ANALYSIS OF THE OPERATION OF A DC CONVERTER WITH A SWITCHING INDUCTOR ON A CORE WITH A RECTANGULAR HYSTERESIS LOOP

V. N. MISHIN, V. P. LAIER

Pulsed thyristor DC converters with commutation by means of an oscillatory circuit whose inductance is made on a core with a rectangular hysteresis loop (RHL) (Fig. 3) have a number of advantages in comparison with similar converters having a linear commutating inductance.

Fig. 1

The converter circuit of Fig. 1 operates as follows. In the initial state the commutating capacitor \(C_k\) is charged to the voltage \(U_{c0}\). After the thyristor \(T\) is turned on by a control pulse, the load current \(I_n\) flows through the thyristor, and remagnetization of the saturating choke \(L_k\) begins along the circuit \(C_k—T—L_k—C_k\). The saturating choke \(L_k\) and the capacitor \(C_k\) provide commutation of the power thyristor \(T\), since the saturating choke may be regarded as a switch. When it “closes,” the capacitor \(C_k\) is connected in parallel with the thyristor and is recharged through the inductance \(L_k\) to the voltage \(U_{c0}\), creating reverse polarity on the thyristor \(T\). After recharging, the switch is “opened” and “closed” after remagnetization of the choke. The capacitor \(C_k\) is discharged by a countercurrent through the thyristor and, when the current equals the thyristor current, turns it off, while the capacitor is recharged by the load current to the initial voltage \(U_{c0}\). In principle, the conducting period \(T\) is determined by the time required for remagnetizing \(L_k\) from positive saturation to negative saturation and back.

We shall analyze the commutation processes under the following assumptions:

1) for an unsaturated core \(|B|<B_s\)

\[ R(B)=\infty,\quad l=I_0,\quad |I_0 R_k|\ll U_c; \]

2) for a saturated core, \(|B|=B_s\),

\(R(B_s)=0,\ L_k=L_{ks}\);

3) the load current \(I_k\) over the interval of commutation processes is constant, where \(B_s\) and \(B\) are the magnetic induction at core saturation and the induction in the absence of saturation;

\(R(B)\) is the resistance determined by the magnetic state of the core [2].

\(I_0\) is the constant component of the current, depending on the magnitude of the threshold field strength \(H_0\) [2].

\(L_{kn}=L_{ks}\) is the inductance of the saturated core at \(|B|=B_s\).

Then, for the equivalent circuit of Fig. 2, representing the commutating circuit of the thyristor \(T\), after application of a turn-off pulse to the thyristor the following differential equations hold:

Fig. 2

\[ W_k S\,\frac{dB}{dt}=-U_c \qquad (1)\quad \text{for } |B|<B_s \]

\[ L_{ks}C_k\,\frac{d^2U_c}{dt^2}+R_kC_k+U_c=0 \qquad (2)\quad \text{for } |B|<B_s . \]

The initial values of the core induction and the voltage across the commutating capacitor \(C_k\) at \(t=0\), respectively, are:

\[ B(0)=B_0 \quad \text{and} \quad U_c(0)=U_{c0}. \]

The current of the commutating capacitor \(C_k\) is

\[ i=C_k\,\frac{dU_c}{dt}. \tag{4} \]

Solving (4) with respect to \(U_c\), taking (3) into account,

\[ U_c=U_{c0}+\frac{I_0t}{C_k}. \tag{5} \]

Integrating (1), taking (5) into account, we determine

\[ B=\int dB=-\frac{1}{W_kS}\int U_c(t)\,dt \]

\[ B=B_0-\frac{U_{c0}t}{W_kS}-\frac{I_0t^2}{2W_kSC_k}. \tag{6} \]

Taking into account that

\[ -\frac{I_0t^2}{2W_kSC_k}\simeq 0, \]

and assuming that during the time \(t=t_i\) the core is remagnetized to the induction \(B=-B_s\) at \(t=t_i\), we determine the remagnetization time of the core from \(B_0\) to \(-B_s\)

\[ t_{u1}=\frac{(B_o+B_s)W_kS}{U_{co}} . \tag{7} \]

