Preliminaries - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 68

function, with infinitely many branches denoted W k , k ∈ Z . The function W k may be defined

for z ∈ C\ ( −∞ , 0] as the (unique) solution of

W k ( z ) = ln z + j 2 k π − ln W k ( z ) , (2.4)

where ln is the principal branch of the logarithm. Note that (2.4) implies that if w = W k ( z )

e w =

then w z , which is the basic equation satisfied by all the branches W k .For z > 0, W 0 ( z )

e w =

is the only positive solution of w z . The equation (2.4) can be thought of as a fixed-point

equation for the function

T k , z ( w ) = ln z + j 2 k π − ln w.

For k w n ) converge fast to W k ( z ).

The approximation formula (4.11) given in (Corless et al. 1996) can be written as W k ( z )

= 0 and large | z | , the iterations defined by w n + 1 = T k , z (

≈
T k , z ( T k , z (1)). Since, for large real z , W 0 ( z ) is a better initial approximation of W k ( z ) than 1,
the following approximation is used for W k ( z ):

W k ( z ) ≈ T k , z ( T k , z ( W 0 ( z ))) = ln z + j 2 k π − ln (ln z + j 2 k π − ln W 0 ( z )) . (2.5)

From (2.3) and the definition of the Lambert W function, there is

τ d s k = W k ( a ) ( k ∈ Z ) , τ d ω c = W 0 ( a ) ,

and

W k ( a ) − τ d ω c

τ d

s k = . (2.6)

Since a is normally very large, W k ( a ) can be well approximated by (2.5), i.e.,

W k ( a ) ≈ ln( τ d ω c ) + τ d ω c + j 2 k π − ln ( τ d ω c + j 2 k π ) .

Now from (2.6),

j 2 k ln (

π
τ d ω c

2 k

tan − 1

τ d s k ≈ ln( τ d ω c ) − τ d ω c ) 2 + (2 k π ) 2 + π − .

Hence, the real part and the imaginary part of the poles s k are

ln 1

2

2 k

1
τ d

τ d ω c 1

2

π
τ d ω c

Re s k ≈ ln ( =− + , (2.7)

τ d

τ d ω c ) 2 + (2 k π ) 2

1

2 k π
τ d ω c

tan − 1

2 k π −

2 k π

τ d

1
τ d ω c

Im s k ≈ ≈ − . (2.8)

τ d