PROJECT #1 SINE-∆ PWM INVERTER
JIN-WOO JUNG, PH.D STUDENT
E-mail: jung.103@osu.edu
Tel.: (614) 292-3633
ADVISOR: PROF. ALI KEYHANI
DATE: FEBRUARY 20, 2005
MECHATRONIC SYSTEMS LABORATORY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
THE OHIO STATE UNIVERSITY
1. Problem Description
In this simulation, we will study Sine-∆ Pulse Width Modulation (PWM) technique. We will
use the SEMIKRON® IGBT Flexible Power Converter for this purpose. The system
configuration is given below:
Fig. 1 Circuit model of three-phase PWM inverter with a center-taped grounded DC bus.
The system parameters for this converter are as follows:
ƒ
IGBTs: SEMIKRON SKM 50 GB 123D, Max ratings: VCES = 600 V, IC = 80 A
ƒ
DC- link voltage: Vdc = 400 V
ƒ
Fundamental frequency: f = 60 Hz
ƒ
PWM (carrier) frequency: fz = 3 kHz
ƒ
Modulation index: m = 0.8
ƒ
Output filter: Lf = 800 µH and Cf = 400 µF
ƒ
Load: Lload = 2 mH and Rload = 5 Ω
Using Matlab/Simulink, simulate the circuit model described in Fig. 1 and plot the
waveforms of Vi (= [ViAB ViBC ViCA]), Ii (= [iiA iiB iiC]), VL (= [VLAB VLBC VLCA]), and IL (= [iLA
iLB iLC]).
2
2. Sine-∆ PWM
2.1 Principle of Pulse Width Modulation (PWM)
Fig. 2 shows circuit model of a single-phase inverter with a center-taped grounded DC bus,
and Fig 3 illustrates principle of pulse width modulation.
Fig. 2 Circuit model of a single-phase inverter.
Fig. 3 Pulse width modulation.
As depicted in Fig. 3, the inverter output voltage is determined in the following:
ƒ
When Vcontrol > Vtri, VA0 = Vdc/2
ƒ
When Vcontrol < Vtri, VA0 = −Vdc/2
3
Also, the inverter output voltage has the following features:
ƒ
PWM frequency is the same as the frequency of Vtri
ƒ
Amplitude is controlled by the peak value of Vcontrol
ƒ
Fundamental frequency is controlled by the frequency of Vcontrol
Modulation index (m) is defined as:
∴m =
vcontrol
peak of (V A0 )1
=
,
vtri
Vdc / 2
where, (VA0 )1 : fundamental frequecny component of
VA0
2.2 Three-Phase Sine-∆ PWM Inverter
Fig. 4 shows circuit model of three-phase PWM inverter and Fig. 5 shows waveforms of
carrier wave signal (Vtri) and control signal (Vcontrol), inverter output line to neutral voltage (VA0,
VB0, VC0), inverter output line to line voltages (VAB, VBC, VCA), respectively.
Fig. 4 Three-phase PWM Inverter.
4
Fig. 5 Waveforms of three-phase sine-∆ PWM inverter.
As described in Fig. 5, the frequency of Vtri and Vcontrol is:
ƒ
Frequency of Vtri = fs
ƒ
Frequency of Vcontrol = f1
where, fs = PWM frequency and f1 = Fundamental frequency
The inverter output voltages are determined as follows:
ƒ
When Vcontrol > Vtri, VA0 = Vdc/2
ƒ
When Vcontrol < Vtri, VA0 = −Vdc/2
where, VAB = VA0 – VB0, VBC = VB0 – VC0, VCA = VC0 – VA0
5
3. State-Space Model
Fig. 6 shows L-C output filter to obtain current and voltage equations.
Fig. 6 L-C output filter for current/voltage equations.
By applying Kirchoff’s current law to nodes a, b, and c, respectively, the following current
equations are derived:
c node “a”:
iiA + ica = i ab + i LA
⇒ iiA + C f
dV LCA
dV LAB
=Cf
+ i LA .
dt
dt
(1)
iiB + i ab = ibc + i LB
⇒ iiB + C f
dV LBC
dV LAB
=Cf
+ i LB .
dt
dt
(2)
d node “b”:
6
e node “c”:
iiC + ibc = ica + i LC
where, iab = C f
⇒ iiC + C f
dV LBC
dV LCA
=Cf
+ i LC .
dt
dt
(3)
dVLBC
dVLCA
dVLAB
, ibc = C f
, ica = C f
.
dt
dt
dt
Also, (1) to (3) can be rewritten as the following equations, respectively:
c subtracting (2) from (1):
dV ⎞
dV ⎞
⎛ dV
⎛ dV
iiA − iiB + C f ⎜ LCA − LAB ⎟ = C f ⎜ LAB − LBC ⎟ + i LA − i LB
dt ⎠
dt ⎠
⎝ dt
⎝ dt
dV
dV ⎞
⎛ dV
⇒ C f ⎜ LCA + LBC − 2 ⋅ LAB ⎟ = −iiA + iiB + i LA − i LB
dt
dt ⎠
⎝ dt
.
