According to the H-bridge inverter model in
Figure 2 and the welding transformer with output rectifier model in
Figure 3, the Matlab/Simulink model of the RSW system can be made, as shown in
Figure 5. The model of the RSW system can be made using any other appropriate software that allows time-dependent simulations of electrical circuits. On the left side of
Figure 5 is the model of an H-bridge inverter with a constant DC link voltage
, the three-phase input rectifier is not modeled. The transistors or switches are modeled as voltage-controlled switches with a neglected resistance in a closed state (
) and very high resistance in an open state (
). Each switch has a backward diode that protects it against large, induced voltages when switching off inductive loads. The parameters of the backward diode model are set on neutral values, due to the neglected impact on the currents and voltages in the model. The forward voltage is set to
, the on-resistance is
, and the off-resistance is
. The welding transformer with the central tape on secondary winding is in Simulink (
Figure 5), modeled as two separate ideal one-phase transformers, with primary windings connected in parallel and secondary windings connected in series. The only parameter of both transformers is the turn ratio, set to 55. All the other parameters of the RSW model in
Figure 5, the resistances, leakage inductances, and diodes are set according to the values in
Table 1.
According to the Simulink model in
Figure 5 and the parameters set, the numerical calculations were made at the PWM frequency
and the duty cycle
, for example. The numerically calculated time series for 20 periods are shown in
Figure 6. In the first graph, the time series of the PWM voltage
is shown in the second graph, the time series of welding current
and both secondary windings (
,
) are shown, and, in the last, third graph, the time series is shown of the primary current. In
Figure 6, the welding current
reaches the steady state value in approximately 20 time periods. Due to the relatively long time series of the transformer’s voltages and currents in
Figure 6, their analysis is difficult, so, in
Figure 7, the same time series is shown for only one period of time for more straightforward observation.
In
Figure 7, the RSW voltage and currents are shown, where it can be seen that the welding current increases relatively slowly when the positive or negative voltage
is applied and decreases relatively slowly when the voltage
. The welding current
is changing according to two different time constants. The primary current
in the third graph in
Figure 7 increases relatively slowly when the positive or negative voltage
is applied but when the voltage
changes its value, the primary current increases or decreases relatively fast. The primary current
is changing according to two very different time constants. The time intervals when the primary current increases or decreases very fast are relatively short, but significantly impact the steady-state welding current. The time series of welding
and primary current
in separate time intervals look like straight lines define them and can be described analytically with linear equations; that is not true. The currents in each time interval can be described analytically as a step response of a first-order system, as explained below.
In
Figure 7, some of the time intervals are very short, so it is not easy to mark the beginnings and ends of these intervals. For that reason, in
Figure 8, the time series are sketched again with intentionally longer time intervals, and denotations are added at the beginning and end of each time interval for a more straightforward explanation. The time series in
Figure 8 are shown for one and a half of the periods, due to the aforementioned exceptionally shorter
time interval in the first half period to prevent iron core saturation. If the analytical solution of the welding and primary current is described for one and a half of the periods, the description for all subsequent periods is the same.
3.4.2. Time Interval , Increasing Welding and Primary Current
In this time interval, the switches
and
are closed, and the PWM voltage
is applied to the welding transformer. The time
is the beginning of the voltage pulse
, the time
is the end of the voltage pulse defined by Equation (6). Due to the applied voltage,
, the primary
and welding current
starts to increase, and the energy from an inverter is delivered to the weld. According to the induced voltages on the transformer’s secondary, the diode
in
Figure 3 prevents the current
from flowing. Therefore, the equivalent circuit can be simplified, as shown in
Figure 9. The diode
is modeled as a serial connected resistance
and voltage source
in the simplified equivalent circuit.
In an equivalent circuit, in
Figure 9, all resistances and leakage inductances are connected in series; therefore, the equivalent inductance
by Equation (12) and resistance
by Equation (13), can be calculated to define the time constant
, defined by Equation (14).
