Simulation of a discharge device for the spacecraft power supply system

One of the main systems of a spacecraft is the power supply system [1; 2] – this is an interconnected set of equipment used for the production and conversion, transmission, accumulation, distribution and consumption of electrical energy. The service life of the power supply system determines the active life of the spacecraft. One of the important aspects to consider when designing such systems are the standards regarding on-board network types [3] and voltage levels. Currently, two main power supply standards are accepted in the space industry: 27 and 100 V. Typically 27V is used to power utility systems and low-power consumers. Switching to a 100V bus for high-power consumers allows to reduce the current required to transmit the same power and to use a smaller cable cross-section, resulting in savings of materials and reduction in the weight of a system.

Most spacecraft use solar panels as their primary energy source [4; 5], but the position of the spacecraft relative to the Sun varies, which can affect the amount of solar energy they receive. For example, when passing through the planet's shadow, solar panels may receive less energy, which necessitates the use of a battery in the shadow portion of the orbit.

The battery contains battery cells connected in series to increase the overall voltage. The research considers the battery operating in the voltage range from 62.1 to 96.6 V, while the consumer bus voltage is 100 V, which determines the need to use a step-up converter in the discharge device.

To stabilize the output voltage of the discharge device at the required level, it is necessary to perform the following:

• –    static analysis and synthesis, including the selection of a structural diagram, calculation of its ratings, calculation of the error in voltage stabilization in the steady-state mode;

• –    dynamic analysis and synthesis, based on the results determining the parameters of the corrective device, ensuring the required stability reserves and quality parameters of transient processes;

• –    development of a simulation model of a discharge device layout;

• –    computational experiments with a simulation model, confirming the correctness of previously performed calculations;

• –    adjustment of calculations if necessary.

The MathCAD 12 computer algebra package [6] was chosen as the program for mathematical calculations. It is related to automated design systems and allows to to perform the calculations with sufficient accuracy, deal with equations and systems of equations, plot graphs, and create interactive documents.

To analyze the parameters and test the operation of the discharge device with the element ratings selected based on the calculation results, the Micro-Cap 12 circuit simulation package is selected, as it obtains the following features:

• –    easy edition of SPICE models [7; 8] of components and their parameters;

• –    simulation of analog and digital electrical circuits;

• –    fairly wide range of tools for analyzing electrical circuits;

• –    multivariate analysis of circuits (analysis of transient processes, frequency analysis, analysis of transfer functions with varying parameters).

Using the developed discharge device model as part of a model of a spacecraft power supply system in the educational process will allow students to visually study the principles of constructing spacecraft power supply systems, the processes occurring in them, and telemetry information.

Static calculation of the discharge device

When developing the layout of the discharge device, a static analysis was carried out; it resulted in determining the block diagram presented in Fig. 1.

Рис. 1. Структурная схема разрядного устройства:

УС – усилитель напряжения; ИПН – импульсный преобразователь напряжения; ДН – датчик напряжения; ШИМ – широтно-импульсный модулятор; γ – коэффициент заполнения; U _УС – напряжение усилителя-сумматора; U _АБ – напряжение аккумуляторной батареи; Z _АБ – сопротивление аккумуляторной батареи; U _ВХ – входное напряжение преобразователя; I _ВХ – входной ток преобразователя;

U Д _Н – напряжение датчика напряжения; U_ε – напряжение рассогласований; U _ОП – опорное напряжение;

U _Н, I _Н и Z _Н – напряжение, ток и сопротивление нагрузки

• Fig. 1.    Block diagram of the discharge device:

VAS – voltage amplifier-summer; PVC – pulse voltage converter; VS – voltage sensor;

PWM – pulse width modulator; γ – duty cycle; U_VAS – voltage amplifier-summer; U_B – battery voltage; Z _B – battery impedance; U_in – input voltage of the converter; I_in – input current of the converter; U _VS – voltage of the voltage sensor;

U _m – mismatch voltage; U _re f – reference voltage; U_load , I_load and Z_load – voltage, current and load resistance

The discharge device has got the following technical requirements:

– voltage change ranges from 62.1 to 96.6 V;

• –    current ranges from 1 to 10 A;

• –    the amplitude of output voltage pulsations is no more than 1 V;

• –    output voltage stabilization error is no more than 1 %;

• –    ensured voltage stabilization is at 100 V;

• –    adjusting time for the output voltage of the battery cell is no more than 1300 ms;

• –    overshoot of the output voltage of the battery cell is no more than 5 %.

To ensure voltage stabilization mode, the basis of the discharge device is an output voltage stabilizer. The output voltage stabilizer must contain a controlled pulse converter and a control system. The controlled pulse converter [9; 10] can be made in the form of a step-up pulse converter, the circuit of which is shown in Fig. 2.