(4)
d subtracting (3) from (2):
dV ⎞
dV ⎞
⎛ dV
⎛ dV
iiB − iiC + C f ⎜ LAB − LBC ⎟ = C f ⎜ LBC − LCA ⎟ + i LB − i LC
dt ⎠
dt ⎠
⎝ dt
⎝ dt
dV
dV ⎞
⎛ dV
⇒ C f ⎜ LAB + LCA − 2 ⋅ LBC ⎟ = −iiB + iiC + i LB − i LC
dt
dt ⎠
⎝ dt
.
(5)
e subtracting (1) from (3):
dV ⎞
dV ⎞
⎛ dV
⎛ dV
iiC − iiA + C f ⎜ LBC − LCA ⎟ = C f ⎜ LCA − LAB ⎟ + i LC − i LA
dt ⎠
dt ⎠
⎝ dt
⎝ dt
dV
dV ⎞
⎛ dV
⇒ C f ⎜ LAB + LBC − 2 ⋅ LCA ⎟ = −iiC + iiA + i LC − i LA
dt
dt ⎠
⎝ dt
.
(6)
To simplify (4) to (6), we use the following relationship that an algebraic sum of line to line load
voltages is equal to zero:
7
VLAB + VLBC + VLCA = 0.
(7)
Based on (7), the (4) to (6) can be modified to a first-order differential equation, respectively:
⎧ dV LAB
1
1
(i LAB )
iiAB −
=
⎪
3C f
3C f
⎪ dt
⎪ dV LBC
1
1
iiBC −
(i LB C ) ,
=
⎨
dt
3
C
3
C
f
f
⎪
⎪ dV LCA
1
1
(i LCA )
iiCA −
=
⎪
3
3
dt
C
C
f
f
⎩
(8)
where, iiAB = iiA ─ iiB, iiBC = iiB ─ iiC, iiCA = iiC ─ iiA and iLAB = iLA ─ iLB, iLBC = iLB ─ iLC,
iLCA = iLC ─ iLA.
By applying Kirchoff’s voltage law on the side of inverter output, the following voltage
equations can be derived:
⎧ diiAB
1
1
V LAB +
ViAB
=−
⎪
Lf
Lf
⎪ dt
⎪ diiBC
1
1
V LBC +
ViBC .
=−
⎨
Lf
Lf
⎪ dt
⎪ diiCA
1
1
⎪ dt = − L V LCA + L ViCA
f
f
⎩
(9)
By applying Kirchoff’s voltage law on the load side, the following voltage equations can be
derived:
⎧
⎪V LAB = Lload
⎪
⎪
⎨V LBC = Lload
⎪
⎪
⎪V LCA = Lload
⎩
di LA
di
+ Rload i LA − Lload LB − Rload i LB
dt
dt
di LC
di LB
+ Rload i LB − Lload
− Rload i LC .
dt
dt
di LC
di
+ Rload i LC − Lload LA − Rload i LA
dt
dt
8
(10)
Equation (10) can be rewritten as:
⎧ di LAB
R
1
V LAB
= − load i LAB +
⎪
Lload
Lload
⎪ dt
⎪⎪ di LBC
R
1
V LBC .
= − load i LBC +
⎨
Lload
Lload
⎪ dt
⎪ di LCA
R
1
V LCA
= − load i LCA +
⎪
⎪⎩ dt
Lload
Lload
(11)
Therefore, we can rewrite (8), (9) and (11) into a matrix form, respectively:
dVL
1
1
=
Ii −
IL
3C f
3C f
dt
dI i
1
1
=−
VL +
Vi
dt
Lf
Lf
,
(12)
R
dI L
1
=
VL − load I L
dt
Lload
Lload
where, VL = [VLAB VLBC VLCA]T , Ii = [iiAB iiBC iiCA]T = [iiA-iiB iiB-iiC iiC-iiA]T , Vi = [ViAB ViBC ViCA]T ,
IL = = [iLAB iLBC iLCA]T = [iLA-iLB iLB-iLC iLC-iLA]T.
