Root
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Conference Proceedings
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3rd IFAC Workshop "PERIODIC CONTROL SYSTEMS" (PSYCO'07)
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Self-excited oscillations systems with switchings: theory, calculations and investigations.
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Self-excited Oscillations Systems with Switchings:

Theory, Calculating,Investigating.

Yurii Dusavitskii

Russia 129090 Moskva

by-street Vasnetsova 3 - 25

688-35-79 Dusav@yandex.ru

Abstract

It is accepted for the systems: autonomy, quick and slow motions, the feedback switchings with hysteresis, the variable structure on the intervals.

The mathematical description is m differential equations of the n-order, which change one to another periodically

Some values of the intervals during (tk ) are put under the cycles equations, which are changed into algebraic instead of trascendentale.

Then the surfaces for cycles (SC) are calculated and constructed. The switching conditions are put under SC. The intersections of the curves make the parameters of oscillations.

It is proved the existence and the unique of SC.

They have special points and lines and may by

Multisheeted. It is possible to find the

required switching conditions. There are two numerical examples.

Key words: self- excited oscillations, cycles, switching.

The self-excited oscillations system is the system in the regime of the self-excited oscillations.

The self-excited oscillations are the nondamping oscillations, which exist in physical system, when the external influence as function of time is absent.(Russian academician A.A.Andronov 1928). There is very much manuscripts, which show the physical nature of self-oscillations, but all of them gave the qualitative character. The methods of calculations are: the points transformations and harmonic balance. The first of it is very low-powered, the second is approximative and cannot take into conesideration the variable structure.

The investigation of big number of self-oscillations systems of variable principles shows some properties of them: 1) division of periodic motion to quick and slow motions , 2) existence of the histeresis quick motions. The nonstable element or the nonstable system in itself are the reason of it. 3) There is the variable structure on the intervals. 4)The constant connecting up the source of energy. 5) The motion of slow intervals can have description in the form of system of liner differential equations with constant coefficients. In this paper for calculation and investigation of self-excited oscillations was presented the METHOD OF THE SURFASES FOR CYCLES.

The surfaces for cycles represent dense sets of points whose coordinates are equal to the final conditions for each from partial uninterruptible motions which form a possible periodic motion.

Superpositions on these surfaces concrete feedback switching conditions give the parameters of the concrete self-exited oscillations. The feature of the method is the refusal from the current time operations and from current coordinates

In accordance with this determination we have coordinates of the surfaces for cycles:

x11 -the value of the coordinate x1 at the end of the first interval

x21 - the value of the coordinate x2 at the end of the first interval

x12 - the value of the coordinate x1 at end of the second interval

x21 (t) - the coordinate x2 as a function of time at the first interval

x12 (t) - the coordinate x1 as a function of time at second interval and so on.

In a common case a system’s mathematical description is

X˙=AX+C

X˙=BX+D m XRn (1)

X˙=GX+Q

Where m is the quantity of intervals of the uninterruptable motion.

A,B…G are constant ordinary square matrixes; C,D…Q are constant columns, at least two of them are not equal one to other. S1(X),S2(X),Sm(X) are the surfaces of rapid switchings. The final values of one-interval coordinates are equal to the initial values of subsequent interval coordinates; i.e., "the sewing together" of solutions is occurred. For all possible periodic solutions of system (1) we have the equations of cycles in the matrixe form

Х1 = ехрАt1 (Xm + A-1 C) – A-1 C the first interval

X2 = expВt2 (X1 + B-1 D) – B-1 D the second interval

Xm = expGtm (Xm-1 + G-1 Q) – G-1 Q m-interval

which solution may be very complicated, therefore it is necessary to use the other classical form.

xik(tk)=nj=1 xj(k-1)* fijk(tk)+ik(tk) (2)

Where i = 1…n is the number of the coordinate, k is the number of the interval, k = 1…m, xik is the value of the coordinate i at the end of the interval k, xj(k-1) is the value of the coordinate j at the end of the interval (k-1)(foregoing), tk – the duration of the interval, f and may be taken from the handbook

