Simulation Study for a Transformer Based Voltage Regulator
Report (Ref: PCSR‐SC 01‐2011)
Professor Jihong Wang
School of Engineering, University of Warwick, Coventry CV4 7AL, UK
Sept. – Dec. 2011
Executive Summary
EMSC (UK) Limited requested support from the Science City Research Alliance at the
University of Warwick under the Energy Efficiency and Demand Reduction Project to
undertake a programme of work involving mathematical modelling and a simulation
study of a transformer technology based voltage regulator being developed by the
company. The objective of the study is to gain a clearer picture of the working principles
of the voltage regulator and explain the induced power flow relationship of the voltage
regulator. The agreed work has completed and is summarised in this report.
The main findings of the study are:
1) The mathematical and simulation model is presented in the report, which explains
how the voltage regulator works.
2) The theoretical analysis and simulation results prove that the voltage regulator can
lead to energy saving.
3) The overall power consumption is reduced because the negative power is induced
as feedback power to the source. Virtually, this power can be considered as power
“generated” from the load side.
1.
Background
EMSC (UK) Ltd wished to take advantage of the Science City Energy Efficiency and
Demand Reduction initiative to develop a programme of collaborative evaluation of the
company’s new design for a voltage regulator/controller device.
EMSC had tested units and had identified a reverse power flow from the controller
winding towards the electrical supply. The reverse power flow was directly proportional
to the applied load. The company requested support utilizing expertise and resources
from the Science City Research Alliance at the University of Warwick to assist the
company through mathematical modelling of power flows within the regulator/controller
unit followed by a simulation study.
The voltage regulator/controller to be developed at EMSC (UK) is based on transformer
technologies including autotransformers. The transformer can be arranged to work either
with a fixed turn ratio or variable turn ratios as shown in Figure 1. With the designed
transformer polarity arrangement, the electrical current passing through the primary
winding has almost 1800 phase shift from the secondary winding’s current. This explains
why the total power is reduced by connecting the regulator between the power source and
the load, which are explained through modelling and simulation study as described in the
following sections.
2.
Mathematical modelling
The voltage regulator to be developed at EMSC (UK) has the circuit structure as shown
in Figure 1. Without considering any losses, the ideal transformer can be explained by the
structure shown in Figure 2.
(a)
(b)
Figure 1. Autotransformer with a fixed turn ratio (a) and tapped changeable turn ratios (b).
ip (t)
+
+
Primary
Winding
is (t )
v p (t)
vs (t)
Secondary
Winding
-
Figure 2. Illustration of an ideal single-phase transformer
The relationships between the primary and secondary (or input and output) voltages and
currents can be described by the following equations:
v p (t)
v s (t)
=
Np
Ns
= a , N p i p (t ) = N s is (t ) and
i p(t )
is (t )
=
Ns 1
=
Np a
where N p is the number of primary winding turns, N s is the number of secondary
winding turns and a defined as the turn ratio: a =
In terms of phasor quantities, we have:
Np
Ns
.
Vp (t )
Vs (t )
= a and
I p (t )
I s (t )
=
1
a
The transformer polarity is marked by a “dot” and the polarity relationships are described
as follows:
1) If the primary voltage is positive at the dotted end of the winding with respect to
the undotted end, then the secondary voltage will also be positive at the dotted
end. Voltage polarities are the same with respect to the dots on each side of the
core.
2) If the primary current of the transformer flows into the dotted end of the primary
winding, the secondary current will flow out of the dotted end of the secondary
winding.
The power in an ideal transformer can be calculated by:
‐
The power supplied to the transformer by the primary circuit:
Pin = V p I p cos ϕ p ,
where ϕ p is the phase angle between the primary voltage and the primary current.
‐
The power supplied to the transformer by the secondary circuit to its load:
Pout = Vs I s cosϕ s ,
where ϕ s is the phase angle between the secondary voltage and the secondary
current.
Since the phase angles for the voltage and current are not affected by the ideal
transformer, the primary and the secondary windings of an ideal transformer have the
same power factor.
