The equivalent circuit model of the considered toroidal transformer connected to the “HV-Discharge” system is presented in Figure 5. The transformer is supplied from a capacitor bank *C*_{0} via feeder resistance *R*_{f} and feeder inductance *L*_{f}. The transformer incorporates two coupled windings: primary denoted by index “P” and secondary denoted by index “S”.

Figure 5 The equivalent circuit model of the transformer system (parameters are described in chapter 3)

The primary winding is divided in two parts: the main part modelled by resistance *R*_{p1} & inductance *M*_{p1} and the short-circuited part modelled by resistance *R*_{p2} & inductance *M*_{p2}. An electrical fault which short-circuits the turns of the primary winding is modelled as resistance *R*_{sc} and inductance *L*_{sc}. The eddy current effects and excess losses in the core are modelled with an additional circuit defined by inductance *M*_{e} and resistance *R*_{Fe}. Resistance *R*_{Fe} represents both the classical and anomalous eddy current losses (which may be non-linear) [8].

The Lagrange energy method is proposed to determine the mathematical model of the transformer. The generalized coordinates – loops of currents: *i*_{P1}, *i*_{P2} and *i*_{Fe} are proposed in Figure 5. It is assumed that the electrical currents are denoted by: *i* = *dq*/*dt* = *q̇*. The Lagrange function is defined as follows:
$$\begin{array}{}{\displaystyle L(\dot{{q}_{P1}},\dot{{q}_{P2}},\dot{{q}_{Fe}},{q}_{P1},{q}_{P2},{q}_{Fe})=\frac{1}{2}{L}_{0}{\dot{{q}_{P1}}}^{2}}\\ {\displaystyle +\underset{0}{\overset{\dot{{q}_{P1}}}{\int}}{\mathrm{\Psi}}_{P1}(\dot{{q}_{P1}},\dot{{q}_{P2}},\dot{{q}_{Fe}})d\dot{{q}_{P1}}+\underset{0}{\overset{\dot{{q}_{P2}}}{\int}}{\mathrm{\Psi}}_{P2}(\dot{{q}_{P1}},\dot{{q}_{P2}},\dot{{q}_{Fe}})d\dot{{q}_{P2}}}\\ {\displaystyle +\underset{0}{\overset{\dot{{q}_{Fe}}}{\int}}{\mathrm{\Psi}}_{Fe}(\dot{{q}_{P1}},\dot{{q}_{P2}},\dot{{q}_{Fe}})d\dot{{q}_{Fe}}-\frac{1}{2}\frac{{q}_{P1}}{{C}_{0}}+\frac{1}{2}{M}_{sc}(\dot{{q}_{P1}}-\dot{{q}_{P2}}{)}^{2}}\end{array}$$(3)

The Rayleigh dissipation function is defined as:
$$\begin{array}{}{\displaystyle {P}_{e}(\dot{{q}_{P1}},\dot{{q}_{P2}},\dot{{q}_{Fe}})=\frac{1}{2}{R}_{0}{\dot{{q}_{P1}}}^{2}+\frac{1}{2}{R}_{P1}{\dot{{q}_{P1}}}^{2}}\\ {\displaystyle +\frac{1}{2}{R}_{P2}{\dot{{q}_{P2}}}^{2}+\frac{1}{2}{R}_{sc}(\dot{{q}_{P1}}-\dot{{q}_{P2}}{)}^{2}+\frac{1}{2}{R}_{Fe}{\dot{{q}_{Fe}}}^{2}}\end{array}$$(4)

The general form of the Euler-Lagrange equation system takes the form of:
$$\begin{array}{}{\displaystyle \frac{d}{dt}\left[\frac{\mathrm{\partial}L}{\mathrm{\partial}\dot{{q}_{P1}}}\right]-\frac{\mathrm{\partial}L}{\mathrm{\partial}{q}_{P1}}+\frac{\mathrm{\partial}{P}_{e}}{\mathrm{\partial}\dot{{q}_{P1}}}=0}\\ {\displaystyle \frac{d}{dt}\left[\frac{\mathrm{\partial}L}{\mathrm{\partial}\dot{{q}_{P2}}}\right]-\frac{\mathrm{\partial}L}{\mathrm{\partial}{q}_{P2}}+\frac{\mathrm{\partial}{P}_{e}}{\mathrm{\partial}\dot{{q}_{P2}}}=0}\\ {\displaystyle \frac{d}{dt}\left[\frac{\mathrm{\partial}L}{\mathrm{\partial}\dot{{q}_{Fe}}}\right]-\frac{\mathrm{\partial}L}{\mathrm{\partial}{q}_{Fe}}+\frac{\mathrm{\partial}{P}_{e}}{\mathrm{\partial}\dot{{q}_{Fe}}}=0}\end{array}$$(5)

