__
UI-Equations for a Small Transformer with Galvanically Separated
Secondary Windings and Autotransformer Primary Tap Switching__

Author:
Dipl.Ing. Hadzimanovic Ratibor, RALE Engineering*

The calculation of a small transformer with
galvanically separated secondary windings and autotransformer primary tap
switching is very complex and time consuming. For this reason these small
transformers are normally calculated with the help of computer programs.

The classical system of UI-equations for transformers
having only galvanically separated windings has been extended by the author **{R.1, R.2}** to introduce autotransformer
primary tap switching and has been prepared in the form of an algorithm for
integration into the computer programs.

Since the arrangement of the partial windings of a
small transformer is very variable, the classical method for calculating the
short-circuit impedance is not very suitable. For this reason the method of
root mean squares of the distances has been proposed **{R.3}**.

The system of UI-equations presented here has been
integrated into the computer programs of the company RALE Engineering **{R.4}** since 1988 and has been used in
practical applications.

The principal characteristics of
a small transformer can be specified as follows:

Figure 1

·
The
windings are normally wound on a single-section or a double-section bobbin.

·
The
core is made of stamped metal plates or ferrite. Many types of cores are used:

EI, EE, UI, UU, 3PEI, SE, SM,
SG, SU, 3SU, M, PM, ETD, EC, ER, P, RM, PQ, EP,
EFD, ...

·
The
power rating at 50 and 60 Hz is up to a maximum of 10 kVA per phase.

Figure 2 shows a universal schematic of a small
transformer with galvanically separated secondary windings and autotransformer
primary tap switching..

Figure 2

On the left primary side are Ni tapping points. Power supply to the transformer is
always via one tap only. Each tap is designed for a certain supply voltage
level. When the supply voltage is switched the outlet voltage should normally
remain unchanged.

On the left primary side one
also sees Na tapping points. Na consumers are supplied via these
autotransformer taps. The Ns secondary windings are used for providing
galvanically separated supplies to the consumers.

Purely resistive consumers are almost always connected
directly or via rectifiers to the secondary windings. In practice the input
voltages (Ui...), the output voltages (Ua..., Us...)
and the output currents (Ia..., Is...) are known.

The transformer is separated into

n=Ni+Na+Ns (1)

partial windings (see Figure 3).

Figure 3

The transformer can be supplied via one tap only. Accordingly,
the system of UI-equations for supplying the transformer via a specified tap is
written as follows:

u1 = r1*i1 + L11*(di1/dt) + L12*(di2/dt) +...+ L1n*(din/dt) u2 = L21*(di1/dt) + r2*i2 + L22*(di2/dt) +...+ L2n*(din/dt) (2) .................................................................................... un = Ln1*(di1/dt) + Ln2*(di2/dt) +...+ rn*in +Lnn*(din/dt) |

or:

up = |

p=1...n

q=1...n

n=Ni+Na+Ns

where the operator zpq is defined as follows:

zpq = rpq + Lpq*(d/dt) (4)

·
p,q Number of
the winding

·
L_{pq}=L_{qp }Mutual inductance of windings p and q.

·
L_{pp},_{Lqq} Self
inductance of windings p and q.

·
r_{pp}=r_{p }Ohmic resistance of winding p (r_{pq}=0)

·
u_{p} Voltage
at winding p (p=q : u_{p}=u_{q})

·
i_{q} Current
through winding q (p=q : i_{p}=i_{q})

The system of UI equations
(2 or 3) is particularly influenced by the n self inductances and the n*(n-1)/2 mutual
inductances. In the case of a small transformer with a ferrite core or a core
made from stamped or wound laminations it is also strongly non-linear, since
the self- and mutual inductances are dependent on the core temperature and the
induction. Measurements show that the calculation of the self- and mutual
inductances has an accuracy of about +-50%. For this reason the system of UI
equations (2 or 3) cannot be used. It must be transformed as follows **{R.1, R.2}**.

