Christoph Haederli - Les thèses en ligne de l'INP - Institut National ...

96 NP Control with Carrier based PWM
5.2.1 Constrained DC and 6 th harmonic injection
It is proposed to only inject either DC or a 6 th harmonic depending on the operating point of
the converter, as a pure injection of a 6 th harmonic is almost as effective as the optimized
combination of higher harmonic terms in the case of reactive power; DC is best in the case of
active power. A 6 th harmonic injection has been proposed by Tallam in [77]. Note that the two
methods are not related, as the 6 th harmonic injection by Tallam is equivalent to driving DC CM
injection into saturation (physical limitation) in predominantly active power operation (motor
application).
In case of 6 th harmonic injection care needs to be taken with the amplitude. Without proper
constraints, the DC NP current will increase with rising 6 th harmonic amplitude, but then decrease
again for even higher amplitudes. This is due to the non monotonic shape of the NP current
function in reactive power operation. The CM voltage needs to be constrained to the range
between the CM voltages generating minimum and maximum NP currents. A simple clamping to
the minimum and maximum values is proposed. With rising amplitudes of the 6 th harmonic, the
switching function based on fundamental, 3 rd and 6 th harmonic will therefore be distorted, with the
effect that additional harmonics are injected and the same endpoint can be reached as with the realtime
NP current function. There is a linear region without saturation. Simulation shows that the NP
current is linear to the injected 6 th harmonic as expected. An implementation of a linear feedback
controller is straight forward. Starting from the clamping region, the gain is reduced and the NP
current asymptotically reaches its maximum value.
TABLE 32 shows 3-D plots of the NP current function over one period with different levels of
constrained 6 th harmonic CM injection. Note that the minimum and maximum NP current curves
are not unique for pure reactive power operation and low modulation depth. NP currents are zero
for both high and low CM voltage. If the intermediate sections of the piecewise linear NP current
function are positive, the midpoint represents the maximum possible value, any of the adjacent
singularities and the two outer sections of the piecewise linear function represent the minimum
possible value. In case of negative values in the intermediate sections, the midpoint represents the
minimum value and the adjacent singularities and the two outer sections of the piecewise linear
function represent the maximum value. The transitions indicated in the U CM(Phi U) functions for
maximum NP current (see TABLE 32, lowest columns) could in fact take place in different positions
and in different numbers. To be compatible with 6 th harmonic injection, the CM functions for
minimum and maximum NP current need to be defined as follows:
1. The CM function for minimum and maximum NP current have to be on the
singularities of the piecewise linear function rather than within the outer sections to
avoid useless application of CM jumps.
2. The ideal NP current function over time must not contain any DC component
3. The ideal NP current function over time must have the same zero crossings as a 6 th
harmonic with zero phase shift, leading to 9 vertical transitions in the U CM function
for maximum NP current as shown in TABLE 32, lowest columns.
The scheme proposed also corresponds qualitatively with the limiting function for higher
modulation depth (see TABLE 33).