1) I think you asked, first of all, how to solve the matrix:
Vds=(R1+L1p+1/pC+Mp-wM]*ids
Vqs=(R1+L1p+1/pC+wM+Mp)*iqs
Vdr=(Mp+R2+L2p-wL2)*idr
Vqr=(Mp+wL2+R2+L2p)*iqr
Rearranging per rank:
Vds=[R1-wM+(L1+M)p+1/pC]*ids
Vqs=[R1+wM+(L1+M)p+1/pC]*iqs
Vdr=[R2-wL2+(L2+M)p]*idr
Vqr=[R2+wL2+(L2+M)p]*iqr
2)From here you'll find the current[still in Laplace mode]:
Let's say: Vds=VDSeff*sin(w*t+psids), for instance
So
ids=VDSeff*sin(w*t+psids)/[R1-wM+(L1+M)p+1/pC]
Further let's split ids=idsa+idst where
idsa=unattenuated sinusoidal current
idst=attenuated sinusoidal current
If Z={(R1-wM)^2+[w(L1+M)-1/wC]^2} then :
idsa=VDSeff*sin(w*t+psids-fi)/Z where tan(fi)=(w(L1+M)-1/wC)/(R1-wM)
Don't ask me how to calculate idst! It's a very boring calculation and the result is something like this:
idst=-VDSeff/Z*exp(-alpha*t)*Function(psi,fi,L,C,w,alpha,wo)
where :
alpha= (R1-wM)/(L1+M)/2 and wo= wo=sqrt(1/(L1+M)/C-alpha^2)
Good Luck!