For an ideal autotransformer as shown in Figure 3, we can derive the current and voltage
relationships below.
i p (t)
vs1 (t)
v p (t)
ip1(t)
is (t )
vs (t)
Figure 3. Autotransformer
From Figure 3, we have:
v p (t) = vs1 (t ) + v s (t ) =
1
Ns
v p (t) + v s (t) = v p (t ) + vs (t )
Np
a
1
a −1
v s (t) = (1 − )v p (t) =
v p (t )
a
a
i p (t ) = i p1 (t ) + is (t) = −
is (t ) =
a
i p (t)
a −1
Vp (t )
=
a
a −1
=
a −1
a
Vs (t )
I p (t)
I s (t)
Ns
1
is (t ) + is (t) = (1 − )is (t)
Np
a
For a single-phase transformer in the real world, not all the flux produced in the primary
coil passes through the secondary coil, as shown in Figure 4. The portion of the flux that
goes through one of the transformer coils but not the other one is called leakage flux. The
flux in the primary coil of the transformer can thus be divided into two components: a
mutual flux, which remains in the core and links both windings, and a small leakage flux,
which passes through the primary winding but returns through the air, bypassing the
secondary winding.
Figure 4. Illustration of a non-ideal transformer
The voltages on the primary and secondary windings are:
v p (t ) = e p (t ) + vLp (t )
vs (t) = es (t) + vLs (t)
and
e p (t)
es (t)
=
Np
Ns
=a
On the other hand, when an AC power is connected to a transformer, the current flows in
its primary winding, even when the secondary circuit is open circuited. This current is the
current required to produce flux in a real ferromagnetic core. It consists of two
components:
1) The magnetization current iM which is the current to produce the flux in the
transformer core.
2) The core-loss current ih + e which is the current required to make up for hysteresis
and eddy current losses.
The total no-load current in the core is called the excitation current of the transformer and
it is:
iex = im + ih + e
Considering all the losses of a transformer, a no-ideal transformer has the equivalent
circuit as shown in Figure 5.
IS
Ip
RP
Vp
XP
Rc
RS
X M NP
NS
XS
VS
Figure 5. The equivalent circuit of a no-ideal transformer
Then, the relationship between the primary and the secondary voltage can be described
by:
Vp = ( R p + jX p )I p + aVs + aI s ( Rs + jX s )
or
V p = ( R p + jX p )I p + aVs +
Ip =
1
I s a 2 ( Rs + jX s )
a
1
Is + I M
a
The simulation study will be based on the above equivalent circuit.
3.
Simulation study
A single phase transformer is used in the simulation study and the model is implemented
in Simulink/SimPower simulation platform as shown in Figure 6. The transformer’s turn
ratio is chosen as a =
Np
N
= 5 . Different loads will be connected to the transformer for
s
simulation study.
Primary
i
-
i
+ -
C urrent Measurement
1
AC Voltage Source
2
Linear Transformer
i
+ -
C urrent Measurement1
C
Coonnnn21
+
Secondary
Load
+
- v
Voltage Measurement
Vs, Is
C urrent Measurement2
+
-
v
Voltage Measurement1
Ip, Ip1, Is
Vp, Ip
Figure 6. Simulation implementation
Case I. No load, the secondary winding is connected to an open circuit, as shown in
Figure 7.
Primary
+
Secondary
i
is
-
T o Workspace1
+ i
-
C urrent Measurement
1
2
vs
C urrent Measurement1
AC Voltage SourceLinear Transformer
T o Workspace
+ v
-
Voltage Measurement
+ i
-
Vs, Is
C urrent Measurement2
ip1
T o Workspace5
+
-
v
Ip, Ip1, Is
Voltage Measurement1
Vp, Ip
vp
ip
T o Workspace3
T o Workspace2
t
Cl ock
T o Workspace4
Figure 7. Simulation model diagram with no load connected
The simulation results are shown in Figure 8. It can be seen that the secondary current is
zero but the primary current is not zero as it must have the magnetization current to
generate the magnetic flux.
Figure 8. Simulation results for primary (in purple colour) and secondary winding (in
blue colour) currents (A)
Case II. With a pure resistant load
When a pure resistant load is connected with R = 10Ω, the simulation results are shown
in Figure 9. From the simulation results, it is noticed that the current flow back to the
power source from the primary is negative.
(a) Power supply voltage and current (yellow – voltage (V), purple – current (A) )
(b) Load voltage and current (yellow – voltage (V), purple – current (A))
(c) Supply(yellow), primary winding (purple) and secondary winding (blue) currents (A)
Figure 9. Simulation results while a pure resistant load is connected.
