Calculation of a Switched Inductive-Capacitor Generator

Abstract

A method for calculating a switched inductive-capacitor generator has been developed that makes it possible to determine the voltages, currents, power, and efficiency of the generator taking into account the parameters of its elements in a pulse-frequency mode of power supply of an active-inductive load. To increase the efficiency of the generator, the voltage across the capacitor and the power of the generated current pulses in the load, it is necessary to increase the source voltage, the switching voltage of the dynistors (Shockley diodes), the switching frequency and duty cycle, the number of capacitor-charging periods, and the time constant of the inductive storage, as well as reduce the capacitance of the capacitor. A generator with a source voltage of up to 48 V and a current-pulse frequency in a load of 1 Hz can have a pulse energy of up to 12 kJ, an average power of up to 12 kW with an efficiency of up to 0.89 and a pulse power of up to 35 MW. It is possible to connect the generator to a source with an insufficient dc voltage level for charging the capacitor, with the voltage across the charged capacitor (639–3060 V) being able to be tens of times higher than the source voltage (12–48 V).

High-power current pulses with a repetition rate of up to 100 Hz or higher are required by a number of active-inductive consumers. Such consumers include technological laser equipment, pulse electric welding devices, and equipment for electrohydraulic, magnetic-pulse, and electrical-discharge machining of metals. Capacitors charged from a dc voltage source (accumulator, rectifier) are widely used in generators for power supply of such consumers. A situation often occurs, especially at autonomous facilities, in which the dc voltage of a source is insufficient for direct charging of capacitors even with the use of a current limiting choke when the voltage across the capacitors may not exceed twice the source voltage [1]. Therefore, to increase the voltage of capacitors, an intermediate circuit is often used consisting of an inverter, transformer, and rectifier from which the capacitors are charged [2, 3]; however, this connection complicates the design of generators and degrades their reliability.

If the source voltage is insufficient to charge a capacitor, one may use an intermediate inductive capacitor storage and commutator, which allow one to increase the voltage on the charged capacitor and, thereby, increase the generated pulse power. The schemes of inductive-capacitor dc pulse voltage converters with an inductive storage device and a semiconductor switch in the form of an IGBT transistor are reported in [4]. These converters operate with parallel connection of the consumer to the output capacitor and are simple and reliable compared to inverter circuits [2, 3], they increase or decrease the voltage of the output capacitor compared to the voltage of the source.

One of the circuits discussed in [4] for charging the capacitor was chosen, which is simple and reliable. This circuit allows using a source with insufficient dc voltage, increasing the voltage on the capacitor being charged. A thyristor switched by dynistors (Shockley diodes) is used to implement the frequency-pulse mode of powering the consumer for its connection to the charged capacitor. This circuit eliminates the possibility of source and commutator current flowing through the consumer, especially in the event of a short circuit at the consumer, which relaxes the specifications for the source and the commutator.

Thus, the switched inductive-capacitor generator (Fig. 1) is designed for reliable supply of consumers by means of powerful pulses of current in the frequency mode even in the case in which the generator is connected to a source with a reduced level of constant voltage insufficient for charging the capacitor. A description is given in [5] of the design elements and the results of experimental studies of such a generator at frequency-pulse power supply of a loaded step-up voltage transformer, which showed the functionality and reliability of the generator. However, the formulas for calculating a commutator inductive-capacitor generator in [5] were not given, and [4] also lacks a method for calculating the frequency-pulse mode. Therefore, the development of a method for designing such a generator in the pulse-frequency mode to determine its efficiency and capabilities is an important task that is of practical interest.

Fig. 1.

