Electric circuit analysis by means of optimization criteria Part I the simple circuits

Transcription

1 BULLETIN OF THE OLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 5, No. 4, 004 Electric circuit analysis by means of optimization criteria art I the simple circuits M. SIWCZYŃSKI* and M. JARACZEWSKI** Institute of Circuits and Control They, Cracow University of Technology, 4 Warszawska St., 3 55 Kraków, oland Abstract. One of the main problems of electrical power quality is to ensure a constant power flux from the supply system to the receiver, keeping in the same time the undisturbed wave fm of the current and voltage signals. Disttion of signals are caused by nonlinear time varying receivers, voltage changes power losses in a supply system. The wave-fm of the voltage of the source may also be defmed. This study seeks the optimal current and voltage wave-fm by means of an optimization criteria. The optimization problem is defined in Hilbert space and the special functionals are minimized. The source inner impedance operat is linear and time-varying. Some examples of calculations are presented. Keywds: electrical circuits, power they, optimization in Hilbert spaces.. Introduction The transmission of electric power from the source to the load is perfmed in one loop circuit, and despite its simplicity there are many unsolved problems related to it. They result from the fact that in the energy transpting process there are two signals involved the voltage and current, and they can produce the same power with various, sometimes unpredictable, wave fms. Applying some quality criteria we can find the optimal wave shape of the voltage and current signals. The power quality discipline usually deals with this crucial problem. Let us consider a simple one loop circuit depicted in Fig.. Fig.. The power transfer circuit Such a circuit serves to transpt power from the source e to the receiver. The problem, which seems to be easy, is to find such a signal i to assure a given prescribed power and it can be solved only by means of optimization technique. Thesourcepowerequation =(e, i ( obviously leads to an abundance of solutions since there is an infinite amount of such signals e and i which lead to the same dot product, equal to active power.inder to choose the unique solution from their infinite set, we * e-3@pk.edu.pl ** jaracz@usk.pk.edu.pl must put one me optimum condition: i =(i, i min ( The task ( ( has only one solution which we get by minimizing the Lagrange functional f(i, λ =(i, i+λ (e, i] min (3 where λ is real. Indeed, calculating the variation of the functional (3 caused by the variation of current δi weget: f(i + δi, λ f(i, λ =(i, δi (λe, δi+(λri, δi+(δi, δi =(i λe, δi+(δi, δi. (4 From the previous equation it follows that the sufficient and necessary condition f the minimum (3 exists i λe =0 i λ =0.5λe. (5 Nowit is necessary to choose one i λ from the abundance of solutions accding to the assumption ( i.e. =(e, 0.5λe =0.5λ(e, e =0.5λ e thus λ = e. So the optimal current signal meeting condition ( ( is i =: i opt = e. (6 e. One loop circuit power distribution In me complex problems the source has a passive linear inner impedance determined by the Z operat. In such a circuit the relation between u and i is determined by U = e Zi 359

2 M. Siwczyński, M. Jaraczewski which we can use to calculate the balance of the circuit power =(u, i =(e, i (Zi,i. If we notice that (Zi,i=(i, iz, where * stands f conjugating operation, we get =(u, i =(e, i (Ri, i (7 where R =0.5(Z + Z (8 is a positively defined linear operat. The power balance consists of three power flux: (e, i the power of the source, Z =(Ri, i thepower losses in the source, the developed absbed power. The (Ri, i is positive f any current signal (is positively defined whereas the two others can be both positive negative it is depicted in Fig.. Fig.. ower