Current harmonics are unwanted components of the current waveform that have frequencies that are integer multiples of the fundamental frequency. They are a common issue in systems with power electronic converters, such as Voltage Source Inverters (VSIs).Let's break down the concepts and equations:

1. RMS Harmonic Current (I_hrms):

The equation $I_hrms = sqrt((1/T) * ∫[i(t) - i_1(t)]^2 dt)$ correctly defines the root-mean-square (RMS) value of the harmonic current. Here:

• $i(t)$ is the total instantaneous current.
• $i_1(t)$ is the fundamental component of the current.
• $T$ is the period of the fundamental waveform.

This equation essentially calculates the RMS value of the difference between the total current and its fundamental component. This difference represents the harmonic content.

2. Normalized Harmonic Current (Distortion Factor - d):

The equation $d = I_hrms / I_hrms_six-step$ defines the distortion factor $d$. This normalization is crucial because the absolute value of $I_hrms$ depends heavily on the load impedance. By normalizing it with respect to the harmonic current of a six-step operation ($I_hrms_six-step$) under the same load, we get a measure of the harmonic content that is independent of the load.

• Six-Step Operation: The six-step operation you describe (u1-u2-u3-u4-u5-u6) is a basic, unmodulated operation of the VSI. It results in a line-to-line voltage waveform with six distinct steps. It serves as a reference point for comparing the harmonic performance of other modulation techniques.
• d = 1 for Six-Step: As you point out, $d = 1$ for six-step operation, because $I_hrms$ will be equal to $I_hrms_six-step$ in this case.
• d > 1 is Possible: It's important to note that $d$ can be greater than 1. This means that a particular modulation strategy might introduce more harmonic current than the basic six-step operation. This can happen, for example, if the modulation strategy is poorly designed or if it's operating under certain load conditions.

3. Complex Conjugate Form:

The equation $I_hrms = sqrt((1/T) * ∫[i(t) - i_1(t)][i(t) - i_1(t)]* dt)$ is an alternative way of expressing the RMS harmonic current using complex notation. The asterisk (*) denotes the complex conjugate. This form is useful when dealing with phase relationships between the current harmonics. It's mathematically equivalent to the first equation but can be more convenient in some calculations.

4. Loss Factor (d^2):

You correctly state that the copper losses in the load are proportional to the square of the harmonic current. Therefore, $d^2$ represents the loss factor. Minimizing $d$ (and therefore $d^2$) is a key objective in VSI modulation design, as it directly reduces power losses and improves efficiency.

In summary:

• $I_hrms$ quantifies the absolute level of harmonic current.
• $d$ (distortion factor) normalizes $I_hrms$ with respect to the six-step operation, making it independent of load impedance.
• $d^2$ represents the loss factor, which is directly related to the power losses due to harmonics.