With п = 2 and а11 а12 с1

А = С =

а21 а22 с2 We have, for example

f11=[1/(2-1)][(2-a11)exp1t–(1- a11)exp2 t ]

f12 = [ a12 /( 2 - 1 )] (exp2 t - exp1 t )

f21 = [ a21 /(2 -1 )] ( exp2 t - exp1 t)

f22 = [1/(2- 1)][(2-a22)exp1t- (1-a22)exp2 t]

1 = -(k1 2 + c1 )exp1 t /(2 - 1 ) + [ (k1 1 + c1 )/(2 -1 )]exp2 t + k1

2 = -[(k2 2 + c2 )/(2 - 1)]exp1 t + [ (k2 1 + c2 )/(2 - 1)]exp2 t + k2

k1 = (-a22c1 + a12 c2)/ ( a11 a22 - a12 a21 )

k2 = (-a11c2 + a21 c1)/ ( a11 a22 - a12 a21 )

where λ1,2 are the eigenvalue of A

The systems of equations (1) are closed ( avoidant) systems, because the solution of each equation is the function of the solution of each other equation. Therefore with k = 1 the coordinate xjm must be taken instead of xj(k-1)

With help of (2) we have

Х = -F-1 (3)

where

X T =[x11, x21… xn1,x12, x22…xn2…xnm ]

T =[11, 21…n1, 12, 22…n2 nm ]

If m = 2

│ [-1] [F1 ] -1 0

F = │ [F2 ] [-1 ] [-1] = 0 -1

If m = 3

[-1] [ 0 ] [ F1 ] -1 0 0

F = [F2 ] [-1 ] [0 ] [-1] = 0 -1 0

[0 ] [F3 ] [-1] 0 0 -1

f11k f12k f1nk 0 0 0

Fk = f21k f22k f2nk [0] = 0 0 0

fn1k fn2k fnnk 0 0 0

and so on

In equations (3) several tk must be replaced thus, the system of linear algebraic equations will be obtained. If the number of switching m = 2, the solution(3) gives values of X1 and X2 corresponding to the taken t1 and t2.

XT1 = [x11,x21,x31..xn1],

ХТ2 =[x12,x22,x32..xn2]

To repeat the calculation for other values of t1 and t2.

The surfaces for cycles are done on coordinate systems in which switchings will be.

Further, all points X1 corresponding to the fixed values of t1 (t1 = const) but to the different values of t2 (t2 = var) must be connected among themselves in increasing or decreasing order of t2. Such lines to be obtained are called isochrones t1 . The points X1 corresponding to the fixed values of t2 (t2 = const) and the various values of t1 (t1 = var) are also connected among themselves in increasing or decreasing order of t1 . Such lines to be obtained are also called isochrones t2 . The surface being formed will be the surface for cycles P1 . The coordinate grid of t1, t2 forms the second coordinate system!!! The surface P2 is constructed analogously.

If the obtained parts of the surfaces P1 and P2 have intersections with S1 and S2 , then they are sufficient for finding the periodic solution. Otherwise, it is necessary to broaden the row of t1 and t2 or reduce the step between t1 and t2. To reduce cumbersome of the calculations, it is important to select an optimal quantity and values for t1 and t2 .We can recommend t1 =(0.5, 1, 2 )/λ and t2 = (0.5, 1, 2 )/μ, where λ and μ are mean square values for λi and μi (the eigenvalue of A,B λi = α±jβ, µi = γ ±jξ ).

For starting of the calculation it is sufficient to take 3 or 4 values for t1 and t2 and construct the surfaces xk(xl),xp(xq).

On the received surfaces the switching conditions must be superposited.

Further, find the periodic solution. Take the first switching condition as the function of xk and xl (curve v11), and the second it as function of xp , xq curve v21. Curves v11 and v21 with the help of the grid t1 t2 are depicted on the surfaces P2 and P1 so that the curves w12 and w22 are obtained. The intersections of v11 and w22 or v21 and w12 give parameters of the periodic solution(t1n, t2n, xkn, xln, xpn, xqn) . To obtain the required calculation accuracy we can use several new pairs of t1 and t2 located in the vicinity of the first approximation and repeat the calculations. In the case of a greater dimention

( n 2 ) we propose using of the PARTIAL surfaces for cycles which may be constructed in the system of any two or three coordinates (components of the vector X1 and X2 ).

The solutions obtained with any switching conditions, because it is done graphically.

