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

 

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

1.     Overview

 

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.

 

2.     How Large is a Small Transformer ?

 

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.

3.     Development of the UI-equations

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 =                                                                                                                                           (3)

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 p²

r´pp = r´p = rpp/k p²

L´pq = Lpq/k p /k q

z´pp = zpp/k p²

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´ =                                                                                                                                      (6)

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) +                    + 0.5* u3´ - u1´ = r3´*i3´ - r1´*i1´ +0.5*(L33´ - L11´- Lk13)* (dio/dt) +                    + 0.5* ......................................................................................          (14) un´ - u1´ = rn´*in´ - r1´*i1´ +0.5*(Lnn´ - L11´- Lk1n)* (dio/dt) +                    + 0.5* .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 k₂...k_j, the sum of the voltages in the first j partial windings is formed:

 - 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 k₁...k_k the sum of the voltages in the first k partial windings is formed:

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:

 

Na-Equations Ui - Uam  =*(io* )+ *(*()) Ni-1 Equations   i´q/kq = i´q+1/kq+1   Ns - Equations Ui - up/kp = io* + *()                      (21) p=Ni+Na+1...n   One Equation io =

 

4.     Parameters of the UI-equations

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 i_o and the differences in the self-inductances of the partial windings L_pp - L_qq.

4.1     Leakage Inductance of the two Partial Windings L_kpq

 

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)*W²1*P*ln(kc*g²12/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 = h0²/(2*h1*h2)

·      g = hg²/(2*h1*h2)

·      d = hd²/(2*h1*h2)

·      kc= g²14/g13/ g24      The root mean square values of the distances g₁₃, g₁₄ and g₂₄ are calculated using the same algorithm as for g₁₂.

4.2     No-load Current i_o

The no-load current i_o consists of two components:

 

ioa = PFe/(uij-)

and

ior = QFe/(uij-)

where

·      PFe             Active core losses (W)

·      QFe                Reactive core losses(VAr)

 

4.3     Differences in the Self-inductances of two Partial Windings

 

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