Serial-connected equivalent resistance
, inductance
and the diode’s voltage source
define the first-order system supplied by a step voltage
. Therefore, the time-dependent welding current
can be described analytically by Equation (15). In Equation (15) the voltage step is the H-bridge voltage
referred as secondary
, is reduced by the diode’s voltage source
. The initial value of the welding current in Equation (15) is zero, because the welding current at the time
. The variable
in Equation (15) is general, but to calculate the welding current
for any time in the interval
to
, the variable
can be only between
and
, where time
is denoted in
Figure 8.
At the end of this interval,
is also the beginning of the next time interval
. Therefore, it is essential to calculate the initial value of welding and primary current at the time
. The initial value of welding current
is defined by Equation (15) if considering
, as it is defined by Equation (16).
In
Figure 9, all three currents are equal
. Therefore, Equation (15) also defines the primary current
, and Equation (16) also defines the initial value of the primary current
, if they are referred to as primary (
).
3.4.3. Time Interval , Fast Decreasing Primary Current
In this time interval, the inverter is in the
stage, so all the switches (
) are open. The welding current
decreases from the initial value
, the secondary current
decreases fast from the initial value
, the secondary current
increases fast from the initial value 0. The primary current decreases fast from the initial value
to zero. Although all the switches (
) are open, the primary current
cannot fall to zero instantly, due to the accumulated magnetic energy in the leakage inductances. The primary current now flows through the integrated antiparallel diodes of the inverter (
and
) in
Figure 2. Therefore, the H-bridge voltage is opposite
. If the beginning of this time interval
is defined by Equation (6), the end of this interval
is not so trivial to determine. The end of this interval
is defined by the primary current
, and how long it is decreasing from the initial value
to the zero
; this time interval is denoted by
in
Figure 8. To derive an analytical solution for primary current
, the equivalent circuit must be defined, that describes the fast decreasing primary current in this time interval. The analysis shows that the equivalent circuit in
Figure 10 defines the dynamic behavior of the primary current, where only specific resistances and leakage inductances have influence. The circuit parameters
,
,
,
and diode’s voltage source
do not influence the dynamic behavior of the primary current in this interval.
In
Figure 10 it can be seen that the inductances
and
are connected in parallel and then connected in series with
and
. Therefore, the equivalent inductance
can be defined by Equation (17). The equivalent resistance
of the circuit can also be defined by Equation (18).
The equivalent inductance
and resistance
define the time constant
by Equation (19):
The serial-connected
and
present the first-order system with the step voltage supply
in this time interval. Therefore, the analytical solution for primary current can be defined by Equation (20):
The end of the time interval
can now be derived by Equation (20). Previously, it has been mentioned that the time interval
is over when the primary current
decreases from the initial value
to zero. After substituting the primary current value
,
, and
into Equation (20), we obtain a new expression from which the time interval
can be derived as:
The end of the analyzed time interval
can now be defined by Equation (22)
Equation (20) is an analytical solution for the dynamic behavior of the primary current
in time interval
, where the starting time
is at the beginning of the time interval
. The initial condition
for the next time interval
is zero
but is defined by Equation (23), where considering
and voltage step
:
The welding current
is decreasing from the initial value
to
in the analyzed time interval
, according to the equivalent circuit in
Figure 11. From the circuit, only the parameters from the secondary define the dynamic behavior of the welding current
. At the same time, the primary current
is also flowing, and voltage is induced at the secondary, but it does not influence the welding current
, due to the symmetrical secondary branches. The secondary current
is decreasing from the initial value
and
is increasing from the initial value
(see
Figure 8). Both currents are equal at the end of this time interval
, and their values are half of the welding current
.
According to the circuit in
Figure 11 the equivalent inductance
and resistance
can be determined by Equations (24) and (25), that define the time constant
by Equation (26).
Serial-connected equivalent resistance
and inductance
present the first-order system with the step voltage supply
. Therefore, an analytical solution for welding current can be defined by Equation (27). In Equation (27)
is the voltage source of the diode, which always works as a load; therefore, the welding current decreases a little faster, and the initial value of the welding current
is defined by Equation (16).