The back problem: find the required xkn, xln, xpn, xqn for receiving t1n, t2n

Take the surfaces xkn(xln) and xpn(xqn) and isochrones t1n t2n. The intersections of isochrones give xkn, xln, xpn, xqn.

There are equations of the limit isochrones, which determine the regions of existence of the surfaces for cycles.

There are the examples of the surfaces for systems of 2, 3, 4 order.

If m = 2 ,n=2 SC allow verify the stability of received periodic solution.

Appendix 1

To the paper ”Hybrid Mode-Switched Control of DC-DC Boost Converter Circuits” Pawan Gupta and Amit Patra, TCAS2 11 2005

The converter’s circuits are the wonderful example for using of the “Method of surfaces for cycles”

The problem: find x(t).

Mark: Uoutput = x1, the inductor current = x2 (I got used it) m = 3, n = 2

For Mode 1(the first interval), S is ON.

X11 = x13 expλ1t1

X21 = -c2/λ2 (1-expλ2t1)

t1- the duration of Mode1

λ1 =-1/C(R+rc), λ2 = -(rL+rS)/L, c2 =U1/L

For Mode2(S is OFF,D is ON)

(the second interval)

b11 b12 d1 Х•= ВХ+D B = D = . b21 b22 d2

x12

X =

X22

b11 = -1,58E4, b12 = 3E5, b21= -2,12E5, b22 = -0,318E5

µ1,2 =-2,88E4 ± 1,84E5j d1 = 6,4E4, d2 = 1,06E5

For Mode3(S is OFF, D is OFF)(the third interval) x13 = x12 expνt3 ν = -1/C(R+rC)

We have the equations of cycles:

X11 = x13 f111

X21 = g21

-x12 + x11 f112 + x21 f122 = - g12

X11 f212 + x21 f222 = -g22

The switching conditions are:

From Mode1 to Mode2 x21 = iref

Mode2 Mode3 x22 = 0

Mode3 Mode1 x13 = Uref

For construction of the surface for cycles we take the several values of t1 and t2, which are near 1/ µ (1,2,3…µsec). After the replacements t1 and t2 into equations and solving them we have the surface for cycles x12(x21), which is formed with special lines – isochrones t1 = const and t2 = const

For finding the periodic solution we plot the switching conditions ( for example iref = 1,11A, Uref=12v).

We have:x11n = 11.892, x12n =11.978, x13n =11.93, x21n =1.105, t1n =1.06µsec, t2n = 0.75µsec

t3 = ln(x11n/x13n) = 1.338µsec

The equations of cycles are correct for current time too, therefore we have:

Mode1 U11(t) = 11.93 exp(-3*103 t1) 0 < t1 < 1.06µsec

Mode2 U12(t) = 4.929 + exp(-2.88 104 t2) ( 6.963 cos 1.84 105 t2 +

2.087 sin1.84 105 t2) 1.06µsec < t2 < 1.81 µsec

Mode3 U13(t) = 11.978 exp(-3 103 t3) 1.81µsec < t3 < 3.148µsec

Appendix 2.

The example of the solving problem.

Given:

X• = AX (1.1)-the first interval

X• = BX +D (1.2)- the second interval

п = 3, m = 2 , λ1,2 = - 260 ± j1000 , λ3 = -500, μ1 = -100, μ2 = -300, μ3 =-500, α = -260, β = 1000

0 1 0 0

A = 0 0 1 D = 0

-5,34Е8-1.32Е6-1,02Е3 15E6

0 1 0

B = 0 0 1

-15Е6 -2,3Е5 -900

The switching conditions (voluntary, in table form)

From(1.2)to(1.1)

x12= 0.217 0.2175 0.225 0.2375

x22 = 54 59 61.6 63.5

From (1.1) to (1.2)

x11 = 0.1 0.2 0.3 0.36 0.5

x21 = -85 -54 -35 -32 -30

Find: x11(t), x12(t)

The answer: may be a chaos with limited cycle

x11 (t) = expα t (0.1832 Sinβt – 0.03 Cosβt ) +0.255 expλ3 t

х12(t) = 1 – 3,14 expµ1t + 4,18expµ2t -1,811expµ3t

The during of the first interval

1.3mc, the second- 13mc

Yu.Ya. Dusavitskii “A switched Stabilizer: an Investigation Usinig the Method of Surfaces for Cycles”

Electricity

12 2003 (in Russian)page50-57

Theory, Calculating,Investigating.