After saturation of the core (point c, Fig. 3), the recharge process is described by differential equation (2), while the inductance of the core is

\[ L_{ks}=\mu_o\cdot \mu_{so}\frac{W_k^2S}{l}, \tag{8} \]

where \(\mu_o=4\pi\cdot10^{-7}\ \mathrm{H/m}\) is the magnetic permeability of vacuum,
\(\mu_{so}\) is the apparent saturation permeability [2],
\(s\) and \(l\) are the cross section of the core and the mean length of the magnetic field line.

The magnetic-field strength after saturation of the core is so large that the inductance \(L_s\) may be calculated from the geometrical dimensions of the winding as for a winding without a core, i.e., \(\mu_{so}=1\) [1].

Solving (2) for \(U_c\), taking into account that

\[ i_c=C\frac{dU_c}{dt} \tag{9} \]

\[ U_c=e^{\beta t}\left(A_1\cos\omega_s t+A_2\sin\omega_s t\right) \]

\[ i_c=ce^{\beta t}\left(\beta A_1\cos\omega_s t+\beta A_2\sin\omega_s t+\omega_sA_2\cos\omega_s t-\omega_sA_1\sin\omega_s t\right). \]

Determining \(A_1\) and \(A_2\) from the initial conditions \(U_c(0)=U_{co}\) and \(i_c(0)=0\), we obtain expressions for the voltage and current of the capacitance

\[ U_c=U_{co}e^{\beta t}\left(\cos\omega_s t-\frac{\beta}{\omega_s}\sin\omega_s t\right) \tag{11} \]

and

\[ i_c=C_k\omega_sU_{co}\cdot e^{\beta t}\left(\frac{\beta^2}{\omega_s^2}+1\right)\sin\omega_s t, \tag{12} \]

where

\[ \beta=-\frac{R_k}{2L_{ks}} \]

is the damping coefficient,

\(L_{ks}\) is the inductance of the saturated choke, determined by (8),
\(R_k\) is the resistance of the switching circuit at the switching frequency,

\[ \omega_s=\sqrt{\frac{1}{C_kL_{ks}}-\frac{R_k^2}{4L_{ks}^2}} \tag{13} \]

is the angular frequency of the circuit’s natural oscillations.

The time of resonant recharge with a saturated core may be taken equal to one half-period of the circuit’s natural oscillations, i.e.,

\[ t_n=\frac{T_{ks}}{2}=\pi\sqrt{L_{ks}C_k}. \tag{14} \]

The voltage across the capacitance after recharge, taking into account that \(t=t_n\), is

\[ U_{c2}=U_{co}\cdot e^{-\frac{\pi}{2a}}=K_n\cdot U_{co}. \tag{15} \]

where \(K_n\) is a coefficient depending on the quality factor \(Q_n\) of the recharge circuit;

\[ Q_n=\frac{\sqrt{\frac{L_{ks}}{C_k}}}{R_k} \]

is the quality factor of the circuit of the recharge chain of the switching capacitance.

The resonant recharge corresponds to the saturated segment \(a, b, c\) at \(|B|=-B_s\) (Fig. 3), after which remagnetization of the core begins up to the induction \(|B|=-B_s\) by the voltage \(U_c=U_{c2}\). The transient process is described by equation (6), and remagnetization of the core occurs on the segment \(d, f\) (Fig. 3) from the induction \(|B|=-B_s\) to \(|B|=+B_s\).