Case III. Connecting with an inductive load
Two load branches are connected to the secondary winding. For each load branch, we
have the load at R=1 Ω and L=0.1 H. The load connection is shown in Figure 10.
Primary
i
+
Secondary
is
T o Workspace1
+
C urrent Measurement
1
2
i
-
vs
C urrent Measurement1
AC Voltage SourceLinear Transformer
Series RLS
CeB
rireasnR
ch
CLBranch1
T o Workspace
+
- v
Voltage Measurement
+ i
-
Vs, Is
C urrent Measurement2
ip1
T o Workspace5
+
-
v
Ip, Ip1, Is
Voltage Measurement1
Vp, Ip
vp
T o Workspace3
ip
T o Workspace2
t
Clock
T o Workspace4
Figure 10. Simulation circuit with inductive loads
The simulation results are shown in Figure 11. From the simulation results, it can be seen
that there is an 1800 phase shift between the primary and secondary currents. Also, the
primary has a negative current flow back to the power supply and the voltage has a phase
lead compared with the current.
Supply input voltage
Supply input c urrent
300
200
100
0
-100
-200
-300
0
0.01
0.02
0.03
0.04
0.05
time (s)
0.06
0.07
0.08
0.09
0.1
(a) Power supply voltage (V) and current (A)
300
Load voltage
Load/Secondary current
200
100
0
-100
-200
-300
0
0.01
0.02
0.03
0.04
0.05
time (s)
0.06
0.07
0.08
0.09
0.1
(b) Load voltage (V) and current (A)
20
Supply input current
Load/Secondary current
Primary current
15
10
5
0
-5
-10
-15
-20
0
0.01
0.02
0.03
0.04
0.05
time (s)
0.06
0.07
0.08
0.09
0.1
(c) Supply, primary winding, and secondary winding currents (A)
Figure 11. Simulation results while an inductive load is connected
Case IV. With a capacitive load
Two load branches are connected to the load side. For each load branch, we have the load
at R=1 Ω and C=0.1 mF. The load connection is shown in Figure 12.
Primary
i
+
Secondary
i
+ -
C urrent Measurement
C urrent Measurement1
1
AC Voltage Source
2
Linear Transformer
Series RLS
CeB
r ireasnR
ch
LC Bra nch1
i
+ -
+
- v
Voltage Measurement
Vs, Is
C urrent Measurement2
+
-
v
Voltage Measurement1
Ip, Ip1, Is
Vp, Ip
Figure 12. Simulation circuit with a capacitive load
The simulation results are shown in Figure 13.
400
Voltage ( V)
Cur r ent ( A)
300
200
100
0
-100
-200
-300
-400
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
time(s)
(a) Power supply voltage (V) and current (A)
300
Voltage ( V)
Cur r ent ( A)
200
100
0
- 100
- 200
- 300
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
time(s)
(b) Load voltage (V) and current (A) (yellow – voltage, purple – current)
20
Primary winding current ( A)
Supply current ( A)
Load current ( A)
15
10
5
0
-5
-10
-15
-20
0
0.01
0.02
0.03
0.04
0.05
time(s)
0.06
0.07
0.08
0.09
0.1
(c) Supply, primary winding and secondary winding currents (A)
Figure 13 Simulation results while a capacitive load is connected
Case V. Mixed load connection
A mixed load, that is, the load with resistance, capacitance, inductors and a motor is
connected to the system. The purpose is to verify the current and voltage phase
relationships while a mixed electrical machinery load connected in.
Primary
+
Secondary
i
is
-
T o Workspace1
C urrent Measurement
1
2
+ i
C urrent Meas urement1
vs
To Workspa ce
0.5
Constant
AC Voltage Source Linear Transformer
+ i
-
S eries RLS
CeB
r ireasnR
ch
L
C Bra
nch1
+
Voltage Meas urement
M+
C urrent Measurement2
-
+ v
-
Tm
M
split
phase
Vs, Is
m
i p1
Single Phase
Asynchronous Machine1
T o Wo rksp ace5
v
Ip, Ip1, Is
Voltage Measurement1
Vp, Ip
vp
ip
T o Wo rksp ace3
T o Wo rkspa ce2
Figure 14. Simulation circuit with a mixed load
The simulation results are shown in Figure 15.
t
Cl ock
T o Wo rksp ace4
400
Voltage (V)
Current (A)
300
200
100
0
-100
-200
-300
-400
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(a) Power supply voltage (V) and current (A)
300
Voltage ( V)
Current ( A)
200
100
0
-100
-200
-300
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(b) Load voltage (V) and current (A)
300
Primary winding current (A)
Supply current (A)
Secondary current (A)
200
100
0
-100
-200
-300
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
time(s)
(c) Supply, primary winding and secondary winding currents (A)
Figure 15 Simulation results while a mixed load is connected
4.