Yurii Dusavitskii

Russia 129090 Moskva

by-street Vasnetsova 3 - 25

688-35-79 Dusav@yandex.ru

Abstract

It is accepted for the systems: autonomy, quick and slow motions, the feedback switchings with hysteresis, the variable structure on the intervals.

The mathematical description is m differential equations of the n-order, which change one to another periodically

Some values of the intervals during (tk ) are put under the cycles equations, which are changed into algebraic instead of trascendentale.

Then the surfaces for cycles (SC) are calculated and constructed. The switching conditions are put under SC. The intersections of the curves make the parameters of oscillations.

It is proved the existence and the unique of SC.

They have special points and lines and may by

Multisheeted. It is possible to find the

required switching conditions. There are two numerical examples.

Key words: self- excited oscillations, cycles, switching.

The self-excited oscillations system is the system in the regime of the self-excited oscillations.

The self-excited oscillations are the nondamping oscillations, which exist in physical system, when the external influence as function of time is absent.(Russian academician A.A.Andronov 1928). There is very much manuscripts, which show the physical nature of self-oscillations, but all of them gave the qualitative character. The methods of calculations are: the points transformations and harmonic balance. The first of it is very low-powered, the second is approximative and cannot take into conesideration the variable structure.

The investigation of big number of self-oscillations systems of variable principles shows some properties of them: 1) division of periodic motion to quick and slow motions , 2) existence of the histeresis quick motions. The nonstable element or the nonstable system in itself are the reason of it. 3) There is the variable structure on the intervals. 4)The constant connecting up the source of energy. 5) The motion of slow intervals can have description in the form of system of liner differential equations with constant coefficients. In this paper for calculation and investigation of self-excited oscillations was presented the METHOD OF THE SURFASES FOR CYCLES.

The surfaces for cycles represent dense sets of points whose coordinates are equal to the final conditions for each from partial uninterruptible motions which form a possible periodic motion.

Superpositions on these surfaces concrete feedback switching conditions give the parameters of the concrete self-exited oscillations. The feature of the method is the refusal from the current time operations and from current coordinates

In accordance with this determination we have coordinates of the surfaces for cycles:

x11 -the value of the coordinate x1 at the end of the first interval

x21 - the value of the coordinate x2 at the end of the first interval

x12 - the value of the coordinate x1 at end of the second interval

x21 (t) - the coordinate x2 as a function of time at the first interval

x12 (t) - the coordinate x1 as a function of time at second interval and so on.

In a common case a system’s mathematical description is

X˙=AX+C

X˙=BX+D m XRn (1)

X˙=GX+Q

Where m is the quantity of intervals of the uninterruptable motion.

A,B…G are constant ordinary square matrixes; C,D…Q are constant columns, at least two of them are not equal one to other. S1(X),S2(X),Sm(X) are the surfaces of rapid switchings. The final values of one-interval coordinates are equal to the initial values of subsequent interval coordinates; i.e., "the sewing together" of solutions is occurred. For all possible periodic solutions of system (1) we have the equations of cycles in the matrixe form

Х1 = ехрАt1 (Xm + A-1 C) – A-1 C the first interval

X2 = expВt2 (X1 + B-1 D) – B-1 D the second interval

Xm = expGtm (Xm-1 + G-1 Q) – G-1 Q m-interval

which solution may be very complicated, therefore it is necessary to use the other classical form.

xik(tk)=nj=1 xj(k-1)* fijk(tk)+ik(tk) (2)

Where i = 1…n is the number of the coordinate, k is the number of the interval, k = 1…m, xik is the value of the coordinate i at the end of the interval k, xj(k-1) is the value of the coordinate j at the end of the interval (k-1)(foregoing), tk – the duration of the interval, f and may be taken from the handbook

With п = 2 and а11 а12 с1

А = С =

а21 а22 с2 We have, for example

f11=[1/(2-1)][(2-a11)exp1t–(1- a11)exp2 t ]

f12 = [ a12 /( 2 - 1 )] (exp2 t - exp1 t )

f21 = [ a21 /(2 -1 )] ( exp2 t - exp1 t)

f22 = [1/(2- 1)][(2-a22)exp1t- (1-a22)exp2 t]

