Modeling Method for Autonomous Current Inverters

f (I
i 
, U
j 
, n/3ꞷ) = 0 
 i = 1… k , j=1…n 
 (1) 

where k is the number of inductances; 
n - number of capacitances in the stage substitution 
circuit; 
n/mꞷ - the duration of the repeatability interval. 
The system of equations (1) is linear, which is 
typical for any valve transducers with one substitution 
circuit (one stage) on the repeatability interval. The solution 
of (1) does not represent any special difficulties [22] and 
may be found on a computer with standard hardware. 
During the analysis of electromagnetic processes in 
inverters of the second group, in which there are already two 
stages on the repeatability interval, we obtain a system of 
equations of the form: 
 f
1
 (I
beg
, U
beg
,I
τ 
,τ) = 0 
 f
2
 (I
end
, U
end
, I
τ
 , U
τ 
,[(n/mꞷ) - τ]) = 0 (2) 
with the first subsystem derived from the equations of 
step 1 with duration τ and the second subsystem derived 
from the equations of step 2 with duration [(n/mꞷ) - τ], 
similar to system (2). 
Substituting from f
1
(...) into f
2
(...) and considering that 
I
beg 
= I
end
= I
τ
, U
beg
= U
end
= U
τ
we obtain a non-linear 
system: 
f (I
τ
, U
τ
, τ) = 0, (3) 
where the parameter τ determining the moment of transition 
from one stage to another enters this system implicitly, i.e. 
makes it transcendental. Transcendence is determined by the 
presence in (3) of coefficients in the form of exponential or 
trigonometric functions on τ. 
When analyzing electromagnetic processes in inverters 
of the third group, we obtain a system of equations of the 
form 
 f
1
 (I
beg
, U
beg
, I 
τ1
, U
τ1
, τ
1
) = 0, 
 f
2
 (I
τ1
, U 
τ1
, I 
τ2
, U 
τ2 
, τ
2
 ) = 0,
 
(4) 
f {I
τ2
, U
τ2
, I
end
, U
end
, [(n/mꞷ) - τ
1
 - τ
2
]} = 0 
with the first, second and third subsystems being obtained 
from the equations of (3) of the first (τ1), second (τ2) and 
third [(n/mꞷ)-τ1-τ2] steps in a similar way to system (1). 
By successive substitution of the second subsystem into 
the third one and then the first one into the result, 
considering: 
 I
beg.i 
= I
end. i
 = I
inter.i
 
U
beg.j
 = U
end. j
 = U
inter. j
, 
we get a system of non-linear equations of the form 
 f (I
inter.i
 , U
inter.j
 , τ
1
 , τ
2
) = 0. (5) 
In this system, two parameters τ