Finally, the given plant model (12) can be expressed as the following continuous-time state space
equation
& (t ) = AX(t ) + Bu(t ) ,
X
⎡
⎢ 0 3×3
⎢
V
⎡ L⎤
1
⎢
⎥
, A = ⎢⎢−
where, X = I i
I 3×3
⎢ ⎥
Lf
I
⎢
⎣⎢ L ⎦⎥ 9×1
⎢ 1
I 3×3
⎢L
⎣ load
(13)
⎤
1
I 3×3 ⎥
⎡ 0 3×3 ⎤
3C f
⎥
⎢
⎥
⎥ , B = ⎢ 1 I 3×3 ⎥ , u = [V ] .
0 3×3
i 3×1
⎥
⎢Lf
⎥
⎥
⎢0
⎥
Rload
⎥
⎣ 3×3
⎦ 9×3
−
I 3×3 ⎥
Lload
⎦ 9×9
1
I 3×3
3C f
−
0 3×3
0 3×3
9
Note that load line to line voltage VL, inverter output current Ii, and the load current IL are the
state variables of the system, and the inverter output line-to-line voltage Vi is the control input
(u).
4. Simulation Steps
1). Initialize system parameters using Matlab
2). Build Simulink Model
ƒ
Generate carrier wave (Vtri) and control signal (Vcontrol) based on modulation index (m)
ƒ
Compare Vtri to Vcontrol to get ViAn, ViBn, ViCn.
ƒ
Generate the inverter output voltages (ViAB, ViBC, ViCA,) for control input (u)
ƒ
Build state-space model
ƒ
Send data to Workspace
3). Plot simulation results using Matlab
10
5. Simulation results
Vtri, Vsin [V]
Vtri and Vsin and ViAn
1
ViAn [V]
Vsin
0
-1
0.9
500
0.902
0.904
0.906
0.908
0.91
0.912
0.914
0.916
0.918
0.92
0.902
0.904
0.906
0.908
0.91
0.912
0.914
0.916
0.918
0.92
0
Vtri, Vsin [V]
-500
0.9
1
Vtri
Vsin
0
-1
0.9
500
ViAn [V]
Vtri
0.901
0.902
0.903
0.904
0.905
0.906
0.907
0.908
0.909
0.901
0.902
0.903
0.904
0.905
Time [Sec]
0.906
0.907
0.908
0.909
0
-500
0.9
Fig. 7 Waveforms of carrier wave, control signal, and inverter output line to neutral voltage.
(a) Carrier wave (Vtri) and control signal (Vsin)
(b) Inverter output line to neutral voltage (ViAn)
(c) Enlarged carrier wave (Vtri) and control signal (Vsin)
(d) Enlarged inverter output line to neutral voltage (ViAn)
11
Inverter output line to line voltages (ViAB, ViBC, ViCA)
ViAB [V]
500
0
-500
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
Time [Sec]
0.96
0.97
0.98
0.99
1
ViBC [V]
500
0
-500
0.9
ViCA [V]
500
0
-500
0.9
Fig. 8 Simulation results of inverter output line to line voltages (ViAB, ViBC, ViCA)
12
Inverter output currents (iiA, iiB, iiC)
100
iiA [A]
50
0
-50
-100
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
Time [Sec]
0.96
0.97
0.98
0.99
1
100
iiB [A]
50
0
-50
-100
0.9
100
iiC [A]
50
0
-50
-100
0.9
Fig. 9 Simulation results of inverter output currents (iiA, iiB, iiC)
13
Load line to line voltages (VLAB, VLBC, VLCA)
400
VLAB [V]
200
0
-200
-400
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
Time [Sec]
0.96
0.97
0.98
0.99
1
400
VLBC [V]
200
0
-200
-400
0.9
400
VLCA [V]
200
0
-200
-400
0.9
Fig. 10 Simulation results of load line to line voltages (VLAB, VLBC, VLCA)
14
Load phase currents (iLA, iLB, iLC)
iLA [A]
50
0
-50
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.91
0.92
0.93
0.94
0.95
Time [Sec]
0.96
0.97
0.98
0.99
1
iLB [A]
50
0
-50
0.9
iLC [A]
50
0
-50
0.9
Fig. 11 Simulation results of load phase currents (iLA, iLB, iLC)
15
ViAB [V]
500
0
iLA, iLB, iLC [A]
VLAB, VLBC, VLCA [V]
iiA, iiB, iiC [A]
-500
0.9
100
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
iiA
iiB
0
-100
0.9
400
iiC
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
200
VLAB
0
VLBC
VLCA
-200
-400
0.9
50
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
iLA
iLB
0
-50
0.9
iLC
0.91
0.92
0.93
0.94
0.95
Time [Sec]
0.96
0.97
Fig. 12 Simulation waveforms.