1 = -(k1 2 + c1 )exp1 t /(2 - 1 ) + [ (k1 1 + c1 )/(2 -1 )]exp2 t + k1

2 = -[(k2 2 + c2 )/(2 - 1)]exp1 t + [ (k2 1 + c2 )/(2 - 1)]exp2 t + k2

k1 = (-a22c1 + a12 c2)/ ( a11 a22 - a12 a21 )

k2 = (-a11c2 + a21 c1)/ ( a11 a22 - a12 a21 )

where λ1,2 are the eigenvalue of A

The systems of equations (1) are closed ( avoidant) systems, because the solution of each equation is the function of the solution of each other equation. Therefore with k = 1 the coordinate xjm must be taken instead of xj(k-1)

With help of (2) we have

Х = -F-1 (3)

where

X T =[x11, x21… xn1,x12, x22…xn2…xnm ]

T =[11, 21…n1, 12, 22…n2 nm ]

If m = 2

│ [-1] [F1 ] -1 0

F = │ [F2 ] [-1 ] [-1] = 0 -1

If m = 3

[-1] [ 0 ] [ F1 ] -1 0 0

F = [F2 ] [-1 ] [0 ] [-1] = 0 -1 0

[0 ] [F3 ] [-1] 0 0 -1

f11k f12k f1nk 0 0 0

Fk = f21k f22k f2nk [0] = 0 0 0

fn1k fn2k fnnk 0 0 0

and so on

In equations (3) several tk must be replaced thus, the system of linear algebraic equations will be obtained. If the number of switching m = 2, the solution(3) gives values of X1 and X2 corresponding to the taken t1 and t2.

XT1 = [x11,x21,x31..xn1],

ХТ2 =[x12,x22,x32..xn2]

To repeat the calculation for other values of t1 and t2.

The surfaces for cycles are done on coordinate systems in which switchings will be.

Further, all points X1 corresponding to the fixed values of t1 (t1 = const) but to the different values of t2 (t2 = var) must be connected among themselves in increasing or decreasing order of t2. Such lines to be obtained are called isochrones t1 . The points X1 corresponding to the fixed values of t2 (t2 = const) and the various values of t1 (t1 = var) are also connected among themselves in increasing or decreasing order of t1 . Such lines to be obtained are also called isochrones t2 . The surface being formed will be the surface for cycles P1 . The coordinate grid of t1, t2 forms the second coordinate system!!! The surface P2 is constructed analogously.

If the obtained parts of the surfaces P1 and P2 have intersections with S1 and S2 , then they are sufficient for finding the periodic solution. Otherwise, it is necessary to broaden the row of t1 and t2 or reduce the step between t1 and t2. To reduce cumbersome of the calculations, it is important to select an optimal quantity and values for t1 and t2 .We can recommend t1 =(0.5, 1, 2 )/λ and t2 = (0.5, 1, 2 )/μ, where λ and μ are mean square values for λi and μi (the eigenvalue of A,B λi = α±jβ, µi = γ ±jξ ).

For starting of the calculation it is sufficient to take 3 or 4 values for t1 and t2 and construct the surfaces xk(xl),xp(xq).

On the received surfaces the switching conditions must be superposited.

Further, find the periodic solution. Take the first switching condition as the function of xk and xl (curve v11), and the second it as function of xp , xq curve v21. Curves v11 and v21 with the help of the grid t1 t2 are depicted on the surfaces P2 and P1 so that the curves w12 and w22 are obtained. The intersections of v11 and w22 or v21 and w12 give parameters of the periodic solution(t1n, t2n, xkn, xln, xpn, xqn) . To obtain the required calculation accuracy we can use several new pairs of t1 and t2 located in the vicinity of the first approximation and repeat the calculations. In the case of a greater dimention

( n 2 ) we propose using of the PARTIAL surfaces for cycles which may be constructed in the system of any two or three coordinates (components of the vector X1 and X2 ).

The solutions obtained with any switching conditions, because it is done graphically.

The back problem: find the required xkn, xln, xpn, xqn for receiving t1n, t2n

Take the surfaces xkn(xln) and xpn(xqn) and isochrones t1n t2n. The intersections of isochrones give xkn, xln, xpn, xqn.