Рис. 2. Схема повышающего импульсного преобразователя

• Fig. 2.    Diagram of the boost pulse converter

There is a mathematical description illustrating the currents and voltages flowing in the discharge device, presented in the form of systems of equations for static (1) and dynamic small-signal (2) modes; functional diagrams are developed (Fig. 3, 4), the construction of which provided the most visual representation of the sequence of processes occurring in the device; a calculation of the stabilization error of the output voltage of the step-up pulse voltage converter is performed.

Uladd = U_ln - R L l_m + U_l0 a _{d 0} Y 0 + U_{ш 0} A ₇ + _7o A U_lo a _d ,

I in = I ln₀ Y 0 ⁺ I ln₀ ^A Y ⁺ Y 0 ^{A I}in ⁺ I load ,

m = U b — R b I_m ,                                 ,                   (1)

^Um = ( ^Uref ^{- U}load K VS ) K VAS ,

, ^Y = ^Um K PWM ,

R_L – resistance of the winding; U_{load 0} – value of the load voltage at the linearization point.; γ₀ – value of the fill factor at the linearization point.; Δγ – increment of the fill factor value; Δ U_load – increment of the load voltage value; I in 0 – value of the input current at the linearization point.; Δ I in – increment of the input current value; R_B – active component of the battery resistance; K_VS – voltage sensor coefficient; K_VAS – static transmission coefficient of the VAS; K_PWM – pulse width modulator coefficient.

A Uload ( s ) = A Un ( s ) - ( RL + Ls ) AIin ( s ) + Uload0 AY( s ) + Yo A Uload ( s ),

AIin (s) = Iin0 AY(s) + Y0 AIin (s) + AIload (s) + CsAUload (s),

A Um ( s ) = (A Uref ( s ) — A Uload (s ) KyS ) WvAS ( s),

^Ay('s ) = ^{A U}m⁽s) ^Kpwm ,

C – capacitor capacity of the input filter; L – inductance of the winding; W_VAS(s) – transfer function of the; KPWM – pulse width modulation coefficient.

To estimate the methodological error of stabilization [10; 11] of the output voltage, expression (3) of the relative error was used:

^{a( U}B, KVAS ) =

^Ur^{- U}load

Ur

100 %,

σ – a methodical error in voltage was used; Ur – required value of the output voltage; Uload – calculated value of the load voltage.

Рис. 3. Функциональная схема разрядного устройства в статическом режиме

• Fig. 3.    Functional diagram of a discharge device in static mode

Рис. 4. Функциональная схема разрядного устройства в динамическом режиме для приращений

• Fig. 4.    Functional diagram of a discharge device in dynamic mode for increments

Fig. 5 shows the dependence of the methodical error σ (3) on the transfer coefficient KVAS for different values of the input voltage of the pulse voltage converter UB.

Рис. 5. Зависимость методической погрешности δ(U) от коэффициента передачи K_УС при различных значениях входного напряжения импульсного преобразователя напряжения U_АБ

• Fig. 5.    The dependence of the methodological error σ on the transmission coefficient KVAS at different input voltage values of the pulse voltage converter U_B

The graph in Fig. 5 demonstrates that the methodical error σ decreases with increasing coefficient K_VAS and increasing voltage U_B. The worst operating condition is observed at the minimum voltage U_B, equal to 62.1 V.

Dynamic calculation of the discharge device

A dynamic calculation was performed; it includes the analysis and synthesis of the dynamic properties of the discharge device, such as regulation time, overshoot, stability margins, and oscillation.

Due to the functional diagram (Fig.4) of the discharge device for the dynamic mode and the solution of the equation system (2), the transfer function of the open loop stabilization of the output voltage of the PVC was determined as the voltage function UB and the output current Iload:

Wol (s ) = %r¥\ = f (^UB ■ load ’ s) ^ЛUref⁽s )

Logarithmic amplitude-frequency characteristic (LAFC) and phase-frequency characteristic (PFC) of an open-loop voltage stabilization circuit:

Lol (f ) = 20lg(WOL (J2nf )|)

ФOL ( ^f ) = arg (WOL ( ^J2nf ))                                 ⁽⁶⁾

The stability analysis of the discharge device must be performed at different values of voltage U_B and output current Iload, since the transfer function WOL(s) depends on these parameters. Fig. 6 shows the frequency response LAFC LOL(f) (5) and PFC and φOL(f) (6) of the open-loop circuit (OLC) of the discharge device. At the minimum output current of the PVC Iload = 1 A, the average value of the output current Iload = 5A and at the maximum output current of the DC power supply Iload = 10A, with the following parameters: KVAS = 5536.99; C = 47 μF; L = 147.5 μH; RL = 0.061 Ohm.

Fig. 6 demonstrates, the uncorrected system does not have stability reserves (there are no phase or amplitude reserves).

