted by (
8).
To simplify the description of the port voltage, the coupling coefficient
k is introduced. The coupling coefficient represents the geometric mean of the ratio of mutual inductance to the self-induced inductance chain of the two coils and is expressed by Equation (
9).
Substituting the magnetic flux linkages separately gives the coupling coefficient expression (
10).
Quantitatively describing coupled coils in terms of coupling coefficients and leakage inductance allows the modelling of coupled coils to be directly embedded in the port voltages of coil1 and coil2, which can be expressed as (
11) and (
12), respectively.
Based on the above analysis, it can be seen that the coupling coefficient is less than or equal to 1, i.e.,
, and the leakage inductance of the two coils can be expressed as
and
, respectively. The coupled inductance voltage-current relationship is reconstructed to create a symmetrical coupled inductance model. Assuming that the coupled inductance has equal values in terms of excitation inductance, Equations (
11) and (
12) can be expressed as (
13) and (
14).
The equivalent circuit of the two coils is shown in
Figure 3, where the controlled voltage source represents the coupling effect between the two coils, and the inductance is the respective leakage inductance of the coupled coils. The voltage of the coupled coils in the converter secondary side can be expressed by Equation (
15).
Since the mismatch is only affected by the secondary side, the following analysis focuses on it. Embedding coupled inductance models into the topological secondary side, the equivalent circuit is shown in
Figure 4. The reference points
and
for the voltages of flux mutual and flux cancellation can be expressed by (
16) and (
17), respectively.
Unifying the equivalent voltage source generated by the coupled inductance into the voltage source of the secondary excitation inductance, the complex model of coupled inductance is simplified into the equivalent model of the voltage source and the leakage inductance.
Based on the coupled inductance equivalent model established above, the output current change rate of the converter flux mutual aid and flux cancellation coupled inductance is expressed by Equations (
18) and (
19).
The peak value of the current during the steady-state operation of the converter can be obtained according to the converter operating principle, and the peak value of the current for flux mutual and flux cancellation can be expressed by Equations (
20) and (
21), respectively.
From the above analysis, it can be seen that the output current ripple suppression effect is positively correlated with the coupling coefficient in the flux mutual; and the output current ripple suppression effect is negatively correlated with the coupling coefficient in the flux cancellation.
The main reasons for the mismatch in current distribution include the mismatch of on-resistance and parasitic inductance at the device level, the passive components at the circuit level, and the parasitic mismatch of the layout. The above mismatches can be expressed by correcting the device model, where
denotes the different on-resistance of the two branches and
denotes the different parasitic inductance of the two branches. Embedding the modified device model into the output model, the secondary side equivalent circuit of the conventional flyback topology, the flux mutual coupled inductance, and the flux cancellation are shown in
Figure 5.
Referring to
Figure 5a, the mismatch resistance can be expressed as
and the mismatch inductance can be expressed as
. Since the on-resistance mismatch of the MOSFET is at the
level, its parasitic inductance and that of the circuit layout are at the level of a few nH, while the filtering inductance is at the level of a few tens of μH, and the non-ideal effect can be ignored when performing loop current calculations. Under these conditions, the converter’s secondary side current is consistent with the typical current of the converter. The total current at the secondary side in this case can be used in (
22).
This current is split between the two branches, and according to Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL), the current distribution between the two branches is inversely proportional to the total impedance of the branches, which can be expressed as (
23).
The mismatch current in the absence of coupling inductance can be expressed by (
23), where the parasitic inductance size is roughly at the level of a few nH, while the filtering inductance is in the order of tens to tens of μH and
. Therefore, a further simplified representation of the mismatch current can be made.
From the above analysis, it can be seen that the current distribution between the two MOSFETs is independent of filter inductance, and only of the resistance and parasitic inductance of the MOSFETs and the layout. The current mismatch is directly determined by the MOSFETs resistance and the parasitic inductance mismatch. Referring to the flux mutual coupling inductance model shown in
Figure 5a, the voltage equations for the two branches can be listed as (
25) and (
26), respectively.
where
and
represent the voltage drop across the MOSFET, which can be expressed by Equation (
27).
denotes the equivalent voltage source generated by the transformer coupling to the secondary side, which is determined by the converter parameters and is a constant in steady-state operation.
and
denote the voltage drops generated by the coupled excitation inductance and leakage inductance, respectively, which can be expressed by the Equation (
28).
denotes the voltage drop corresponding to the mutual inductance generated by inductance
over inductance
, and
denotes the voltage drop corresponding to the mutual inductance generated by inductance
over inductance
, which can be expressed by (
29).
Since the coupled inductance is wound by the PCB, its consistency and symmetry are extremely high, and the difference generated by the leakage inductance is negligible compared with the excitation inductance and mutual inductance, so it can be assumed that and .
Substituting (
28) and (
29) into Equations (
25) and (
26) yields (
30) and (
31).
Subtract (
30) from (
31) using the formula (
32).
Then, Equation (
32) can be reduced to (
33).
By defining
according to the difference subtraction relation, we can denote (
33) by (
34).
Due to the intrinsic properties of the coupled inductance, . When , there are , and the coupling inductance suppresses the mismatch current with a suppression rate of . When with , the coupled inductance will suppress the mismatch current, and the suppression rate is still . When with , the coupling inductance will increase the mismatch current to equalize the voltage drop of the two branches at a rate of . When with , the coupling inductance increases the mismatch current to equalize the voltage drops of the two branches, and the rate of increase remains .
The difference between a flux cancellation coupled inductance and a flux mutual coupled inductance is in the polarity of the mutual inductance. The two-branch voltage relationships of flux cancellation coupled inductance are shown in (
25) and (
26), the voltage drop and self-inductance voltage relationships are shown in (
27) and (
28), and the mutual inductance voltage drop is different from that of flux mutual coupling, which can be expressed as (
35).
The solution process is the same as the flux mutual approach, which will not be repeated in this paper, and the obtained mismatch current transformation rate can be expressed as (
36)
In the flux cancellation, when , there are , and the coupled inductance suppresses the mismatch currents with a suppression rate of . When with , the coupled inductance will suppress the mismatch current, and the suppression rate is still . When with , the coupling inductance will increase the mismatch current to equalize the voltage drop of the two branches at a rate of . When , there are , the coupling inductance will increase the mismatch current to realize the voltage drop of the two branches are equal, and the rate of increase is still .
To summarize the above, the coupled inductance scheme mismatch current to the two-branch MOSFET on the voltage drop is equally as critical when the two-way MOSFET current size and the voltage drop size trend are the same, which suppresses the mismatch current; when the two-way MOSFET current size and the voltage drop size of the opposite, the mismatch current is increased. The unbalanced voltage drop is suppressed, centered around the two MOSFET voltage drops being equal, and the suppression speed is in the flux mutual and in the flux cancellation.