Schematic of switched inductive-capacitor generator: S, constant voltage source; L, current pulses i₂ of the consumer (active-inductive load); E and R_E, constant EMF and resistance of dc energy source S; S_w, semiconductor switch (commutator) on IGBT transistor; L₁ and R₁, inductance and resistance of inductive energy-storage device; C, capacitance of charged capacitor; D₁ and D₄, semiconductor diodes; D₂, thyristor; D₃, one or several dynistors connected in series; L₂ and R₂, inductance and resistance of load L; R₃, current limiting resistor for dynistors D₃; and i_E, i₁, i₂ and u_S, u_C, currents and voltages as a function of time.

Let us consider frequency-pulse power supply to a consumer represented by active-inductive load L. Figure 1 shows a schematic diagram of an inductive-capacitor generator. Let us assume that the inductances and resistances of the source, inductive accumulator, and consumer are roughly fixed and semiconductor diodes D₁ and D₄, thyristor D₂, and dynistors D₃ are ideal devices. In this case, we will assume that semiconductor IGBT transistor switch S_w in the “open” state has an infinite resistance and, in the “closed” state, is characterized by constant voltage U_CE between the collector and emitter (U_CE = 1–3 V). Let us also assume that the inductances and capacitances of the dc voltage and the connecting wires have no appreciable influence on the currents and voltages of the generator. Suppose that, at the time interval associated with current pulse i₂, switch S_w is “open” and diode D₁ is off, that is, current i₂ does not affect currents i_E and i₁.

The influence of external factors on the operation of the generator (temperature, humidity, vibration, interference, etc.) is beyond the scope of this article. Switch S_w (IGBT transistor) periodically closes and opens the generator supply circuit with frequency f₁ and duty cycle Q > 1. If switch S_w is open and the voltage at the terminals of source S is approximately constant and equal to U_E, then the calculated value of EMF E, switching period T₁, and time intervals t₁ (S_w “closed”) and t₂ (S_w “open”) will be considered equal:

$$\begin{gathered} E \approx {{U}_{E}};\,\,\,\,{{T}_{1}} = {1 \mathord{\left/ {\vphantom {1 {{{f}_{1}}}}} \right. \kern-0em} {{{f}_{1}}}};\,\,\,\,{{t}_{1}} = {{{{T}_{1}}} \mathord{\left/ {\vphantom {{{{T}_{1}}} Q}} \right. \kern-0em} Q}; \\ {{t}_{2}} = {{T}_{1}} - {{t}_{1}} = {{{{T}_{1}}\left( {Q - 1} \right)} \mathord{\left/ {\vphantom {{{{T}_{1}}\left( {Q - 1} \right)} Q}} \right. \kern-0em} Q}. \\ \end{gathered} $$

(1)

According to Fig. 1, the generator operates as follows. When switch S_w is closed and source S is connected, currents i_E and i₁ increase and the energy of the magnetic field of the inductive storage device (L₁, R₁) increases. At open commutator S_w and disconnected source S the energy of the inductive storage is transferred to capacitor C, which is charged to a certain maximum voltage U_mC level during several switching periods T₁ of S_w key and U_mC is equal to dynistor D₃ turn-on voltage. When capacitor voltage u_C reaches value U_mC, dynistor D₃ turns on together with the thyristor D₂ and capacitor C is discharged into load L (L₂, R₂), forming current pulse i₂. Generation of current pulses i₂ is repeated with period T₂ and frequency f₂ = 1/T₂ at f₂ ≤ f₁, whereby, to close thyristor D₂, it is necessary to reduce its current i₂ to zero. In order to separate in time the processes of energy storage in an inductive storage device (L₁, R₁) and capacitor C, semiconductor diode D₁ is added, and, to prevent negative voltage at capacitor C during its discharge, semiconductor diode D₄ is connected.

Three modes are possible when charging the capacitor:

— continuous current mode i₁ when this current is nonzero at all periods of T₁ of commutation S_w;

— intermittent current mode i₁ when this current reaches zero for a certain number of T₁ periods of commutation S_w; and

— pulsed current operation i₁ when this current reaches zero in all T₁ commutation periods of S_w.