distribution The Eq. (7 with the unknown current vect also has infinitely many solutions. But the optimization task (i, i min (9 (e, i (Ri, i = as it is easy to prove, will have the unique solution i. We can find it by minimizing the functional f(i, λ =(i, i+λ(ri, i (e, i+ ] min (0 which difference δf(i,λ=f(i + δi,λ f(i,λ =(i,δi +(λri,δi (λe,δi +(δi,δi +(λrδi,δi =((i + λri λe,δi+(δi + λrδi,δi ( is positive f all δi. It will be true when the current signal i meets (i + λri λe =0 ( + λri =0.5λe ( where: denotes an identity operat, 0 zero signal. The Eq. ( includes ( + λr operat w ith a λ parameter. The same operat appears in a quadratic fm fming the second part of the increment (. F λ that ( + λr operat is positively defined, there exists a solution of ( and simultaneously it is the minimum point of (0. The λ f which the solution of ( i λ exists also determines the function F (λ =(e, i λ (Ri λ, i λ (3 But there are also such values of λ f which F (λ is undefined. To find them we must use the operat spectrum of R. The operat spectrum of R is the set of λ such that the ( + λr operat is non-reversible. It means that the equation (λ Rx =0 has a non-zero solution. Hence multiplying this equation by the scalar x weget: λ x =(Rx, x (4 then the operat R spectrum consists uniquely of the positive real numbers, because the R is a self-adjoint, positively defined operat. It results from (4 that the spectrum is a non-empty closed and bounded set because the R is bounded. In der to study F (λ function we must calculate the derivative of i λ signal (the i λ exists in the determinate set of F (λ, by differentiate equation ( in the λ direction ( + λri λ =0.5(e Ri λ (5 From (3 follows the derivative fmula f F (λ: F (λ =(e, i λ (Ri λ, i λ=(e Ri λ, i λ =(( + λri λ, i λ the resulting quadratic fm is positively negatively definite, only when ( + λr is positively negatively definite i.e. when the functional (0 has maximum minimum value. It follows that the F (λ function has two branches: ascending and decreasing. F the prescribed power < max drawn from the source the equation F (λ = (6 has two solutions of λ multipliers f which the functional (0 has respectively: a minimum f λ from the ascending branch of F (λ and maximum from decreasing one. The function F (λ with the power fluxes f appropriate λ ranges is depicted in Fig. 3. It entails the existence of two extreme source currents: i min and i max, which minimize maximize respectively the functional (. From the practical point of viewme imptant is the i min current. The F (λ function shows the relation between the delivered power of the source and the λ coefficient and it can be called a power characteristics of the source. 360 Bull. ol. Ac.: Tech. 5(4 004

3 Electric circuit analysis by means of optimization criteria art I Fig. 3. The power characteristics of the source Monotonic branch a ascending, b decreasing, c singular The me detailed examination of the F (λ function shows that f λ =0 (i λ =0,F (0 = 0, az (i λ = 0.5e, F (0 = 0.5 e the function is zero and has a positive derivation. Whereas f λ ± (i λ =0.5R e, F (λ =0.5(R e, e = max which also results from the limits of the functional (0 f(i,λ (Ri, i (e, i+ ] min f(i,λ (Ri, i (e, i+ ] max and using the variation method we get Ri =0.5e. It is equivalent to the solution of the so called receiver matching problem accding to the maximum power given and it is discussed in the literature f various particular current spaces and impedance operats of the source. F the lossless source with the zero loss operat R : R =0, Eq. ( gives i λ =0.5λe and F (λ =0.5 e λ which means that the power characteristics of the source is linear and unlimited. The lossy source has always the power characteristics limited by the maximum power max. Example. Let us find the power characteristic of the source with the time varying inner impedance (Fig. 4. Fig. 4. Theequivalent