First, each equation is
multiplied by its transformation factor 1/k_{p}, where

k_{p}=Wp/Wb

·
Wp Number of turns in the p'th winding

·
Wb The freely selectable number of turns (Wb=W1+W2+...Wj).

·
j Number of the partial winding to which the supply voltage is
connected.

With the help of the following transformations::

u´p = up/ k p

i´p = ip*k p

L´pp = Lpp/k p^{2}

r´pp =
r´p = rpp/k p^{2}

L´pq = Lpq/k p /k q

z´pp = zpp/k p^{2}

z´pq = zpq/k p /k q

the system of UI equations **(2 or 3)** evolves as follows:

u1´ = r1´*i1´ + L11´*(di1´/dt) + L12´*(di2´/dt) +... L1n´*(din´/dt) u2´ = L21´*(di1´/dt) + r2´*i2´ + L22´*(di2´/dt) +... L2n´*(din´/dt) ................................................................................................ (5) un´ = Ln1´*(di1´/dt) + Ln2´*(di2´/dt) +... rn´*in´ +Lnn´*(din´/dt) |

or:

up´ = |

p=1...n

q=1...n

n=Ni+Na+Ns

In Figure 4 the short-circuit impedance Z_{kpq}
of two windings is measured.

Figure 4

It can however also be calculated as follows:

Lkpq =Lpp´+Lqq´-2*Lpq´ (7)

or:

zkpq =zpp´+zqq´-2*zpq´ (8)

From this an important conclusion follows:

Lpq´ =
0.5*(Lpp´+Lqq´-Lkpq) (9)

and:

zpq´ = 0.5*(zpp´+zqq´- zkpq) (10)

Equation (9) is now substituted into an equation of
the UI-equation system (5):

up´ = rp´*ip´ +0.5*Lpp´*_{} + 0.5* _{} -

-0.5* _{} (11)

With the help of the equation of the Ampere-turns the
no-load current **i _{o}**
is introduced:

.io = i1´ + i2´ +...+ in´ = _{} (12)

The no-load current plays an important role in a small
transformer, especially in the area of power outputs up to 10 VA, and cannot be
ignored.

Equation (11) now evolves as follows:

up´ = rp´*ip´ +0.5*Lpp´* io + 0.5*_{} -

-0.5* _{} (13)

The difference of the two equations is now formed. After
some transformations the new system of UI-equations is produced:

u2´
- u1´
= r2´*i2´ - r1´*i1´ +0.5*(L22´ - L11´- Lk12)* (dio/dt) + u3´
- u1´
= r3´*i3´ - r1´*i1´ +0.5*(L33´ - L11´- Lk13)* (dio/dt) + ...................................................................................... (14) un´
- u1´
= rn´*in´ - r1´*i1´ +0.5*(Lnn´ - L11´- Lk1n)* (dio/dt) + .io =
i1´ + i2´ +...+ in´ = |

The system of UI-equations (14) consists of **n** equations and **n** unknown currents. In comparison with the system of UI-equations
(3) it is much less nonlinear and can be calculated with a good degree of
accuracy. The non-linear elements or "constants" are the no-load
current and the differences in the self-inductances of the windings. They are
present in the following part of each UI-equation:

0.5*(Lnn´ - Lqq´- Lkqn)* (dio/dt)

After introducing the
following relations:

L0pq´ =
0.5*(Lpp´ -
Lqq´- Lkpq) (16)

and

Lipq =
0.5*(Lkip + Lkiq - Lkpq) (17)

the system of the UI equations reads as follows:

u2´
- u1´
= r2´*i2´ - r1´*i1´ + L021´* (dio/dt) + u3´
- u1´
= r3´*i3´ - r1´*i1´ + L031´* (dio/dt) + ............................................................................................ (18) un´ - u1´ = rn´*in´ - r1´*i1´ + L0n1´* (dio/dt) + .io = |

or

u2´ - u1´ = z021´* io + u3´ - u1´ = z031´* io + ............................................................................................ (19) un´ - u1´ = z0n1´* io + .io = |

where

·
z1i1= -r1

·
z1i1= ri + Lk1i*(d/dt)

·
z1iq = 0.5*(Lk1i + Lk1q - Lkiq) *(d/dt)

The system of UI-equations (**19**) can be used for a transformer with **n** galvanically separated secondary
windings. In the case of a transformer with galvanically separated secondary
windings and autotransformer primary winding switching, two additional
transformations of the system of UI-equations (19) are required.