Equation (27) is an analytical solution for the dynamic behavior of the welding current
in time interval
, where the starting time
is at the beginning of the time interval
. The initial condition
for the next time interval
is defined by Equation (28), where, considering
and the voltage step is the diode’s voltage source
:
3.4.4. Time Interval , Decreasing Welding Current
In this time interval the inverter is still in the
stage, so all the switches (
) are open, and the primary current does not flow anymore
. The welding current
still decreases according to the time constant
and Equation (27). The only difference is the initial value of the welding current
defined by Equation (28), and the starting time
is at the beginning of the time interval
.
The initial condition
for the next time interval
is defined by Equation (29), where considering
and voltage step is still the diode’s voltage source
, so we get Equation (30). The time
defines the beginning of the voltage pulse
, when the negative
is applied to the welding transformer, the time
is defined by Equation (7) and
by Equation (22).
3.4.5. Time Interval , Fast Increasing Primary Current
In this time interval, the inverter is in the
stage, so the switches
and
are closed, the negative voltage
is applied to the welding transformer, and the primary current increases fast from the initial value
to the welding current
. The secondary currents are the same at the time
with the value
, the current
decreases fast to zero
, and the current
increases fast to the value of the welding current
. Although the voltage
is applied to the transformer, the welding current still decreases, as is shown in
Figure 8. The beginning
of this time interval
is defined by Equation (7), the end time
depends on the time that the primary current needs to increase from the initial value
to the value of the welding current
. This time interval is, in
Figure 8, denoted by
. The welding current in this interval is defined by Equation (27) or (29). The only difference is the initial welding current
, and the starting time
is at the beginning of the time interval
, so we get Equation (31):
The primary current increases from the initial value
according to Equation (20), if the initial value
and step voltage source
are taken into account to get Equation (32):
The time interval
is over when the primary current from the initial value
reaches the value of the welding current
, which is still decreasing in this time interval
. Therefore, Equations (31) and (32) must be equal, and
must be considered to get Equation (33):
The time interval
appears on the left and right sides of Equation (33) and should be derived from it, but it is not possible, because this equation represents a transcendental equation and an analytical solution does not exist for
. A solution for a time interval
in Equation (33) can be found only numerically. When the
is a known value, the time
can be defined by Equation (34):
The initial value of the welding current
for the next time interval
is defined by Equation (31) if
is considered to get Equation (35):
The initial value of the primary current
for the next time interval is defined by Equation (32) if the
is considered to get Equation (36):
3.4.6. Time Interval , Increasing Welding and Primary Current
In this time interval, the inverter is still in the
state, so the switches
and
are closed, the negative voltage
is still applied to the welding transformer. The primary current
, welding current
and
are equal
, the current of the first secondary branch is zero
. This time interval is like the interval
. The difference is that the voltage supply is now negative
and the initial condition is not zero. The analytical solution for the primary and welding currents can be defined based on the equivalent circuits in
Figure 9, due to the symmetrical secondary branches
and the same diode parameters
and
. Equation (15) is the analytical solution for increasing the welding current
in time period
, where the initial current is zero.
Based on Equation (15) the analytical solution for the primary current in this time interval (
) can be defined by Equation (37), if the negative voltage supply
and the initial current condition
defined by (36) are considered:
Due to the negative voltage supply in Equation (37), the initial condition and the current are negative values. The welding current is always positive, due to the output rectifier. Therefore, the analytical solution for the welding current is . In Equation (37), the diode’s negative voltage always works as a load, therefore, it must be subtracted from the negative voltage supply .
Equation (37) is the analytical solution for the dynamic behavior of the welding and primary current in time interval
, where the starting time
is at the beginning of the time interval
. The initial condition
for the next time interval
is defined by Equation (37), where, considering
to get Equation (38). The time
is defined by Equation (8) and time
is defined by Equation (34).