circuit diagram of thesourcewith thetime varying resist-inductance inner impedance In this case Z = r(t+l(t d ] and it affects a current signal accding to the fmula Zi(t = r(ti(t+l(t di ]. In der to define the conjugated operat Z wecalculate the quadratic fm (Zi, i = r(ti(t] + l(t di i(t. After applying the Liouville s transfmation in integration by parts of the second component we get (Zi, i = r(ti(t] dl(ti(t] i(t =(Z i, i of which results the conjugated operat of the fm Z = r(t l(t d dl(t ]. Therefe the loss operat has the fm R =0.5(Z + Z = r(t 0.5 dl(t ]. (7 The operat equation to calculate the optimal current signal ( has nowthe algebraic fm ( +λ r(t 0.5 dl(t ] i(t =0.5λe(t. Thus the power characteristics of the source F (λ is F (λ = λ +0.5ρ(t + λρ(t] e(t] where ρ(t =r(t 0.5 dl(t. Analyzing the fegoing denominat of the integral we find the singularities f λ<0 when f the positively defined two terminal netwks is r(t 0.5 dl(t > 0 f any t. Example. In the case of a stationary circuit the operat (8 is defined with the single variable function of s R(s =0.5(Z(s+Z( s where Z(s is an dinary operat function of the source inner impedance, and Z( s is its conjugated operat. It follows from the Liouville s transfmation of a single difference operat (sx(ty(t = x(t( sy]. Thus f the conjugated operat the following fmula is in fce s = s. The operat Eq. ( has then the following fm + λr(s]i(s = 0.5λE(s (8 Bull. ol. Ac.: Tech. 5(

4 where I(s, E(s stand f the two-sided Laplace transfmation of the source current and voltage. Equation (8 instantly gives the solution I λ (s = 0.5λE(s +λr(s. M. Siwczyński, M. Jaraczewski The power characteristics of the source (3 is defined by the arsevall s fmula: F (λ =(e, i λ (Ri λ, i λ = πj j j πj = πj πj = πj = 8πj E(jωI λ ( jωd(jω j j Re(s=0 Re(s=0 Re(s=0 Re(s=0 R(jωI λ (jωi λ ( jωd(jω E(sI λ ( sds R(sI λ (si λ ( sds ( +λr(s λr(s + λr(s] E(sE( sds ( +λr(s + λr(s] E(sE( sds. From the Jdan lemma the integrals along the imaginary axis are substituted by the integrals along the contour including the right left half plain i.e. the contour consists of an imaginary axis and an appropriate semi circle with the (0,0 center point and a radius approaching to infinity. These contours are marked as and. Thus F (λ = 8πj ( +λr(s + λr(s] E(sE( sds. (9 The fmula (9 is suitable f a rational function in integral, when we can use the Cauchy integral fmula. The derivative of F (λ has the fm F (λ = 4πj ( E(sE( s + λr(s] 3 ds. The root λ of the Eq. (6 f the prescribed power (, max is on the ascending branch of F (λ and determines a particular case of the current signal i.e. the optimal current. The root can be found both graphically analytically using the Newton s method. λ k+ = λ k + F (λ k F (λ k This procedure is always convergent f λ>0 = Γ (λ k. (0 Fig. 5. Γ (λ function subsequent and consecutive iteration points It results from the shape of the Γ (λ function its graph and consecutive iteration points are shown in Fig. 5, ]. 3. ower analysis of the circuit by means of other evaluation functionals In der to calculate the optimal source current other practical functionals, besides (9, can be used. Introducing the prescribed reference signal i 0 we minimize the functional of the current deviation i = i i 0. We can create optimization task in the fm ( i, i min (e, i (Ri, i = ( The task ( has a similar solution to the task (9 considered in the preceding chapter. By analogy we can prove that the minimum condition of ( result from equation ( + λri =0.5λe + i 0. ( The power characteristics of the source (3 is similar to the previous one from the Section. From the practical point of viewit is imptant to minimize the voltage deviation on the source terminals referred to the prescribed signal u 0 : u = u 0 u. It leads to the following minimizing problem: ( u, u min (e, i (Ri, i =. (3 If we note that u = Zi e where: e = e u 0, Z the inner source impedance operat, we can fm the Lagrange functional depended on the current f λ (i =( u, u+λ(ri, i (e, i+ ] min (4 and its deviation (noting that δ( u = Zδi gives δf λ (i =f λ (i + δi f λ (i =(Z u +λri λe,δi +((Z Z + λrδi,δi... (5 36 Bull. ol. Ac.: Tech. 5(4 004