Let us restrict our consideration to the pulsed mode of the current i₁ of the inductive storage device (L₁, R₁) when practically all energy of the inductive storage device during each period of T₁ is transferred to capacitor C and a small part of it is lost as heat, mostly in resistance R₁ of inductive storage device.

The formulas follow to calculate the generator voltages and generator currents obtained considering (1) from solving equations of electric circuit [6] at the following time intervals of the k-th switching period of switch S_w at k = 1, 2, …, N.

(1) Time interval $\left( {k - 1} \right){{T}_{1}} < t < \left( {k - 1} \right){{T}_{1}} + {{t}_{1}}$ when electromagnetic energy is stored in an inductive storage device. Commutator S_w is closed, diodes D₁ and D₄ are locked, and thyristor D₂ is closed. Then,

$$\begin{gathered} {{i}_{E}} = {{i}_{1}} = \frac{{{{U}_{E}} - {{U}_{{CE}}}}}{{{{R}_{E}} + {{R}_{1}}}}\left\langle {1 - \exp \left\{ { - \frac{{\left[ {t - \left( {k - 1} \right){{T}_{1}}} \right]}}{\tau }} \right\}} \right\rangle ; \\ {{i}_{2}} = 0;\,\,\,\,{{u}_{C}} = {{U}_{{k - 1}}};\,\,\,\,{{u}_{S}} = {{U}_{{CE}}}; \\ \tau = \frac{{{{L}_{1}}}}{{{{R}_{E}} + {{R}_{1}}}};\,\,\,\,{{I}_{{m1}}} = \frac{{{{U}_{E}} - {{U}_{{CE}}}}}{{{{R}_{E}} + {{R}_{1}}}}\left[ {1 - \exp \left( { - \frac{{{{t}_{1}}}}{\tau }} \right)} \right]; \\ {{W}_{m}} = \frac{{{{L}_{1}}I_{{m1}}^{2}}}{2};\,\,\,\,{{W}_{U}} = {{U}_{E}}\int_{\left( {k - 1} \right){{T}_{1}}}^{\left( {k - 1} \right)T + {{t}_{1}}} {{{i}_{E}}} dt \\ = \,\,\frac{{{{U}_{E}}\left( {{{U}_{E}} - {{U}_{{CE}}}} \right){{t}_{1}}}}{{R{}_{E} + {{R}_{1}}}}\left\{ {1 - \frac{\tau }{{{{t}_{1}}}}\left[ {1 - \exp \left( { - \frac{{{{t}_{1}}}}{\tau }} \right)} \right]} \right\}; \\ {{P}_{{mU}}} = {{U}_{E}}{{I}_{{m1}}};\,\,\,{{\eta }_{m}} = \frac{{{{W}_{m}}}}{{{{W}_{U}}}} = \left( {1 - \frac{{{{U}_{{CE}}}}}{{{{U}_{E}}}}} \right) \\ \times \,\,\frac{{{{{\left[ {1 - \exp \left( { - {{{{t}_{1}}} \mathord{\left/ {\vphantom {{{{t}_{1}}} \tau }} \right. \kern-0em} \tau }} \right)} \right]}}^{2}}}}{{2\left[ {\left( {{{{{t}_{1}}} \mathord{\left/ {\vphantom {{{{t}_{1}}} \tau }} \right. \kern-0em} \tau }} \right) - 1 + \exp \left( { - {{{{t}_{1}}} \mathord{\left/ {\vphantom {{{{t}_{1}}} \tau }} \right. \kern-0em} \tau }} \right)} \right]}};\,\,\,\,{{P}_{U}} = {{W}_{U}}{{f}_{1}}, \\ \end{gathered} $$

(2)

where U_E, V, is dc voltage at the output of source S when commutator S_w is open; U_CE, V, is dc voltage between the collector and emitter of closed switch S_w; τ, s, is the time constant of currents i_E and i₁; U_k–1, V, is the constant capacitor voltage at the interval in question determined from calculation at the previous time interval; I_m1, A, is the maximum value of currents i_E and i₁ at moment of time $t = \left( {k - 1} \right){{T}_{1}} + {{t}_{1}}$, W_m, J, is the maximum energy stored in the inductive storage; W_U, J, is the energy given by the source; P_mU and P_U, W, are the maximum and average power of the source; and η_m is the efficiency of energy transfer from the source to the inductive storage device.