Merging equations (3)-(5) results in the matrix equation:
$$\begin{array}{}{\displaystyle \left[\begin{array}{ccc}{L}_{0}+{M}_{sc}+\frac{\mathrm{\partial}{\mathrm{\Psi}}_{P1}}{\mathrm{\partial}\dot{{q}_{P1}}}& {M}_{sc}+\frac{\mathrm{\partial}{\mathrm{\Psi}}_{P1}}{\mathrm{\partial}\dot{{q}_{P2}}}& \frac{\mathrm{\partial}{\mathrm{\Psi}}_{P1}}{\mathrm{\partial}\dot{{q}_{Fe}}}\\ -{M}_{sc}+\frac{\mathrm{\partial}{\mathrm{\Psi}}_{P2}}{\mathrm{\partial}\dot{{q}_{P1}}}& {M}_{sc}+\frac{\mathrm{\partial}{\mathrm{\Psi}}_{P2}}{\mathrm{\partial}\dot{{q}_{P2}}}& \frac{\mathrm{\partial}{\mathrm{\Psi}}_{P2}}{\mathrm{\partial}\dot{{q}_{Fe}}}\\ \frac{\mathrm{\partial}{\mathrm{\Psi}}_{Fe}}{\mathrm{\partial}\dot{{q}_{P1}}}& \frac{\mathrm{\partial}{\mathrm{\Psi}}_{Fe}}{\mathrm{\partial}\dot{{q}_{P2}}}& \frac{\mathrm{\partial}{\mathrm{\Psi}}_{Fe}}{\mathrm{\partial}\dot{{q}_{Fe}}}\end{array}\right]\cdot \left[\begin{array}{c}\ddot{{q}_{P1}}\\ \ddot{{q}_{P2}}\\ \ddot{{q}_{Fe}}\end{array}\right]}\\ +\left[\begin{array}{ccc}{R}_{0}+{R}_{P1}+Rsc& -{R}_{sc}& 0\\ -{R}_{sc}& {R}_{P2}+{R}_{sc}& 0\\ 0& 0& {R}_{Fe}\end{array}\right]\cdot \left[\begin{array}{c}\dot{{q}_{P1}}\\ \dot{{q}_{P2}}\\ \dot{{q}_{Fe}}\end{array}\right]\\ =\left[\begin{array}{c}-{\displaystyle \frac{{q}_{P1}}{{C}_{0}}}\\ 0\\ 0\end{array}\right]\end{array}$$(6)

For the considered type of magnetic yoke, it is assumed that there is a flux *Φ*_{c} which is common to all windings. Thus, the left hand side of (6) can be expressed with (7):
$$\begin{array}{}\left[\begin{array}{ccc}{L}_{0}+{M}_{sc}+{M}_{lP1}& {M}_{sc}& 0\\ -{M}_{sc}& {M}_{sc}+{M}_{lP2}& 0\\ 0& 0& {M}_{lFe}\end{array}\right]\cdot \left[\begin{array}{c}\ddot{{q}_{P1}}\\ \ddot{{q}_{P2}}\\ \ddot{{q}_{Fe}}\end{array}\right]\\ +{\displaystyle \frac{\mathrm{\partial}{\mathit{\Phi}}_{c}}{\mathrm{\partial}\mathrm{\Theta}}}\left[\begin{array}{ccc}{N}_{P1}{N}_{P1}& {N}_{P1}{N}_{P2}& {N}_{P1}{N}_{Fe}\\ {N}_{P2}{N}_{P1}& {N}_{P2}{N}_{P2}& {N}_{P2}{N}_{Fe}\\ {N}_{Fe}{N}_{P1}& {N}_{Fe}{N}_{P2}& {N}_{Fe}{N}_{Fe}\end{array}\right]\cdot \left[\begin{array}{c}\ddot{{q}_{P1}}\\ \ddot{{q}_{P2}}\\ \ddot{{q}_{Fe}}\end{array}\right]\\ +\left[\begin{array}{ccc}{R}_{0}+{R}_{P1}+Rsc& -{R}_{sc}& 0\\ -{R}_{sc}& {R}_{P2}+{R}_{sc}& 0\\ 0& 0& {R}_{Fe}\end{array}\right]\cdot \left[\begin{array}{c}\dot{{q}_{P1}}\\ \dot{{q}_{P2}}\\ \dot{{q}_{Fe}}\end{array}\right]\\ =\left[\begin{array}{c}-{\displaystyle \frac{{q}_{P1}}{{C}_{0}}}\\ 0\\ 0\end{array}\right]\end{array}$$(7)

where *Θ* defined by (8) is the total ampere-turns of all coils, *N*_{k} is the number of turns of the *k*-th coil and *M*_{lk} is the leakage inductance.
$$\begin{array}{}\mathit{\Theta}={N}_{P1}\dot{{q}_{P1}}+{N}_{P2}\dot{{q}_{P2}}+{N}_{Fe}\dot{{q}_{Fe}}\end{array}$$(8)

The relation between *Φ*_{c} and *Θ* involves the feedback Preisach model of hysteresis presented in [7, 8].

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.