The device was corrected using the Solodovnikov method. To ensure stability and the required parameters of the transient process according to Solodovnikov's method, it is necessary to ensure a cutoff frequency of at least 10 Hz and a phase margin of about 90°. For this purpose, a compensating device (CD) with a transfer function of the form:

W_cD( s ) = Ts±l,                                  (7)

^v ’ T1 s ± 1

Т₁ – a time constant equal to 0.756 s; Т₂– a time constant of 79.577 microseconds.

PFC, Iload0 = 10А

LAFC, Iload0 = 5А

PFC, Iload0 = 5А

LAFC, Iload0 = 1А

PFC, Iload0 = 1А

Рис. 6. ЛАЧХ и ФЧХ нескорректированного РК стабилизации выходного напряжения ИПН при разных значениях выходного тока

• Fig. 6.    LAFC and PFC of the uncorrected OLC for stabilizing the output voltage of the pulse voltage converter PVC at different output current values

After introducing the compensating device (CD) into the OLC discharge device, the transfer function W_CL(s) of the OLC compensated discharge device and the frequency characteristics L_CL(f) and φ_CL(f) of the OLC acquired the form:

Wcl (s ) = Wol (s) Wcd (s),(8)

Lcl (f ) = 20lg(|Wcl (j2nf )|),(9)

ФCL ( f ) = arg(WCL (j2Пf ))•

Fig.7 presents LAFC L_CL(f) and PFC φ_СL(f) of the compensated OLC of output voltage stabilization Uload^.

Therefore, by implementing a compensating device with the required time constant values, the necessary transient response quality requirements were met. The phase margin was 81.2°, and the cutoff frequency was 21.9 Hz.

Frequency, Hz

LAFC, Iload0 = 10А

PFC, Iload0 = 10А

LAFC, Iload0 = 5А

PFC, Iload0 = 5А

LAFC, Iload0 = 1А

PFC, Iload0 = 1А

Рис. 7. ЛАЧХ и ФЧХ скорректированного РК стабилизации выходного напряжения ИПН при разных значениях выходного тока

• Fig. 7.    The LAFC and PFC of the adjusted OLC output voltage stabilization of the PVC at different values of the output current

Simulation model of a discharge device

The output voltage stabilizer contains a controlled pulse step-up converter and a control system implemented in digital form on the ESP32 Wemos D1 R32 debug board. The following structural circuit elements are implemented on the debug board (Fig.1):

– reference voltage source [12], required to set a stable output voltage;

– an amplifier-summer [13], based on a microcontroller and necessary to generate a signal of mismatch between the reference voltage and the feedback voltage;

– pulse width modulator [14], necessary to convert the error signal into the duty cycle of the rectangular signal controlling the switch;

– analog-to-digital converter for voltage sensor signal.

Fig. 8 demonstrates a simulation model of a discharge device.

In this circuit, the step-up converter contains the following: an input voltage source E1, which sets a time delay for closing the switch X2; an inductance L1 with a resistance R5; two switches S1 and S2 with protective diodes D5 and D6, controlled by the control system (CS); and a resistance R4, used to limit the charging current of the capacitor C1. After the output capacitance C1 is charged, switch X2 shunts resistance R4. The switch X2 is controlled by a photodiode optocoupler consisting of diodes D1, D2, D3, D4, photodiode X1, and constant voltage sources E2 and E3. Resistor R1 serves to limit the optocoupler's input current. The load resistor simulates resistance R7.

Рис. 8. Электрическая схема разрядного устройства

• Fig. 8.    Electrical diagram of the discharge device

The control system of the pulse converter consists of a reference signal source V2; a source V3, which sets the sawtooth voltage; a feedback source E6; a source E4, which controls the switch S1 and generates a PWM; a source E5, which controls the switch S2 and generates a PWM in antiphase under the same conditions as the source E4; an adder X3; an amplifier X4; a multiplier X5; a limiter of the value of the duty cycle X6; a time delay X7, simulating the delay of calculations of the microcontroller; a sequential adjusting device, simulated by the circuit R9R10C2, and a source E7, which blocks the operation of the control system until the switch X2 is closed in order to avoid the accumulation of an error (for 100 ms).

Transient Analysis

Fig. 9 shows the transient processes [15] for control during a stepwise increase and decrease of the output voltage of the PVC from U_load = 80 V to U_load = 100 V and back.

T (Secs)

Рис. 9. Переходные процессы по управлению

• Fig. 9.    Control transients

The graph presented in Fig. 10 allows to draw the following conclusions: the regulation time from 80 to 100 V of the output voltage was 51.645 ms, from 100 to 80 V – 63.152 ms. There is no overshoot, transient processes are aperiodic, which confirms the presence of a phase reserve of about 90° and the correctness of the dynamic calculation. The stabilization error was also calculated from the graph, which amounted to 0.095 %, which does not exceed the required value and confirms the correctness of the static calculation.