3.4.7. Time Interval , Fast Decreasing Primary Current
This time interval is like the interval
, therefore, a detailed explanation is not necessary. The difference is that the voltage supply is positive
because, with all the switches
open, the primary current flows through the integrated antiparallel diodes of the inverter (
and
). The primary current decreases fast from the negative initial value
to zero
, the welding current decreases from the initial value
to a new value
, the secondary current
decreases fast from the initial value
to a new value
, the secondary current
increases fast from the initial value
to a new value
. At the end of this time interval
both secondary currents are half of the welding current
. The beginning of this time interval
is defined by Equation (8), the end time
depends on the time period
denoted in
Figure 8. The time period
is, like the time period
, defined by the primary current, i.e., how long does it take to fall from the negative initial value
to zero? To define the time period
, the analytical solution for the falling primary current must be defined first. The primary current is falling according to the equivalent circuit in
Figure 10 and Equation (20), where the voltage supply
and the initial value of the primary current
, defined by Equation (38), are considered to get Equation (39):
The end of the time interval
can now be derived by Equation (39). Previously, it has been mentioned that the time interval
is over when the primary current
decreases from the initial value
to zero. If, in Equation (39), the primary current value at
is zero
,
and
, Equation (40) can be derived that defines the time interval
The end of the analyzed time interval
can now be defined by Equation (41):
Equation (39) is an analytical solution for the dynamic behavior of the primary current
in the time interval
, where the starting time
is at the beginning of the time interval
. The initial condition
for the next time interval
is zero
, but is defined by Equation (42), where, considering
and voltage step
:
The welding current
is decreasing in this time interval, although
and the primary current
is also flowing, and voltage is induced at the secondary, but it does not influence the welding current due to the symmetrical secondary branches. The welding current is decreasing according to the same Equation (43) as in all previous time intervals between the time
and
, only the initial welding current is different
, defined by Equation (38).
Equation (43) is an analytical solution for the dynamic behavior of the primary current
in the time interval
, where the starting time
is at the beginning of the time interval
. The initial condition
for the next time interval
is defined by Equation (43) if
is considered to get Equation (44)
3.4.9. Time Interval , Fast Increasing Primary Current
This time interval starts at time when the positive voltage supply is applied to the transformer in the second period due to the closed switches and , like in the first half of the first period. The difference between the start of the positive voltage supply in the first and second periods is the initial condition of welding current at and . In the first period, the initial value of the welding current at time is zero . Therefore, there is no time interval for the fast-increasing primary current . In the second period at the time , the welding current is not zero ; therefore, the primary current increases fast from zero to the value of the welding current . The fast increase of the primary current is the difference between the first and second periods at the beginning of the positive voltage supply . Therefore, the explanation and analytical solutions for the welding and primary current are given for one and a half periods instead of only one period.
The time interval
is like
. The only difference is the positive voltage supply
and the initial value of the welding current at the time
; therefore, a detailed explanation is unnecessary. This analyzed time interval
ends when the primary current from the initial value zero
reaches the value of the welding current
in time
, denoted in
Figure 8. To define the
the analytical solution for the decreasing welding current
and fast increasing primary current
must be defined first. The welding current decreases according to Equation (31) if the initial value of the welding current
is considered to get Equation (47)
The primary current
increases fast from the initial value
according to Equation (32), if the initial value
and positive step voltage source
are considered to get Equation (48)
The time interval
is over when the primary current from the initial value
reaches the value of the welding current
, which is still decreasing in this time interval
. Therefore, Equations (47) and (48) must be equal, and
must be considered to get Equation (49)
The time interval
appears on the left and right sides of Equation (49) and should be derived from it, but it is not possible because this equation represents a transcendental equation and an analytical solution for
does not exist. A solution for
can only be found numerically. When the
is a known value, the time
can be defined by Equation (50), where
and
is defined by Equations (5) and (11)
The initial value of the welding current
for the next time interval
is defined by Equation (47) if
is considered to get Equation (51)
The initial value of the primary current
for the next time interval is defined by Equation (48) if the
is considered to get Equation (52)