After multiplying the first **j-1** equations (19) by the factors **k _{2}...k_{j}**, the sum of the voltages in the first

_{} - u1´*_{} = io*_{} + _{}*(_{})

If the supply voltage is connected to the partial
winding **j**, the following is the
case:

_{} = Ui

and

_{} = 1

From this it follows that

Ui - u1´ = io*_{} + _{}*(_{})

or

Ui -
up´ = io* p=1...Ni+Na (20) and Ui -
up´ = io* p=Ni+Na...n |

The final transformation now takes place. After the
multiplication of the first **k**
equations (**20**) by the factors **k _{1}...k_{k}** the sum
of the voltages in the first

Ui _{}-_{} = _{}* (io*_{}) +

+_{}*(_{}*(_{}))

Now **k**
partial windings are selected so that the following condition is fulfilled:

_{}= Uam = Ram*(i´k+1/kk+1 - i´k/kk )

(m=1..Na)

therefore one obtains

Ui _{}- Uam = _{}*(io*_{}) +

+_{}*(_{}*(_{}))

The system of UI-equations is then defined as follows:

Ui
i´q/kq = i´q+1/kq+1
Ui -
up/kp =
io* p=Ni+Na+1...n
io = |

The most important
parameters of the UI-equations (21) are the leakage inductances or
short-circuit inductances of the partial windings **L _{kpq}**, the no-load current

The typical arrangement of the two partial windings 1
and 2 in a small transformer is shown in Figure 5.

Figure 5

The calculation of the short-circuit inductance takes
place by using the method of the root mean squares of the distances **{R.3}**:

Lk12 = .(µo/2/p)*W^{2}1*P*ln(kc*g^{2}12/g1/ g2) (22)

where

·
µo =4* p *10^{-7} **(H/m)**

·
W1 Number of turns in winding 1

·
P Mean turn length of windings 1
and 2 in **m.**.

·
g1 = 0.223*(h1+b1)

·
g2 = 0.223*(h2+b2)

·
g12=(0.223*h1+0.78*d)^{a}*(0.223*ho+0.78*d)^{b}/(0.223*hg+0.78*d)^{g}/(0.223*hd+0.78*d)^{d}

·
a = (h2+hg)^{2}/(2*h1*h2)

·
b = h0^{2}/(2*h1*h2)

·
g = hg^{2}/(2*h1*h2)

·
d = hd^{2}/(2*h1*h2)

·
kc= g^{2}14/g13/ g24 The root mean square values of the distances g_{13}, g_{14} and g_{24} are calculated using the
same algorithm as for g_{12}.

The no-load current **i _{o}** consists of two components:

ioa = **P**Fe/(uij-_{})

and

ior = **Q**Fe/(uij-_{})

where

·
**P**Fe Active
core losses (**W**)

·
**Q**Fe Reactive
core losses(**VAr**)

For a double-section bobbin the following is valid:

|Lnn´ - Lqq´|
~ 0

and

0.5*(Lnn´ -
Lqq´- Lkqn) = - 0.5*Lkqn

For a single-section bobbin two different cases are
considered:

·
Lnn´
- Lqq´ ~ **-**Lkqn

and

0.5*(Lnn´ -
Lqq´- Lkqn) = -Lkqn

·
Lnn´ - Lqq´ ~ **+**Lkqn

and

·
0.5*(Lnn´ - Lqq´- Lkqn) = 0