The remagnetization time \(t_{u2}\) is determined analogously to the first time interval according to expression (7)

\[ t_{u2}=\frac{2B_s\cdot W_k S}{U_{c2}} . \tag{16} \]

After remagnetization of the core to the induction \(|B|=+B_s\), commutation of the thyristor occurs by the current \(i_c\), directed opposite to the load current flowing through the thyristor \(T\).

\[ i_c=C_k\omega_s U_2 e^{\beta t} \left(\frac{\beta_2}{\omega_s^2}+1\right)\sin\omega_s t . \tag{17} \]

At the current \(i_c=I_n\), the thyristor is turned off. The commutation time \(t_k\) can be determined from (17), if it is taken into account that

\[ e^{\beta t}\simeq 1,\quad \text{since } t_k \ll T_{ks}, \]

and then

\[ t_k=\sqrt{C_k L_{ks}}\,\arcsin \frac{4Q_n}{K_i\cdot K_n(1+4Q_n)} , \tag{18} \]

where

\[ K_i=\frac{C_k\omega_s U_{c0}}{I_n} \]

is the load-current ratio with respect to the amplitude of the current in the commutating circuit.

After the thyristor \(T\) is turned off, the commutating capacitance \(C_k\) is recharged by the load current \(I_n\), i.e., \(U_{c2}\) to 0 and then from zero to the voltage of the power source \(E_i\). Over the interval of recharging \(C_k\) from \(U_{c2}\) to 0, a reverse voltage is applied to the thyristor \(T\), the duration of application of which determines the recovery time of the thyristor’s blocking properties

\[ t_z=\frac{K_n U_0 C_k}{I_n}. \tag{19} \]

The charging of \(C_k\) from zero to the voltage of the power source \(E_i\) also proceeds according to a linear law, and the charging time is

\[ t_z=\frac{E_i\cdot C_k}{I_n}. \tag{20} \]

After the voltage on the capacitance \(C_k\) reaches the voltage of the power source \(E_i\), switching of the load current occurs from the circuit \(+E_i—C_k—L_d—L_{dr}—R_n—(-E_i)\) into the circuit \(L_{dr}—R_n—D—L_{dr}\), that is, the current in the load is maintained by the energy of the choke \(L_{dr}\). The duration of the commutation interval of the diode \(D\) is determined by the angular frequency of the natural oscillations of the commutation circuit

\[ \omega_s=\frac{1}{\sqrt{L_{ks}C_k}} . \]

During commutation, the voltage on the capacitance \(C_k\) receives a voltage increment \(\Delta U\), determined by the amount of energy stored in the magnetic field of the commutating inductance, i.e., the voltage on the capacitance

\[ U_c=E_i+\Delta U . \tag{21} \]

To determine the voltage increment, the following equations are valid, describing the diode commutation process.

\[ \left. \begin{aligned} L_s \frac{di}{dt} + iR_{\mathrm p} U + U_c &= E_{\mathrm u},\\ i &= C \frac{dU_c}{dt} \end{aligned} \right\}, \tag{22} \]

where \(R_{\mathrm p}\) is the total active resistance of the inductance \(L_k\), the diode \(D\), and the power supply.

The solution of equations (22) with respect to \(U_c\), for the initial conditions \(U_c(0)=-E_{\mathrm u}\) and \(i_c(0)=I_{\mathrm n}\), has the form

\[ U_c=-\left(E_{\mathrm u}+I_{\mathrm n}\sqrt{\frac{L_{ks}}{C_k}}\, e^{\beta_1 t}\sin \omega_{s1}t\right), \tag{23} \]

where

\[ \beta_1=-\frac{R_{\mathrm p}}{2L_{ks}} \]
is the damping coefficient,

\[ \omega_{s1}=\sqrt{\frac{1}{L_{ks}C_k}-\frac{R_{\mathrm p}}{4L_{ks}}} \]
is the angular frequency of the discharge circuit of the capacitance \(C_k\).

The maximum voltage on the capacitance will occur at

\[ t_g=\frac{1}{4}T_{k1}=\frac{\pi\sqrt{L_{ks}C_k}}{2}, \]

\[ U_{c\max}=-\left(E_{\mathrm u}+J_{\mathrm n}\sqrt{\frac{L_{ks}}{C_k}}\,e^{-\frac{\pi}{Q}}\right). \tag{24} \]

The voltage increment

\[ \Delta U_{\max}=J_{\mathrm n}\sqrt{\frac{L_{ks}}{C_k}}\,e^{-\frac{\pi}{4Q}} \tag{25} \]

is proportional to the load current \(I_{\mathrm n}\) and the leakage inductance of the choke \(L_{ks}\).