5 Electric circuit analysis by means of optimization criteria art I From (5 results the sufficient and necessary condition f the minimum (4 to exist. Z u + λri =0.5λe (Z Z + λri =0.5λe + Z e. (6 The operat Eq. (6 gives the optimal current f the (4 criterion i λ =(Z Z + λr (0.5λe + Z e. (7 The characteristics F (λ is defined in λ points f which the solution of (6 exists i.e. when the invert operat (Z Z + λr exists. The function F (λ has a derivative F (λ =(e Ri λ, i λ where i λ is the current derivative in λ and is calculated by differencing the Eq. (6 (Z Z + λri λ =0.5(e Ri λ. (8 From (7, (8 it results that F (λ is a quadratic fm F (λ =((Z Z + λri λ, i λ (9 with (Z Z + λr operat, which also appear in (5. The positive definition of this operat means the monotonically ascending branch of F (λ and the existence of the minimum of the functional (4 there. Whereas the negative definition of this operat means the monotonically decreasing branch of F (λ and the existence of the minimum of the functional (4 there. The F (λ function has the following quality f λ =0 (Z Z Z e = Z e = Ye F (0 = (e, Y e (Y RY e, e λ F (λ 0.5(R e, e = max The graph 6 shows F (λ function Fig. 6. The power characteristic of a source The calculated optimal current signal (7 f (3 condition can be written in a fm i opt = G opt e + j 0 (30 where: G opt =0.5λe(Z Z + λr, j 0 =(Z Z + λr Z e. G opt self-adjoint, positively defined linear operat, j 0 additional current signal. This result is depicted in Fig. 7. Fig. 7. The equivalent circuit diagram of the optimal current It is wth considering the particular optimal solutions when the source is lossless i.e. when R is the zero operat. In that case the Eq. (6 is reduced to Z Zi =0.5λe + Z e (3 and can be solved directly i λ =0.5λYY e + Y e (3 where Y = Z is the inner admittance of the source. The power characteristics F (λ =0.5λ(Y Y e, e+(y e, e (33 is then an affine function. The solution of (6 is a point λ = (YY (34 e, e where =(Y e, e. So the optimal current of a lossless but flexible source, giving the closest voltage to u 0, is given by the fmula: i opt = (YY e, e YY e + Y e. (35 These results even get reduced f u 0 = e, when e =0 and the fmulas (9 (34 take the fm i λ =0.5λY Y e The F (λ function is linear F (λ =0.5λ(Y Y e, e. (36 Thus the optimal source current giving the minimal voltage drop on terminals is i opt = (YY e, e YY e. (37 It is wth comparing the fegoing current fm to the result (6, f the current nm criterion i min : i opt = (e, e e which exists in the literature as a Fryze current,, 3 8, 9] and it is a particular case of the current (37, which is easy to see after substituting YY e = e. Therefe the optimal current assuring the minimal inner RMS voltage drop(( u, u min is given by i opt = (e,e e. (38 Bull. ol. Ac.: Tech. 5(

6 M. Siwczyński, M. Jaraczewski Example 3. It is easy to showa definite example of the result (37. Let s take the two terminal netwk with a time invariant and reactive elements. Its impedance function is then an odd function Z(s :=sx(s (39 where X(s even rational function. Thus the loss function is R(s =0.5(Z(s+Z( s =0.5sX(s +( sx((s ] = 0. The (39 operat has the inversion (admittance then Y (s = sx(s Y (sy (s =Y (sy ( s = (Y Y e, e = πj = π j j s X(s ] E(sE( s s X(s ] ds E(jω ω X( ω dω. (40 ] Applying (39 and (40 in (37 we get the two sided Laplace transfmation of the optimal current I opt (s = π E(jω ω X( ω ] dω E(s s X(s ]. (4 Given result (4 can be reduced e.g. f the almost periodic signals. In that case the function F (s hasthe values Z(jω=jωX( ω where ω {ω n : n =0,,,...