There are equations of the limit isochrones, which determine the regions of existence of the surfaces for cycles.

There are the examples of the surfaces for systems of 2, 3, 4 order.

If m = 2 ,n=2 SC allow verify the stability of received periodic solution.

Appendix 1

To the paper ”Hybrid Mode-Switched Control of DC-DC Boost Converter Circuits” Pawan Gupta and Amit Patra, TCAS2 11 2005

The converter’s circuits are the wonderful example for using of the “Method of surfaces for cycles”

The problem: find x(t).

Mark: Uoutput = x1, the inductor current = x2 (I got used it) m = 3, n = 2

For Mode 1(the first interval), S is ON.

X11 = x13 expλ1t1

X21 = -c2/λ2 (1-expλ2t1)

t1- the duration of Mode1

λ1 =-1/C(R+rc), λ2 = -(rL+rS)/L, c2 =U1/L

For Mode2(S is OFF,D is ON)

(the second interval)

b11 b12 d1 Х•= ВХ+D B = D = . b21 b22 d2

x12

X =

X22

b11 = -1,58E4, b12 = 3E5, b21= -2,12E5, b22 = -0,318E5

µ1,2 =-2,88E4 ± 1,84E5j d1 = 6,4E4, d2 = 1,06E5

For Mode3(S is OFF, D is OFF)(the third interval) x13 = x12 expνt3 ν = -1/C(R+rC)

We have the equations of cycles:

X11 = x13 f111

X21 = g21

-x12 + x11 f112 + x21 f122 = - g12

X11 f212 + x21 f222 = -g22

The switching conditions are:

From Mode1 to Mode2 x21 = iref

Mode2 Mode3 x22 = 0

Mode3 Mode1 x13 = Uref

For construction of the surface for cycles we take the several values of t1 and t2, which are near 1/ µ (1,2,3…µsec). After the replacements t1 and t2 into equations and solving them we have the surface for cycles x12(x21), which is formed with special lines – isochrones t1 = const and t2 = const

For finding the periodic solution we plot the switching conditions ( for example iref = 1,11A, Uref=12v).

We have:x11n = 11.892, x12n =11.978, x13n =11.93, x21n =1.105, t1n =1.06µsec, t2n = 0.75µsec

t3 = ln(x11n/x13n) = 1.338µsec

The equations of cycles are correct for current time too, therefore we have:

Mode1 U11(t) = 11.93 exp(-3*103 t1) 0 < t1 < 1.06µsec

Mode2 U12(t) = 4.929 + exp(-2.88 104 t2) ( 6.963 cos 1.84 105 t2 +

2.087 sin1.84 105 t2) 1.06µsec < t2 < 1.81 µsec

Mode3 U13(t) = 11.978 exp(-3 103 t3) 1.81µsec < t3 < 3.148µsec

Appendix 2.

The example of the solving problem.

Given:

X• = AX (1.1)-the first interval

X• = BX +D (1.2)- the second interval

п = 3, m = 2 , λ1,2 = - 260 ± j1000 , λ3 = -500, μ1 = -100, μ2 = -300, μ3 =-500, α = -260, β = 1000

0 1 0 0

A = 0 0 1 D = 0

-5,34Е8-1.32Е6-1,02Е3 15E6

0 1 0

B = 0 0 1

-15Е6 -2,3Е5 -900

The switching conditions (voluntary, in table form)

From(1.2)to(1.1)

x12= 0.217 0.2175 0.225 0.2375

x22 = 54 59 61.6 63.5

From (1.1) to (1.2)

x11 = 0.1 0.2 0.3 0.36 0.5

x21 = -85 -54 -35 -32 -30

Find: x11(t), x12(t)

The answer: may be a chaos with limited cycle

x11 (t) = expα t (0.1832 Sinβt – 0.03 Cosβt ) +0.255 expλ3 t

х12(t) = 1 – 3,14 expµ1t + 4,18expµ2t -1,811expµ3t

The during of the first interval

1.3mc, the second- 13mc

Yu.Ya. Dusavitskii “A switched Stabilizer: an Investigation Usinig the Method of Surfaces for Cycles”

Electricity

12 2003 (in Russian)page50-57