This article is a discussion on issue raised in the article of L. S. Czarnecki [1] where the author consider if it is possible to make current decomposition into active, reactive and unbalanced current in time domain and basing on it build reactive compensators. The current decomposition in time domain was presented in the previous articles [2, 3] and in this article is presented the reactive compensators design in time-domain.

In the article [3] was shown that source-receiver current can be decomposed into active and reactive current in “s” domain i.e. for Laplace transform of signals. Reactive current can be compensated with the reactive compensator. The source-receiver current decomposition is given below:

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 [3].

(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 (Figure 1) the reactive current balance in time-domain states that:

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 Figure 2.

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 Figure 3).

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_LT, A_C = a_CT.

For the receiver shown in Figure 4, the operational admittance is

where, and for the positive poles

, Y^o(s) takes form

where, and the coefficients b₁, b₂ are then

thus

The reactive function (12) is shown in Figure 5. It is necessary to distinguish the two cases when b > 0 and b < 0. When → 0 the curve become straight line.

(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], ω₁ = 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 [1], has its counterpart in time-domain. In both approaches the L_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.