Taking into account that

\[ K_i=\frac{U_{co}\sqrt{\frac{L_{ks}}{C_k}}}{J_{\mathrm n}},\qquad K_{\mathrm p}=e^{-\frac{\pi}{2Q}} \quad \text{and} \quad Q\approx Q_1, \]

expression (25) can be represented in a somewhat different form:

\[ \Delta U_{\max}=\frac{\sqrt{K_{\mathrm p}}}{K_i}U_{co}. \tag{27} \]

The occurrence of overvoltages on the commutating capacitance may lead to damped oscillatory processes, during which the energy stored in the electric field of the capacitance \(C_k\) is dissipated; this leads to an increase in energy losses in the circuit \(+E_{\mathrm u}-C_k-L_k-D-(-E_{\mathrm n})\).

The presence of overvoltages makes it necessary to increase the thyristor rating.

In a converter with a commutating inductance made on a core with an RHL, two modes are possible:

1) under the action of \(\Delta U_{c\max}\) during the pause, the core does not enter the region of “negative” saturations \(|B_0|<-B_s\);

2) under the action of \(\Delta U_{c\max}\) during the pause, the core enters the region of “negative” saturations: \(|B_0|=B_s\). In the first mode, a voltage equal to (24) is established on the commutating capacitance,

\[ U_c = U_{c0} = U_{c\max}, \]

but in this case the initial value of the core induction \(B_0\) (Fig. 3) approaches the region of “negative” saturation, which leads to a reduction in the time of the open state of the thyristor (7), since

\[ t_{u1} = - \frac{(B_0 + B_s) W_k S}{U_{c0}} \]

the minimum pulse duration will occur when

\[ |B_0| = -B_s \quad \text{and} \quad t_{u1} \cong 0. \]

Fig. 3

This will lead, if it is necessary to maintain constancy of the voltage across the load, to an increase in the pulse repetition frequency and, consequently, to additional commutation losses.

In the second mode, damped harmonic oscillations arise, and the voltage across the capacitance \(C_k\) is established equal to the voltage of the supply source

\[ U_{c0} = -E_u. \]

The use of diode \(D_1\) (shown by a dashed line in the circuit of Fig. 1) sets the induction at the level \(|B_0| = +B_s\), and the voltage across the commutating capacitance

\[ U_{c0} = U_{c\max} = \frac{E_u}{1 - \dfrac{\sqrt{K_n}}{K_i}} . \tag{28} \]

Figure 4 presents time diagrams explaining the course of the commutation processes in the converter.

The time of the open state of the thyristor

\[ t_T = t_{u1} + t_n + t_{u2} + t_k . \tag{29} \]

In expression (29), \(t_{u1} + t_{u2} \gg t_n + t_k\), therefore in practical calculations one may take

\[ t_T = t_{u1} + t_{u2}. \]

It follows from expressions (7) and (16) that the converter considered provides a change in the pulse duration inversely proportional to the magnitude of the supply voltage, that is, it possesses stabilizing properties.

The duration of the voltage pulse decreases with increasing load current owing to a reduction of the time intervals according to (19) and (20). An increase in the stability of the voltage pulse at the load can be achieved by shunting the thyristor \(T\) with an oppositely connected parallel diode [3].

Fig. 4

The use of a saturating choke makes it possible to reduce the overall dimensions of the commutating circuit and lowers the magnitude of overvoltages on the commutating capacitance.

REFERENCES

1. B. Bedford, R. Hoft. Theory of Autonomous Inverters. “Energiya,” 1969.

2. M. A. Rozenblat. Magnetic Elements of Automation and Computer Technology. “Nauka,” 1966.

3. V. A. Labuntsov, G. A. Belov. Comparative analysis of thyristor pulse DC converters. In: Collection of Reports for 1968–1969, edited by I. L. Kaganov. Moscow Power Engineering Institute, 1969, pp. 38–51.