(2) Time interval $\left( {k - 1} \right){{T}_{1}} + {{t}_{1}} < t < {{t}_{{mk}}}$ when energy W_m is transferred from the inductive storage device to the capacitor. Switch S_w is open, diode D₁ is open, diode D₄ is closed, and thyristor D₂ is closed; then,

$$\begin{gathered} {{i}_{E}} = 0;\,\,\,\,{{i}_{1}} = Cp{}_{1}{{B}_{{1k}}}\exp \left\{ {{{p}_{1}}\left[ {t - {{t}_{1}} - \left( {k - 1} \right){{T}_{1}}} \right]} \right\} \\ + \,\,Cp{}_{2}{{B}_{{2k}}}\exp \left\{ {{{p}_{2}}\left[ {t - {{t}_{1}} - \left( {k - 1} \right){{T}_{1}}} \right]} \right\};\,\,\,\,{{i}_{2}} = 0; \\ {{u}_{C}} = {{B}_{{1k}}}\exp \left\{ {{{p}_{1}}\left[ {t - {{t}_{1}} - \left( {k - 1} \right){{T}_{1}}} \right]} \right\} + {{B}_{{2k}}} \\ \times \,\,\exp \left\{ {{{p}_{2}}\left[ {t - {{t}_{1}} - \left( {k - 1} \right){{T}_{1}}} \right]} \right\};\,\,\,\,{{p}_{{1,2}}} = - \frac{{{{R}_{1}}}}{{2{{L}_{1}}}} \\ \pm \,\,\sqrt {\frac{{R_{1}^{2}}}{{4L_{1}^{2}}} - \frac{1}{{{{L}_{1}}C}}} ;\,\,\,\,{{B}_{{1k}}} = \frac{{{{I}_{{m1}}} - {{p}_{2}}C{{U}_{{k - 1}}}}}{{C\left( {{{p}_{1}} - {{p}_{2}}} \right)}}; \\ {{B}_{{2k}}} = \frac{{{{p}_{1}}C{{U}_{{k - 1}}} - {{I}_{{m1}}}}}{{C\left( {{{p}_{1}} - {{p}_{2}}} \right)}};\,\,\,\,{{t}_{{mk}}} = \left( {k - 1} \right)T + {{t}_{1}} \\ + \,\,\frac{1}{{\left( {{{p}_{1}} - {{p}_{2}}} \right)}}\ln \left( {\frac{{{{p}_{1}}{{p}_{2}}C{{U}_{{k - 1}}} - {{p}_{2}}{{I}_{{m1}}}}}{{{{p}_{1}}{{p}_{2}}C{{U}_{{k - 1}}} - {{p}_{1}}{{I}_{{m1}}}}}} \right); \\ {{U}_{k}} = \frac{{\left( {{{p}_{1}}C{{U}_{{k - 1}}} - {{I}_{{m1}}}} \right)}}{{C{{p}_{1}}}}{{\left[ {\frac{{{{p}_{2}}\left( {{{p}_{1}}C{{U}_{{k - 1}}} - {{I}_{{m1}}}} \right)}}{{{{p}_{1}}\left( {{{p}_{2}}C{{U}_{{k - 1}}} - {{I}_{{m1}}}} \right)}}} \right]}^{{\frac{{{{p}_{2}}}}{{{{p}_{1}} - {{p}_{2}}}}}}}; \\ {{u}_{S}} = {{U}_{E}} + {{u}_{C}};\,\,\,\,{{U}_{{Sk}}} = {{U}_{E}} + {{U}_{k}}, \\ \end{gathered} $$

(3)

where p_1,2, 1/s, are roots of the inductive storage-capacitor characteristic equation; B_1k and B_2k, V, are integration constants; t_mk, s, is the moment of closing of diode D₁ when current i₁ reaches a zero value; and U_k and U_Sk, V, are momentary voltages across the capacitor u_C and commutator u_S.

