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The source reactive-current compensation is crucial in energy transmission efficiency. The compensator design in frequency-domain was already widely discussed and examined. This paper presents results of a study on how to design reactive compensators in time-domain. It’s the first time the reactive compensator have been designed in time domain. The example of compensator design was presented.

This article is a discussion on issue raised in the article of L. S. Czarnecki [

In the article [

where:

stand for the active and reactive parts of receiver admittance operator Y^{o}(s).

The I(s) transform for T-periodic signals is derived using the following relation between non-periodic and periodic signals transform [

(4)

where t Î [0, T), Re(σ) > 0, T-time period.

It can be also calculated directly in time-domain as the T-periodic convolution:

where g^{o}(t), b^{o}(t) stands for T-periodic impulse response of admittance active and reactive part.

Connecting in parallel the compensator (

Admittance of the elementary RLC compensator branch

is:

;;

(We assume later that the compensator is composed of almost lossless elementary branches). And its reactive part is then (3)

which leads to general form

where L(s), M(s)—odd and even polynomials.

The residues meet the relations: if

then:

;;

where:

M′(s) is the derivative of M(s) with respect to s, d— real number.

Thus (7) reduces to

and under (4) and trigonometric identity

where aT = α + jβ.

We get

In the case of almost lossless compensator i.e. for α → 0.

(10)

where:

,—capacitive and inductive reactance for the main frequency.

Residue for B^{k}(s) in can be calculated as

Thus the T-periodic impulse response of reactive part of the elementary RLC branch (without R) has form

(11)

where:

,—angular and relative resonance frequency of m-th branch, and is depicted in

Later, in the article, it assumes that the reactive part of receiver Y^{o}(s) has only real poles, so the receiver is nonoscillatory circuit not as the compensator.

For the single pole receiver e.g. R_{L} or R_{C} type the operational admittance is

thus its impulse response is

where A = aT, Î [0, 1).

The coefficient b can be both positive and negative what is shown below for the RRLC receiver (see

Its operator admittance is

where, , ,

Then the reactive part of Y^{o}(s) is

and its T-periodic impulse response is

where A_{L} = a_{L}T, A_{C} = a_{C}T.

For the receiver shown in

where, and for the positive poles

, Y^{o}(s) takes form

where, and the coefficients b_{1}, b_{2} are then

thus

The reactive function (12) is shown in

(13)

because sh(x) → 0 for x→ 0.

Thus we arrive to the compensatory balance Equation (6) in a new form

where M—total number of compensator branches.

The solution of (14) (see Figures 6 and 7) for the unknowns L_{m} and C_{m}, can by find with optimization method. The (14) can be rewritten in respect to D_{m} and w_{m}

where

,

The relative frequencies of L_{C} compensator branches have to meet the condition

Thus we must choose relative resonance frequency w_{m} of compensator branches as not the even numbers ; p—even number.

The set of Equations (14) can be then solved for D_{m} by minimizing.

where

After equate to zero appropriate partial derivatives

and assuming that

,; (17)

we get necessary minimum condition in form of the set of M linear equations for D_{m}

Then we use relation as to calculate L_{m} of compensator branches._{}

The offset α in (17) must be less then 0.5 as to assure L_{p} positive.

The frequency-domain approach is a well known method (see M. Pasko [4-6]).

The counterpart of (6) in frequency-domain is

where:

,—frequency response of compensator and receiver susceptances,

—harmonic number.

For the elementary compensator branch (L_{C} in series) the branch elementary susceptance is

Thus the formula of reactances balance (19) (see Figures 6 and 7) takes form

The Equation (20) is the counterpart of (11) transformed to optimization task (18).

Then in the particular case of the R_{L} in series receiver we get the set of M linear equations for all L_{m} of compensator branches

where n—harmonic numberm—branch number.

Comparing (18) with (21) we can see that both formulas are the system of linear equations, but in (18) we have

the impulse response instead of the frequency response of the receiver.

Let consider the RL load in series for which: P = 500 [W], T = 0.02 [s], ω_{1} = 314 [rad/s], AL = T/τ = 10, L_{o} = 25.7 [mH] and M = 10, τ—time-constant of the load.

Effective compensation is up to 5-th harmonic (see Figures 6 and 7).

The integral in the right side of (18) was calculated numerically using 21 samples and time samples was shifted by T/21/2 due to singularity problem.

The frequency response method used until now to synthesis L_{C} compensators and considered the only one [_{C} parameters can be found with simple optimization techniques for linear system. The only difference is that in (18) we can use directly the impulse response of the load (differentiated step response) instead of harmonic analysis. Moreover it is the first time in literature that the time-domain reactive compensator design is presented.