} is a countable set of harmonic components making signal e(t: ] e(t =real En e jωnt = n=0 n= E n e jωnt (4 where: E n = En, ω n = ω n. (43 Defining the inner product of the almost periodic signals (u, i = lim T T T 0 u(ti(t (44 and using (4 in (44 we get a particular fm of the arsevall fmula: (u, i = = n,m= n= U ni m lim T T n=0 T 0 e j(ωn+ωmt U n In = real U n In = real UnI n n=0 where I n = I n because lim T T T 0 e j(ωn+ωmt = { dla n + m =0 0 dla n + m 0. In such a case the quadratic fm (40 gives (YY E n e, e = ωnx n n=0 where: X n = X( ωn whereas the spectrum of the almost periodic current signal a counterpart of the current (4 isgivenby I opt n = m=0 E m ω m X m E n ω n X n. (46 Note that the fms (45 and (46 remain valid f periodic signals. 4. ower analysis of the circuit by means of multi criterion optimization tasks In the previous chapter we have fmulated some practical quality criterions concerning power transmission in a one loop circuit. In der to satisfy many individual receivers at the same time, we need to fm some mixed optimization criterions to give reasonable compromises. They can be e.g. the two criterion optimization tasks like: (i, i min (e, i (Ri, i = (47 ( i, i =Q (i, i min (e, i (Ri, i = (48 ( u, u =Q where: i = i i 0, u = u u 0 current and voltage deviations, i 0, u 0 prescribed signals of the current and voltage. In the condition (47 Q is the acceptable disttion measure of i regarding i 0 acceptable disttion measure of u regarding u 0 (48. It is obvious that these are not the unique practical optimization conditions of the source signals. As further examples we can fmulate ( i, i min (e, i (Ri, i = (49 ( i, i =Q ( i, i min (e, i (Ri, i = (50 ( u, u =Q These tasks are fmulated crectly and all of theme have a unique solution 60]. The task (47 is equivalent to 364 Bull. ol. Ac.: Tech. 5(4 004

7 Electric circuit analysis by means of optimization criteria art I minimize the Lagrange s functional f λ,µ (i =(i, i+λ(ri, i (e, i] + µ( i, i min (5 which difference in the minimum point is f λ,µ (i =f λ,µ (i + δi f λ,µ (i =(i + λri +µ i λe,δi + (( + µ+λrδi,δi > 0 (5 f any variation of the current signal δi. Thuswegetthe necessary minimum condition ( + µ+λri =0.5λe + µi 0. (53 F the positively defined operat A λ,µ =: + µ+λr (54 the equation has the solution i λ,µ. Then the vect function is defined ] Fλ (λ,µ F (λ, µ = F µ (λ, µ ] (e, iλ,µ (Ri = λ,µ,i λ,µ. (55 (i λ,µ i 0,i λ,µ i 0 The first component of the fegoing function is the source active power and the second is the deviation nm of the current. This function can be called a power-disttion characteristics. The search of λ and µ which determine the optimal current i opt := i λ,µ is perfmed using the following set of equations F (λ, µ = Q = A λ,µ (0.5λ e + µ i 0 (56 ]. (57 It is the counterpart of the power Eq. (6, which appear in the one criterion optimization task. It is proved ] that there exist such and Q that above equation has the unique solution. Example 4. The optimization equation which solves the problem (47 has fm (53 but f lossless source it get reduced to ( + µi =0.5λe + µi 0 (58 and it can be solved directly i =0.5 λ +µ e +µ i 0 + i 0 (59 therefe i = 0.5λ +µ e +µ i 0. (60 The component functions in (55 gives the fm F λ (λ, µ =0.5 e +µ λ +µ (e, i 0+(e, i 0 F µ (λ, µ =0.5 e λ 4λ(e, i 0 +4 i 0 ( + µ. (6 It turns out that f the lossless source the set of Eq. (57 can be unequivocally solved to λ and µ. Ittakes the following fm 0.5 e λ (e, i 0 =(+µ(e, i e λ 4(e, i 0 λ +4 i 0 =4 i ( + µ. (6 The solution of Eq. (6 is ] λ = e (e, i 0 ± (e, i K 0 K µ = ± K 0 K (63 where K 0 and K meet the Schwarz inequality K = e i (e, i > 0 (64 K0 = e i 0 (e, i 0 > 0. (65 These coefficients have a characteristic fm. They stand f the residual nonnative powers in the geometric differences between the active and apparent powers. They can be associated with the reactive powers of the