(3) Time interval t_mk < t < kT₁ when energy from the inductive storage device is not transferred to the capacitor. Commutator S_w is open, diodes D₁ and D₄ are off, and thyristor D₂ is closed. Then,

$${{i}_{E}} = {{i}_{1}} = {{i}_{2}} = 0;\,\,\,\,{{u}_{C}} = {{U}_{k}};\,\,\,\,{{u}_{S}} = {{U}_{E}}.$$

(4)

(4) Time interval ${{t}_{{mN}}} < t < N{{T}_{1}}$ when k = N and at t = t_mN, voltage u_C on the capacitor has reached maximum value U_mC = U_N in which dynistor D₃ and thyristor D₂ open up and capacitor C discharges into the load L₂ and R₂. Thus, switch S_w is open, diodes D₁ and D₄ are off, and thyristor D₂ is on. Then,

$$\begin{gathered} {{i}_{E}} = {{i}_{1}} = 0;\,\,\,{{i}_{2}} = C{{p}_{3}}{{B}_{3}}\exp \left\{ {{{p}_{3}}\left[ {t - {{t}_{{mN}}}} \right]} \right\} \\ + \,\,C{{p}_{4}}{{B}_{4}}\exp \left\{ {{{p}_{4}}\left[ {t - {{t}_{{mN}}}} \right]} \right\};\,\,\,\,{{u}_{C}} = {{B}_{3}} \\ \times \,\,\exp \left\{ {{{p}_{3}}\left[ {t - {{t}_{{mN}}}} \right]} \right\} + {{B}_{4}}\exp \left\{ {{{p}_{4}}\left[ {t - {{t}_{{mN}}}} \right]} \right\}; \\ {{W}_{{mC}}} = \frac{{CU_{{mC}}^{2}}}{2};\,\,\,\,{{f}_{2}} = \frac{{{{f}_{1}}}}{N};\,\,\,\,{{P}_{R}} = {{W}_{{mC}}}{{f}_{2}}; \\ {{p}_{{3,4}}} = - \frac{{{{R}_{2}}}}{{2{{L}_{2}}}} \pm \sqrt {\frac{{R_{2}^{2}}}{{4L_{2}^{2}}} - \frac{1}{{{{L}_{2}}C}}} ;\,\,\,\,{{B}_{3}} = \frac{{{{p}_{4}}{{U}_{{mC}}}}}{{{{p}_{3}} - {{p}_{4}}}}; \\ {{B}_{4}} = - \frac{{{{p}_{3}}{{U}_{{mC}}}}}{{{{p}_{3}} - {{p}_{4}}}};\,\,\,\,{{I}_{{m2}}} = - C{{p}_{3}}{{U}_{{mC}}}{{\left( {\frac{{{{p}_{4}}}}{{{{p}_{3}}}}} \right)}^{{\frac{{{{p}_{3}}}}{{{{p}_{3}} - {{p}_{4}}}}}}}; \\ {{P}_{{mR}}} = I_{{m2}}^{2}R{}_{2};\,\,\,\,{{t}_{{m2}}} = {{t}_{{mN}}} + \frac{{\ln \left( {{{{{p}_{4}}} \mathord{\left/ {\vphantom {{{{p}_{4}}} {{{p}_{3}}}}} \right. \kern-0em} {{{p}_{3}}}}} \right)}}{{{{p}_{3}} - {{p}_{4}}}}; \\ {{\eta }_{C}} = \frac{{0.5C\left( {U_{{mC}}^{2} - U_{0}^{2}} \right)}}{{N{\kern 1pt} {{W}_{U}}}};\,\,\,\,{{U}_{{mS}}} = {{U}_{E}} + {{U}_{{mC}}}, \\ \end{gathered} $$

