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Comparative study of the design and simulation of an AC to high DC voltage generation circuit
Journal of Electrical Systems and Information Technology volume 9, Article number: 10 (2022)
Abstract
Decades after the invention of the Cockcroft–Walton voltage multiplier, it is still being used in broad range of high voltage and ac to dc applications. High voltage ratio and high efficiency are its main features. Due to the limitations of original circuit, reducing the output ripple and increasing the accessible voltage level motivated scientists to propose new topologies. However, over the worldwide bibliography, most of CockcroftWalton voltage designers persist in using equal capacitances in every stage without considering an optimal design. The aim of this paper is to do a comparative study of the design and simulation of a fixed model (conventional model used by most authors) and a variable model (new design) of generating High Voltage Direct Current (HVDC) based on Cockcroft–Walton voltage multiplier that stresses on the choice of the adequate capacitance values to reduce the output voltage drop, produce less ripple and the calculations of the optimal number of stages that is necessary to produce the desired output voltage with a better performance. The generation of HVDC based on Cockcroft–Walton voltage multiplier and an eight stage was used for simulations and theoretical analysis which yielded up to 4.4 kV DC from an input voltage of 230 V, 50 Hz ac supply. The results are compiled from the simulations done on MATLAB/SIMULINK, by the designs and simulations characteristics of the models the performances, output voltages and ripple voltages per stage have been compared.
Introduction
In this modern world, HVDC system provides a secure and stable asynchronous interconnection of power grids that operate on different frequencies. In addition, HVDC provides instant and precise control of power flow. HVDC transmission is becoming more popular in the present scenario of bulk power transmission over longdistances, it is necessary to study the testing of various insulation materials. To get HVDC output from a smaller input voltage many methods have been utilized to perform this task. Some of the most common methods used to produce a voltage larger than the power supply voltage are stepup transformers [1], voltage doublers [2, 3], multiplier circuits [4,5,6], charge pump circuits [7], switchedcapacitor circuits [8] and boost or stepup converters [9]. High voltage conversion ratio is becoming increasingly essential in a multitude of industrial and laboratory research applications [10, 11], such as frontend stages for batteries or photovoltaic sources, DC backup systems, UPS devices, step down inverters and many more. Voltage Multipliers and especially the halfwave Cockcroft–Walton voltage multiplier are at the forefront of these applications and methods [12,13,14]. The diode capacitor topologies are more suitable [15]. In 1920 Greinacher, a young physicist published a circuit [16] which was improved in 1932 by a British physicists John Douglas Cockcroft and Irish physicists Ernest Thomas Sinton Walton to produce highenergy positive ions [17]. Cockcroft and Walton invented the CockcroftWalton voltage multiplier (CWVM).
Cockcroft–Walton voltage multiplier
Voltage multiplier circuits are primarily used to develop high voltages where low current is required. The output voltage of voltage multiplier circuits may be several times more than the input voltage. For this reason, voltage multipliers are used in special applications where load is constant and has high impedance or where the stability of the input voltage is not critical. The advancement in studies focus on different hardware circuits that can offer high voltage conversion ratio, small output voltage ripple, high efficiency, simple design and low cost. Among these, the Half Wave CWVM is highly used as AC to DC convertor [18]. A half wave voltage doubler is shown in Fig. 1.
During the first half of the supply period, diode D_{1} is turned on and diode D_{2} is turned off, capacitor C_{1} is being charged to the peak value of input voltage. During the second half cycle, diode D_{2} is turned on and diode D_{1} is turned off, charging capacitor C_{2} to the twice the peak value of input voltage because capacitor C_{1} (charged to Vs) and input voltage (Vs) now act as series aiding voltage source. When input voltage returns to its original polarity, diode D_{2} is again reverse biased (off), and then the capacitor C_{2} will be discharged through the load R. The time constant RC_{2} is so adjusted that C_{2} has little time to lose any of its charge before the input polarity reverses again. During the negative half cycle, diode D_{2} is turned on, capacitor C_{2} will be recharged again until voltage across it is again equal to 2Vs. CockcroftWalton Voltage value increases as it goes through each stage of the voltage multiplier. Multipliers are made up of multiple stages, they may be classified as voltage doublers, triplers, quadruplers, etc. The classification depends on the ratio between the output voltage and the input voltage.
Design of the multiplier circuit
Diode selection
Diode at each level of the voltage multiplier performs two major operations as firstly, it conducts to discharge every capacitor at each level and secondly it blocks the reverse voltage to prevent the unwanted conduction [20]. To choose our diode the following basic device parameter must be considered.