Fig. 10 demonstrates transient disturbance processes caused by a step change in load resistance from 10 to 100 Ohm at 1000 ms and back at 1200 ms.

Рис. 10. Переходные процессы по возмущению

Fig. 10. Transient perturbation processes

The graph in Fig. 10 permits to draw the following conclusion: the output voltage regulation time was 9.744 ms.

For educational purposes, students can be shown how the output voltage ripple will change when the value of inductance L1 changes (Fig. 11).

Рис. 11. Пульсации выходного напряжения при изменении индуктивности в диапазоне от 14,75 до 1475 мкГн

Fig. 11. The output voltage ripples when the inductance changes in the range from 14,75 to 1475 μH

Table 1 shows the dependence of the output voltage pulsation amplitude on the inductance value of the discharge device choke.

Table 1

Dependence of the pulsation amplitude on the value of the choke inductance

Inductance (µH)	Pulsation amplitude (V)
14,75	0,451
147,5	0,453
1475	0,454

Fig. 11 and Table 1 allows to conclude that in a step-up converter at a switching frequency f K = 154170.8 Hz, the amplitude of the output voltage pulsations remains practically unchanged when the inductance changes.

As Fig. 12 illustrates, when changing the capacity of the output capacitor, the output voltage pulsations in transient disturbance processes will change.

Рис. 12. Пульсации выходного напряжения при изменении емкости выходного конденсатора в диапазоне от 23,5 до 94 мкФ

Fig. 12. Pulsations of the output voltage when changing the capacitance of the output capacitor in the range from 23,5 to 94 μF

Table 2 shows the dependence of the output voltage pulsation amplitude on the value of the capacitance of the output capacitor of the discharge device.

Table 2

Dependence of the pulsation amplitude on the value of the output capacitor capacitance

Capacitance (μF)	Pulsation amplitude (V)
23,5	0,784
47	0,440
94	0,187

Due to Table 2, we can conclude that the larger the value of the output capacitor, the smaller the pulsation amplitude.

We are constructing the impedance-frequency characteristic (IFC) in Fig. 13 for different values of the inductance of the choke L1.

Рис. 13. Графики ИЧХ при изменении индуктивности в диапазоне от 14,75 до 1475 мкГн

Fig. 13. Graphs of the impedance-frequency characteristic (IFC) with a change in inductance in the range from 14,75 to 1475 μH

Based on Fig. 13, we can conclude that the lower the inductance of the choke, the higher the resonant surge frequency.

Table 3 presents the dependence of the resonant emission frequency to IFC on the inductance of the discharge device choke.

Table 3

Inductance (µH)	Frequency (kHz)
14,75	3,896
147,5	1,234
1475	0,370

Dependence of the resonant emission frequency on the inductance of the discharge device choke

We are constructing the impedance-frequency characteristic (IFC) in Fig. 14 for different values of the output capacitor C1.

Рис. 14. Графики ИЧХ при изменении емкости выходного конденсатора в диапазоне от 4,7 до 470 мкФ

Fig. 14. IFC graphs with a change in the capacity of the output capacitor in the range from 4,7 to 470 μF

Fig. 14 allows to conclude that the less capacity of the output capacitor, the higher the resonant surge frequency.

Table 4 presents the dependence of the resonant emission frequency to IFC on the capacity of the output capacitor of the discharge device.

Table 4

Dependence of the resonant emission frequency on the value of the capacitance of output capacitor

Capacity (μF)	Frequency (kHz)
4,7	3,842
47	1,230
470	0,418

We construct the IFC in Fig. 15 for different values of the gain factor K VAS .

Рис. 15. Графики ИЧХ при изменении коэффициента усиления K _УС в диапазоне от 553,6 до 55360

Fig. 15. IFC graphs with a change in the gain factor K VAS in the range from 553,6 to 55360

Fig. 15 make it possible to conclude that the higher the gain factor K _VAS, the smaller the active component of the IFC.

Table 5 shows the dependence of the change in the value of the active component of the IFC on the coefficient of the amplifier-summer of the discharge device.

Table 5

Dependence of the change in the active component of the IFC on the coefficient of the amplifier-summer

K VAS	Active component of the IFC (mOhm)
553,6	2,397
5536	0,255
55360	0,025

Conclusion

The developed simulation model of the discharge device allows to verify the correctness of static and dynamic calculations by constructing graphs that determine the cutoff frequency, phase margin, and regulation time. The discharge device model also permits the computational experiments to study the static and dynamic properties, stability and quality of transient processes of the discharge device.

The discharge device model illustrates the capabilities of the simulation model and makes it possible to evaluate the impact of changes in the parameters of key circuit elements: choke inductance, capacitor capacitance, and load on transient processes and IFC.