(5)

where N is the number of current pulses i₁ used to charge capacitor C; t_mN, s, according to (3), is equal to t_mk at k = N; p_3,4, 1/s, are the roots of the characteristic equation of the capacitor-load contour; B₃ and B₄, V, are integration constants; U_mC, V, is the maximum charging voltage of capacitor C at moment t = t_mN; W_mC, J, is the maximum energy stored in capacitor C, which is equal to the energy dissipated as heat in load resistor R₂ during one current pulse i₂; f₂, Hz, is the frequency of repetition of i₂ current pulses; I_m2, А, is the maximum value of current i₂ at moment of time t_m2; P_mR and P_R, W, maximum and average power of current i₂ in load resistance R₂; η_C is the net efficiency of charging of capacitor C during N periods of T₁; U₀, V, is the initial voltage on capacitor C at moment t = 0; and U_mS, V, is maximum voltage at open switch S_w at moment t_mN.

For stable operation of the generator, it is necessary to close thyristor D₂ reliably when decreasing its current i₂ to zero when commutator S_w is open. Therefore, during an aperiodic transient of current i₂ when roots p_3,4 (5) are real, negative, and nondegenerate, the following conditions must be fulfilled:

$$C > \frac{{4{{L}_{2}}}}{{R_{2}^{2}}};\,\,\,\,{{t}_{p}} \approx \frac{5}{{\left| {{{p}_{3}}} \right|}};\,\,\,\,{{t}_{{mN}}} + {{t}_{p}} < N{{T}_{1}},$$

(6)

at complex-conjugate roots p_3,4 (oscillatory transient process),

$$\begin{gathered} C < \frac{{4{{L}_{2}}}}{{R_{2}^{2}}};\,\,\,\,\omega = \sqrt {\frac{1}{{{{L}_{2}}C}} - \frac{{R_{2}^{2}}}{{4L_{2}^{2}}}} ;\,\,\,\,{{t}_{p}} \approx \frac{\pi }{\omega }; \\ {{t}_{{mN}}} + {{t}_{p}} < N{{T}_{1}}, \\ \end{gathered} $$

(7)

where t_p, s, is the duration of the current pulse i₂ and ω, 1/s, is the angular frequency of current i₂.

In turn, to realize the pulsed mode of current i₁ according to (1), (3) at k = 1, when t_mk = t_m1 has the greatest value, it is necessary to fulfill the inequality

$${{t}_{{m1}}} = {{t}_{1}} + \frac{1}{{\left( {{{p}_{1}} - {{p}_{2}}} \right)}}\ln \left( {\frac{{{{p}_{1}}{{p}_{2}}C{{U}_{0}} - {{p}_{2}}{{I}_{{m1}}}}}{{{{p}_{1}}{{p}_{2}}C{{U}_{0}} - {{p}_{1}}{{I}_{{m1}}}}}} \right) < {{T}_{1}},$$

from which, at U₀ ≈ 0 and R₁ ≈ 0, we can obtain condition for capacitor C:

$${{C}_{m}} = \frac{{4t_{2}^{2}}}{{{{\pi }^{2}}{{L}_{1}}}} > C.$$

(8)

For stable operation of the generator in the pulse-frequency mode, the values of C, L₁, L₂, and R₂ must satisfy the inequalities (6)–(8).

To assess the efficiency of the generator considered calculations were carried out using formulas (1)–(8). Figure 2 shows the calculated curves of energy-transfer efficiency η_m from source S to the inductive storage device as a function of relative duration t₁/τ of closed commutator S_w at different relative values of voltage between the collector and emitter U_CE/U_E = 0, 0.1, and 0.2.