Analysis:
The simulation studies have demonstrated that the primary winding of the
autotransformer has the induced current from the secondary winding. The induced current
has opposite flow direction to the supply current. Therefore, the total current from the
power supply is reduced. This explains the reason why the controller can lead to power
saving and improvement of energy efficiency. Also, this indicates a possibility to
measure the induced current by connecting the primary to a separate AC power source. In
this case, the power output from the source connected to the primary winding will be
negative, that is, the power from the power supply source connected to the secondary
winding is induced, through the primary winding and feedback to the power source. The
simulation for implementation of this idea in Simulink/SimPower is shown in Figure 16
(the separate power source is marked in green background).
Primary
+
Secondary
i
is
-
T o Workspace1
+
C urrent Measurement
1
i
-
vs
C urrent Measurement1
2
T o Workspace
0.5
Constant
AC Voltage SourceLinear Transformer
Series RS
LC
erB
i ersan
RcLhC
AC Voltage Source1 + +
v
-
+
v
-i
Tm
Bran ch1
M+
C urrent Measurement2
m
split
Voltage Measurement
Vs, Is
M phase
ip1
Single Phase
Asynchronous Machine1
Voltage Measurement2
T o Workspace5
Vi ndu
vp
Vi ndu
vp
Vs, Is1
-
Ii ndu
v
Voltage Measurement1
Ip, Ip1, Is
Ii ndu
+
ip
Vp, Ip
ip
C o ntinuo us
po we rg ui
t
Cl ock
T o Workspace4
Figure 16. The primary winding is connected to a separate AC power supply
The induced current is shown in Figure 17. It can be seen from the figure that the current
has opposite sign to the voltage for most of the time periods, which means that the power
output from the second power source (green color) is negative, that is, it absorbs the
power induced.
400
Voltage (V)
Induced current (A)
300
200
100
0
-100
-200
-300
-400
0
0.01
0.02
0.03
0.04
0.05
time (s)
0.06
0.07
0.08
0.09
0.1
Figure 17. Voltage and current on the primary of the transformer.
Furthermore, the power source connected to the primary side of the transformer is
discounted, and then the primary is re-connected to a load (mixed inductance and
resistance), as shown in Figure 18. The induced current will power the load. The induced
voltage and current from simulation are shown in Figure 19.
Primary
+
Secondary
i
is
-
T o Workspace1
+ i
-
C urrent Measurement
1
vs
C urrent Measurement1
2
T o Workspace
0.5
Constant
AC Voltage Source Linear Transformer
LC
erB
iersan
RcLhC Branch1
i Series RS
Series RLC Branch2
+ +
v
-
+ v
-
Tm
M+
C urrent Measurement2
M
split
phase
m
Voltage Measurement
Vs, Is
ip1
Single Phase
Asynchronous Machine1
Voltage Measurement2
T o Workspace5
Vi ndu
vp
Vi ndu
vp
Vs, Is1
-
Ii ndu
v
Voltage Measurement1
Ip, Ip1, Is
Ii ndu
+
ip
Vp, Ip
ip
C o ntinuo us
powe rgui
t
Cl ock
T o Workspace4
Figure 18. The primary winding is connected to a load instead of a source
80
Induced voltage (V)
Induced current (A)
60
40
20
0
-20
-40
-60
-80
0
0.02
0.04
0.06
0.08
0.1
time (s)
Figure 19. Voltage and current on the primary of the transformer.
5.
Conclusions
From the above study, we can conclude the following:
1) The mathematical and simulation model presented in the report explains how the
voltage regulator works.
2) The theoretic analysis and simulation results prove that the voltage regulator can
lead to energy saving.
3) The overall power consumption is reduced because the negative power is induced
and feedbacks to the power source. Virtually, this power can be considered as
power “generated” from the load side.
4) The induced current or power can be measured using a separated power supply by
using an ordinary transformer (not an autotransformer).