Repetitive peak reverse voltage
Multiplier circuit reverse voltage seen by each diode is 2Vm in CockcroftWalton voltage multiplier. So, the device must be selected with reverse voltage (VRRM) setting of at least 2Vm. Therefore, we have to select the diode voltage rating 2Vmax for the safety purpose [4].

Frequency of input signal
While selecting the diodes for the rectifier, the frequency of input voltage to the multiplier circuit must be considered. For symmetrical input signals, the device chosen must be capable of switching at speed faster than the rise and fall times of the input [21]. If the reverse recovery time is too long, the efficiency and regulation of the device will be affected. In the worstcase insufficient recovery speed will result in accessing heating of device. In this case, the device will be permanently damaged. The reverse recovery time is conditioned by the circuit and the conditions used to make the measurement. Reverse recovery Time specification should be used for qualitative, not quantitative purposes since condition specified for the measurement rarely reflects those found in actual real life circuit operation [4]. Decreasing current flow in the multiplier circuit allows higher input frequency to be used. An increase in current flow has been the opposite effect. Ideally, the network load multiplier should draw no current.

Peak forward surge current (Ifsm)
Most rectifier diodes have a peak forward surge current rating. This value (rating) corresponds to the maximum peak value of single sinusoidal halfwave which, when superimposed on the nominal load current of the device, can be conducted without damaging of rectifier. This value becomes important when considering the large capacitance associated with multiplier network. Due to the capacitive loading effects on the rectifier, surge currents can be produced. With a high stepup turn ratio of the transformer, capacitor C_{1} on the secondary side is considered to be the largest one. Its value can be determined as follows:
\({\text{Where}}; C_{1} = {\text{first multiplier circuit capacitance}}\),
\(N = {\text{ turns ratio of high voltage transformer}}\frac{{N_{2} }}{{N_{1} }}\).
When the circuit is turned on, large current develops in the primary side as this effective capacitance begins to charge. On the secondary side, large surge current can flow through the rectifiers during initial capacitor charging at turn on. The addition of a series resistance Rs can greatly reduce these current surges as well as those in the primary circuitry.

Forward current (\({\varvec{I}}_{{\varvec{O}}}\))
In ideal multiplier circuits, the load will draw no current. Ideally, large current flow through the rectifier occurs during capacitor charging. Therefore, device with very low current rating (100 mA) and in case of cables. Micro amperes are also used. It must be noted that forward current and forward surge current rating are related [4]. Both are the function of silicon die area. It is truly speaking that device with a high surge current rating \(I_{fsm}\) will also have high forward current \(I_{O}\) rating and vice versa.