Fig. 2.

Calculated energy transfer efficiency η_m from the source to the inductive storage when U_CE/U_E = 0, 0.1, and 0.2.

From Fig. 2 and formulas (1, 2), it follows that, in order to increase efficiency η_m, it is necessary to increase frequency f₁, duty cycle Q, voltage of source U_E, and the time constant (to increase inductance L₁ and decrease resistance R_E, R₁).

Figure 3 shows resulting efficiency η_C of capacitor C charging as a function of N periods of T₁ of switch S_w and corresponding efficiency η_m. The plots of efficiencies η_C and η_m are obtained when R_E = 0, t₁/τ = 0.1, U_CE/U_E = 0.1, Q = 2, and U₀ = 0 and different capacitances C when coefficient a takes values of a = C_m/C = 1, 5, and 10. Figure 4 shows the dependence of relative maximum voltage on capacitor U_mC/U_E versus number N of switching periods T₁ of switch S_w calculated with the same parameters as in the dependences plotted in Fig. 3. It follows from Figs. 3 and 4 that the resulting charging efficiency of the capacitor η_C and its maximum voltage U_mC increase with increasing number of commutation periods N and reduction of capacitor capacitance C and it is always the case that η_C < η_m.

Fig. 3.

Energy transfer efficiency η_m to the inductive storage and the resulting efficiency of capacitor charging η_C as a function of N periods of switching at a = C_m/C = 1, 5, 10; R_E = 0; t₁/τ = 0.1; U_CE/U_E = 0.1; Q = 2; and U₀ = 0.

Fig. 4.

Relative maximum voltage across capacitor as a function of N commutation periods at a = C_m/C = 1, 5, 10; R_E = 0; t₁/τ = 0.1; U_CE/U_E = 0.1; Q = 2; and U₀ = 0.

To assess the capabilities of the generator, one may look at Table 1, which shows the results of the calculations by formulas (1)–(8) for the following parameters: U_E = 12, 24, and 48 V; U_CE = 2.5 V; f₁ = 200 Hz; Q = 2; L₁ = 0.1 mH; τ = 10t₁ = 25 ms; R_E = R₁ = 2 m Ω; U₀ = 0; C = 0.1C_m ≈ 2500 μF; N = 200; f₂ = 1 Hz; L₂ = 0.1L₁ = 10 μH; and ${{R}_{2}} = \sqrt {{{8{{L}_{2}}} \mathord{\left/ {\vphantom {{8{{L}_{2}}} C}} \right. \kern-0em} C}} = 0.179$ Ω (aperiodic transient current process i₂). According to the calculations increasing the source voltage U_E results in the increase of the following quantities: the voltage U_mC across the capacitor, the amplitudes of currents I_m1, I_m2, energy stored in the capacitor W_mC, efficiency η_C, average P_U, P_R and maximum P_mU, P_mR powers, the relative values of U_mC/U_E, P_mR/P_mU, and P_mR/P_U. Thus, the duration of current pulsei₂ in a load is t_p ≈ 1.909 ms, the rise time from zero to maximum I_m2 is 0.139 ms at allowed voltage U_mS = 651–3108 V at a high-power IGBT transistor, for example, having parameters of 3300 V/1500 A [7]. Thus, it is possible to implement a generator with an average power up to 15 kW and efficiency up to 0.89 for power supply of various technological apparatuses in the frequency-pulse mode [1].