source signals i and i 0 respectively. The minimum point demands that ( + µ > 0 and it assures a positive definition of the operat (54. Then the solution (63 of (6 is unequivocal λ = e (e, i 0 +(e, i K ] 0 K µ =+ K 0 K (66 The optimal current is given by i opt = (0.5λ e i 0 +i 0 +µ = K 0.5λ e + ( KK0 K 0 = ( K K 0 i 0 ] e (e, i 0 + e + ] ( KK0 i 0. Finally it will be convenient to rearrange it to the fm ] i opt = ( KK0 e 0 ( e + KK0 i 0 (67 where: 0 =(e, i 0 power of the ideal source giving the reference signal i 0 ; K 0 = e i 0 0 remaining (reactive power of the ideal source; K = β e i 0 ( 0 (68 β = Q i 0 = i i0 i 0 relative coefficient of the distted current. (69 From the (68 results a certain existence condition of the minimization solution β e i 0 ( 0 0 Bull. ol. Ac.: Tech. 5(

8 M. Siwczyński, M. Jaraczewski β 0 (70 e i 0 From (69 and (70 results one me fm of the condition Q ( 0 e. (7 Inequality (7 can be treated as an existence condition of the solution f the lossless source with the arbitrary assumed signals e, i and scalars, Q. The assumed relative disttion value cannot be too small. The signal (67 is then the minimal RMS current with a prescribed disttion level, giving the prescribed source power. This result can be treated as one me generalization of the so called Fryze current (6 with an additional condition defining the signal disttion level. Example 5. We will consider the minimal RMS current problem of the source with the prescribed voltage disttion (48. The appropriate solution of the optimization equation has the fm ( + µz Z + λri λ =0.5λe + µz e (7 where e = e u 0, u 0 prescribed voltage signals. In a particular case of the lossless source and f u 0 = e (the prescribed voltage equals to the open state voltage we get the problem of finding the minimal RMS current having the prescribed RMS voltage drop in the source. Thus the Eq. (7 reduces to ( + µz Zi =0.5λe. (73 This problem is a little me complex then in the Example 4. In the fegoing equation there is µz Z operat instead of the µ scalar. The Eq. (73 has the solution i λµ =0.5λ( + µz Z e =0.5λK(µe where K(µ =(+µz Z (74 is a linear self-adjoint operat, positively defined f µ>0. F µ =0is K(0 = (identityoperat thus: F λ (λ, µ =0.5λ(K(µe, e F µ (λ, µ =(0.5λ (Z ZK(µe, K(µe (75 From the equation F λ (λ, µ = (76 weget λ (µ = (K(µe, e, from this results the function µ F µ λ (µ,µ]= (K(µZ ZK(µe, e (K(µe, e. (78 A me detailed examination of the function (78 shows that F µ λ (0, 0] = (Z Ze,e e 4 (79 f µ : K(µ µ YY λ (µ (YY e, e µ F µ λ (µ,µ] (YY e, e. (80 Thus we get the function increase F µ λ (0, 0] F µ λ (µ,µ] µ = (Z Ze,e(YY e, e (e, e (YY e, e ( e > 0. (8 The proof of the inequality (8 will be carried out f the convolution operat. In a general situation the use of the spectral they is needed. If the sequence {E n } n=0,±,±,... stands f the appropriate voltage harmonics of the source and {b n } n=0,±,±,... the module components of the Z spectrum then (Z Ze,e(YY e, e (e, e ( ( = b n E n b E m n m m ( ( E n E m n m = a nm E n E m n>m where: ( a nm = b n b + b m b m b = n b m > 0. n b n b m This completes the proof. Then f Q between (YY e, e <Q< (Z Ze,e e 4 (8 there exist a unique root µ which is calculated from (K(µZ ZK(µe, e = Q. (83 (K(µe, e Fig. 8. The root determining process f the equation set ( Bull. ol. Ac.: Tech. 5(4 004