Forward voltage (\({\varvec{V}}_{{\varvec{f}}}\))
In practice the forward voltage drops \({\varvec{V}}_{{\varvec{f}}}\) of the rectifier does not have a significant effect on multiplier networks on the overall efficiency [4]. The calculation of the voltage drop is given by Eq. (3). For a half wave doubler (two stages) multiplier having an output voltage of 8000 V, and considering a forward voltage drop of the rectifier diodes as 2 V (for a forward current of 100 mA), its voltage drop is 0.05%.
Capacitor selection
The size of capacitors used in multiplier circuit is proportional to the frequency of input signal. Capacitor used in off line, 50 Hz application is typically in the range of 1.0 to 200 microfarad. The rated voltage of capacitor is determined by the type of multiplier circuit. The capacitor must be able to withstand a maximum voltage depending upon the numbers of staged used. A good thumb rule is to choose capacitor whose voltage rating is approximately twice that of actual peak applied voltage. Due to the AC impedance of the capacitors, there is a voltage drop \(V_{{{\text{drop}}}}\) and a peaktopeak voltage ripple \(\delta V\) when the circuit is loaded [22,23,24]. Based on the theoretical analysis and the assumption presented in [22, 25, 26], the ratio \(X\) of the output voltage \({V}_{out}\) over \({V}_{max}\) the maximum value of the sinusoidal input supply voltage, is given by the equations;
where \(X_{nl}\) is the noload voltage ratio,\(n\) the number of stages, f the frequency of the voltage supply, \(I_{l}\) the average value of the load current and
For the Fixed selection of capacitors;
and for the Variable selection of capacitors
where \(i\) is the number of every stage and C the capacitance of the last stage (base capacitance). Considering that the total capacitance \(C_{{{\text{tot}}}}\) is the sum of all the capacitances of each topology
Voltage gain for the two models can be calculated as a function of \(C_{{{\text{tot}}}}\).
Figure 2 provides information of the gain \(X = \frac{{V_{{{\text{out}}}} }}{{V_{{{\text{max}}}} }}\) of the Fixed and Variable model against the number of stages for a fixed value of \(\frac{g}{{f \times C_{{{\text{tot}}}} }}\).
Selection of number of stages

Fixed model selection of number of stages
The selection of the number of stages in a Cockcroft Walton voltage multiplier circuit depends the desired output level. For fixed values of capacitances in the various stages, the number of stages should be in accordance with the required output voltage and the voltage drop till last stage [27]. The equation to calculate the number of stages is as follows:
where;
If we gets the value in fraction in the above equation then consider the nearest greater integer for the selection of the number of stages.

Variable model selection of number of stages
CockcroftWalton output voltage as a function of number of stages n, peak input voltage \(V_{max}\), output current \(I_{l}\), and product \(fc\) of frequency \(f\) and capacitance c [23].
From the \(V_{{{\text{out}}}}\) expression, due to the fast growth of the n^{3} term in the negative term (voltage drop relative to the noload value), if n starts from zero and increases without changing other parameters, \(V_{{{\text{out}}}}\) first increases, then reaches a peak, and then decreases [28]. The derivative of \(V_{{{\text{out}}}}\) with respect to n is zero at peak \(V_{out}\) deriving the optimum number of stages (Fig. 3).
Optimum n depends on input and output voltages \(V_{{{\text{max}}}}\), and \(V_{{{\text{out}}}}\) only, not on output current nor frequency or capacitance. Knowing the ratio of \(V_{{{\text{out}}}}\) to \(V_{{{\text{max}}}}\) which is \(X, n_{{{\text{opt}}}}\) can further be simplified as:
Ripple factor and ripple voltage

Ripple voltage for Fixed model
Ripple of the nstage multiplier for the Fixed model will be [4];
For n stage total ripple is given by;
For \(C_{2n} = C_{2n  2} = C_{2n  4} = C_{2n  6} = \ldots = C_{2} = C\)

Ripple voltage for Variable model
Ripple of the nstage multiplier for the Variable model is given as
Ripple voltage is the magnitude of fluctuation in DC output voltage at a specific output current (assuming AC input voltage and AC input frequency are constant) [29]. In addition, ripple is function of number of stages and the switching frequency for a fixed capacitor design. Hence, varying frequency ripple can be reduced and can be treated as high frequency switching of CWVM network [30]. For pertaining the same threshold Fig. 4 provides information regarding variation of ripple with respect to the various frequency level.
Voltage drop for fixed and variable model
Generally, in order to reduce the complexity of the circuit equations, the calculation of the diode voltage drop is neglected. Also, parasitic effect of diode and capacitor is neglected as its contribution is very small in the system. There are two modes of operation of CWVM the no load operation and operation under loaded condition [30]. There is drop due to internal behavior and load applied when the load is connected. This drop (considerably high) reduces the output voltage significantly.
Total drop is:

