# Circulating Current Reduction in MMC-HVDC System Using Average Model
## Appendix A  
The expression in Equations (48) and (49) are derived as:  
\(T=1/f\)  
T (period) = 0.02  
\(\omega_{o}=2\pi f\)  
\(\omega t=\pi\)  
Due to direct modulation arm current is a sinusoidal wave form:  
\[i=V_{m}\ Sincut\] (A1)  
\[Acos(\omega t)+Bcos(2\omega t)+Ccos(4\omega t)+D\] (A2)  
where \(D\) is a constant adapted to become zero in the middle. By taking the 2 nd and 4 th (1 st, 3 rd, and 5 th derivatives are zero) derivatives of Equation (A2):  
\[-Acos(\omega t)+B(-4)cos(2\omega t)-C(16)cos(4\omega t)+D\] (A3)  
\[A-cos(\omega t)+B(16)cos(2\omega t)-C(256)cos(4\omega t)+D\] (A4)  
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