9 Electric circuit analysis by means of optimization criteria art I Therefe the second codinate of the optimum point (λ,µ the root of the power-disttion equations (56 is λ = (K(µ e, e. (84 To complete the F µ λ (µ,µ] investigation it should be proved that it is monotonic and unequivocal, what is carried out in ]. In a periodic steady state µ is the root of the convolution case of Eq. (83 ( Xn En n n,m +µx n = Q (85 E n E m (+µxn (+µx m Then the second codinate of the optimum point (λ,µ is: λ =. (86 n E n +µ X n Where X n stands f the spectrum of the inner source reactance (see ex. 3. The frequency distribution of the optimal current is given by 5. Conclusion In opt = 0.5λ +µ Xn E n. (87 The afe presented results are connected with the problem of matching the load to the source. This issue is presented in many disciplines connected not only with electrical energy transptation. It consists in the appropriate choice of the load to obtain the maximum power from the source. In the electrical domain this problem is reduced to the choice of an appropriate current signal which maximize source power functional i.e. (e, i (Ri, i max. Its solution is well known and leads to the operat equation,, 7 0]: Ri =0.5e. This is used to determine the power balance equation max =(e, i dop 0.5(e, i dop where i dop =0.5R e is the matching current signal. From the fegoing power balance it results that exactly one half of source power is lost. This fact makes such matching method extremely impractical. Nevertheless, the knowledge of possible maximum source power is useful. It is possible to fmulate a lot of me useful matching problems. In the case of e.g. stiff sources keeping invariable voltage signal regardless of the current, there exists the unique current signal giving prescribed source power. This result is of a great practical imptance and is known as the Fryze current. In the present article this signal is defined in the (6. However the problem becomes me sophisticated when we take in consideration the inner impedance Z operat of the source. It causes the source becomes lossy and flexible. Nevertheless we can still get a great variety of appropriate optimal currents using the minimum criteria, concerning the power balance of the circuit shape restriction of signals. These results can be applied to improve power quality when transmitting it to many individual receivers. Obviously, to get the optimal currents, some modification of the load is needed. It is carried out by compensat circuits but this is the problem which is not under the consideration in this article. References ] M. Siwczyński, Optimal methods in the they of electric circuits, Electrical Engineering, ublishing Houseof Cracow University of Technology, Kraków, 995 (in olish. ] M. Siwczyński, ower Engineering Circuit They, Kraków, 003 (in olish. 3] L. S. Czarnecki, Interpretation, identification and modification of power properties of one-phase circuits with deflection course, Scientific Exercises of Silesian University of Technology, Electrotechnics 9, Gliwice, 984 (in olish. 4] L. S. Czarnecki, ower and compensation in circuits with periodic current and voltage courses part 3, ower roperties of Linear One-phase Circuits, 9 40 (998, (in olish. 5] L. S. Czarnecki, ower and compensation in circuits with periodic current and voltage courses part 6, owers in Uncompensated Three-phase Circuits with eriodic Sinusoidal Current and Voltage Courses, 7 0 (000, (in olish. 6] L. S. Czarnecki, ower and compensation in circuits with periodic current and voltage courses part 7, ower roperties of Linear Three-wire, Three-phase Circuits, 8 (000, (in olish. 7] M. asko, Selection of compensats optimizing wk conditions of one-phase and multi-phase voltage sources with periodic deflection courses, Scientific Exercises of Silesian University of Technology, Electrotechnics 35, Gliwice, 994 (in olish. 8] M. asko and J. Walczak, Optimization of power-quality properties of electrical circuits with periodic non-sinusoidal courses, Scientific Exercises of Silesian University of Technology, Electrotechnics 50, Gliwice, 996 (in olish. 9] J. Walczak, Optimization of power-quality properties of electrical circuits in Hilbert spaces, Scientific Exercises of Silesian University of Technology, Electrotechnics 5, Gliwice, 99 (in olish. 0] C. A. Desoer, The maximum power transfer theem f n-pts, IEEE Trans. CT-0, 8 30 (979. Bull. ol. Ac.: Tech. 5(