For the Fixed model
On solving for each stage drop, the Fixed model drop gives us [31];
Adding all n voltage drops gives the total voltage drop on load:

For the Variable model
The total voltage drop [23] for the variable model gives us;
Figure 5 shows us the Variation of Voltage drop against a fixed frequency.
Simulations
The parameters for the simulation are presented in the table below (Table 1):
Circuit diagram
See Fig. 6
Comparison of the fixed capacitors model and the variable capacitors model
Here the output voltage, performance and the ripple voltage of both systems will are compared.

Output voltage comparison
From the simulation results show in Fig. 7, it is clear that the output voltage of the Variable capacitors model is greater than that of the fixed capacitors model. From stages 5 to 8, the increase in the output voltage is from 250 to 500 V.
Performance comparison
From the output waveform, the rising and settling times of the fixed capacitors model and the variable capacitors model can be compared below;
From the simulation results and the values. It is clear that the Variable model is more rapid than the fixed model, both models are stable. Therefore, the Variable model is more performant than the fixed model (Figs. 8, 9, 10, 11 and 12).

Voltage ripple comparison
From the simulation and the output waveform the voltage ripple of the Fixed and Variable model is presented:
Steadystate ripple of the generated HVDC versus time for stage five through stage eight of both models. From the ripple it is clear that the voltage ripple of the Variable model is lesser than that of the fixed model and for both models the higher the stage there is an increase in the voltage ripple.
Conclusion
In this work, the design and simulation of two methods are presented based on CockcroftWalton Voltage multiplier. The output voltage, performance and voltage ripple were compared for the two different models. A theoretical analysis is held that produced new and improved equations about CockcroftWalton’s voltage gain as well as new formulas are introduced that give the Optimal number of stages that provide the proper voltage gain. The theoretical analysis and the simulation in MATLAB/ SIMULINK R2015b reveals that the Variable model is an Optimal design of the CockcroftWalton voltage multiplier due to its higher performance, higher output voltage per stage and a less voltage ripple.
When higher magnitude of output high voltage DC supply is required without changing the input transformer voltage level, the CockcroftWalton Voltage multiplier circuit is used. It is used only in special applications where the input voltage stability is not critical. This kind of high voltage DC power supply test set is of simple control; low cost, portable due to its light weight robust and high reliability. Different high voltage DC output magnitudes are taken from different stages without changing the input voltage.
Our work can be enhanced to generate an HVDC up to the range of 25–50 kV by increasing the number of stages thus it will be very useful for field testing of HV cables of different voltage grade, as a DC source for impulse charging unit of impulse generators.
Data Availability
The datasets analyzed during the current study are available from the corresponding author or reasonable request.
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Acknowledgements
This research was carried out during the end of cycle project at the University of Buea, Cameroon.
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All authors have read and approved the manuscript. Pierre Kenfack made all the simulation work with the help of Edung Fabrice Nkale. Abraham Dandoussou helped Pierre Kenfack with the basics of the experiment and how to use some key functions of MATALB®. Edung Fabrice Nkale helps the team with proofreading of the manuscript to make it academically appropriate. All authors read and approved the final manuscript.
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Kenfack, P., Dandoussou, A. & Nkale, E.F. Comparative study of the design and simulation of an AC to high DC voltage generation circuit. Journal of Electrical Systems and Inf Technol 9, 10 (2022). https://doi.org/10.1186/s43067022000514
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DOI: https://doi.org/10.1186/s43067022000514
Keywords
 Cockcroft–Walton voltage multiplier
 HVDC
 Capacitance
 Performance