Table 1.   Results of generator calculation at different source voltages U_E

For experimental verification of calculated relations (1)–(8), a prototype model of a switched inductive-capacitor generator (Fig. 1) has been implemented with the structural elements indicated in [5]. Source S of dc voltage U_E ≈ 8 V at R_E ≈ 1 Ω (the measuring resistor) was a power unit with adjustable voltage U_E. A solid state dc relay on an IGBT transistor with a switchable current up to 20 A and a maximum open switch voltage of up to 1200 V were used as semiconductor commutator S_w. The datasheet values of the voltage between collector and emitter in closed relay state U_CE < 3 and ≈ 2 V is assumed in calculations. By means of voltage control of the relay, switching frequency f₁ = 500 Hz and duty cycle Q = 2 were set. The inductive storage had a ferrite magnet wire (two W-core) with air gap of 2 mm and the following parameters with a measuring resistor of 1 Ω: L₁ ≈ 10.9 mH, R₁ ≈ 1.452 Ω, and mass 0.7 kg. To fulfill condition (8) when C_m ≈ 37.2 μF, a C = 20.9 μF capacitor was used. A coil was used as active-inductive load L with a measuring resistor (1 Ω) and additional resistors at which L₂ ≈ 0.155 mH and R₂ ≈ 7.24 Ω, aperiodic transient current i₂ was provided, and conditions (6) were fulfilled. As a result, we obtained U_E ≈ 8 V, R_E ≈ 1 Ω, U_CE ≈ 2 V, T₁ = 2 ms, t₁ = 1 ms, τ = 4.445 ms, t₁/τ = 0.225, U_CE/U_E = 0.25, and a = C_m/C = 1.779.

Experiments were carried out using the above-specified parameters of a generator with dynistors D₃ (Fig. 1) having different switching voltages, which corresponded to maximum capacitor voltages U_mC ≈ 36, 72, and 135 V.

To illustrate the processes of charging and discharging of capacitor C, a typical experimental oscillogram is shown in Fig. 5a of voltage u_C on a capacitor with period T₂ ≈ 22 ms corresponding to charging by N = T₂/T₁ ≈ 11 pulses and discharging at U_mC ≈ 36 V and U₀ ≈ 0. Figure 5b shows the calculated time dependence of voltage across capacitor u_C within period T₂ = 22 ms at N = 11 obtained by formulas (1)–(5) for parameters of the experimental generator model in which the calculated maximum voltage across capacitor appeared to be U_mC ≈ 37 V at U₀ = 0.

Fig. 5.

(a) Oscillogram and (b) calculated time dependence of voltage u_C on capacitor C = 20.9 μF at N ≈ 11, T₂ ≈ 22 ms, U_mC ≈ 36–37 V, and U₀ ≈ 0. Vertical scale, 10 V/div; horizontal, 5 ms/div; (a) left mark, zero level.

Table 2 lists the results of experiments and calculations by formulas (1)–(5) for different switching voltages of dynistors for the experimental generator model. According to these results, with an increase in the turn-on voltage of dynistors and the corresponding maximum voltage on capacitor U_mC, frequency f₂ = 1/T₂ of current i₂ in the load decreases with the increase of its amplitude I_m2 and power P_mR. Maximum voltage U_mS at the commutator S_w exceeds maximum voltage at the capacitor U_mC by roughly the value close to voltage of the source U_E. At the same time, maximum voltages U_mC and power P_mR of the load are much higher than corresponding voltages U_E and power P_mU of the source.

Table 2.   Results of experiments and calculations of the experimental generator model

Voltages u_C (Fig. 5), as well as the experimental and calculated values of voltages and currents given in Table 2, approximately correspond to each other, indicating the reliability of the method of calculation, the correct choice of parameters of the generator elements, and that the evaluation of the effectiveness and capabilities of the generator under consideration in the pulse-frequency mode of operation was correct.

CONCLUSIONS

(1) The proposed method of calculating the switched inductive-capacitor generator allows us to determine the voltages, currents, power, and efficiency of the generator taking into account the parameters of its elements in the frequency-pulse mode of power supply of an active-inductive load.

(2) The reliability of the method is confirmed by the approximate coincidence of the calculated and experimental values of voltages and currents of experimental model of the generator. The technique can be used in designing generators with an average power up to 15 kW for power supply of various apparatuses in frequency-pulse mode.