(a) Inverter output line to line voltage (ViAB)
(b) Inverter output current (iiA)
(c) Load line to line voltage (VLAB)
(d) Load phase current (iLA)
16
0.98
0.99
1
Appendix
Matlab/Simulink Codes
17
A.1 Matlab Code for System Parameters
% Written by Jin Woo Jung, Date: 02/20/05
% ECE743, Simulation Project #1 (Sine PWM Inverter)
% Matlab program for Parameter Initialization
clear all % clear workspace
% Input data
Vdc= 400; % DC-link voltage
Lf= 800e-6;% Inductance for output filter
Cf= 400e-6; % Capacitance for output filter
Lload = 2e-3; %Load inductance
Rload= 5; % Load resistance
f= 60; % Fundamental frequency
fz = 3e3; % Switching frequency
m= 0.8; % Modulation index
% Coefficients for State-Space Model
A=[zeros(3,3)
-eye(3)/Lf
eye(3)/(3*Cf)
zeros(3,3)
eye(3,3)/Lload zeros(3,3)
-eye(3)/(3*Cf)
zeros(3,3)
-eye(3)*Rload/Lload]; % system matrix
B=[zeros(3,3)
eye(3)/Lf
zeros(3,3)]; % coefficient for the control variable u
C=[eye(9)]; % coefficient for the output y
D=[zeros(9,3)]; % coefficient for the output y
Ks = 1/3*[-1 0 1; 1 -1 0; 0 1 -1]; % Conversion matrix to transform [iiAB iiBC iiCA] to [iiA iiB
iiC]
18
A.2 Matlab Code for Plotting the Simulation Results
% Written by Jin Woo Jung
% Date: 02/20/05
% ECE743, Simulation Project #1 (Sine-PWM)
% Matlab program for plotting Simulation Results
% using Simulink
ViAB = Vi(:,1);
ViBC = Vi(:,2);
ViCA = Vi(:,3);
VLAB= VL(:,1);
VLBC= VL(:,2);
VLCA= VL(:,3);
iiA= IiABC(:,1);
iiB= IiABC(:,2);
iiC= IiABC(:,3);
iLA= ILABC(:,1);
iLB= ILABC(:,2);
iLC= ILABC(:,3);
figure(1)
subplot(3,1,1)
plot(t,ViAB)
axis([0.9 1 -500 500])
ylabel('V_i_A_B [V]')
title('Inverter output line to line voltages (V_i_A_B, V_i_B_C, V_i_C_A)')
19
grid
subplot(3,1,2)
plot(t,ViBC)
axis([0.9 1 -500 500])
ylabel('V_i_B_C [V]')
grid
subplot(3,1,3)
plot(t,ViCA)
axis([0.9 1 -500 500])
ylabel('V_i_C_A [V]')
xlabel('Time [Sec]')
grid
figure(2)
subplot(3,1,1)
plot(t,iiA)
axis([0.9 1 -100 100])
ylabel('i_i_A [A]')
title('Inverter output currents (i_i_A, i_i_B, i_i_C)')
grid
subplot(3,1,2)
plot(t,iiB)
axis([0.9 1 -100 100])
ylabel('i_i_B [A]')
grid
subplot(3,1,3)
20
plot(t,iiC)
axis([0.9 1 -100 100])
ylabel('i_i_C [A]')
xlabel('Time [Sec]')
grid
figure(3)
subplot(3,1,1)
plot(t,VLAB)
axis([0.9 1 -400 400])
ylabel('V_L_A_B [V]')
title('Load line to line voltages (V_L_A_B, V_L_B_C, V_L_C_A)')
grid
subplot(3,1,2)
plot(t,VLBC)
axis([0.9 1 -400 400])
ylabel('V_L_B_C [V]')
grid
subplot(3,1,3)
plot(t,VLCA)
axis([0.9 1 -400 400])
ylabel('V_L_C_A [V]')
xlabel('Time [Sec]')
grid
figure(4)
subplot(3,1,1)
plot(t,iLA)
axis([0.9 1 -50 50])
21
ylabel('i_L_A [A]')
title('Load phase currents (i_L_A, i_L_B, i_L_C)')
grid
subplot(3,1,2)
plot(t,iLB)
axis([0.9 1 -50 50])
ylabel('i_L_B [A]')
grid
subplot(3,1,3)
plot(t,iLC)
axis([0.9 1 -50 50])
ylabel('i_L_C [A]')
xlabel('Time [Sec]')
grid
figure(5)
subplot(4,1,1)
plot(t,ViAB)
axis([0.9 1 -500 500])
ylabel('V_i_A_B [V]')
grid
subplot(4,1,2)
plot(t,iiA,'-', t,iiB,'-.',t,iiC,':')
axis([0.9 1 -100 100])
ylabel('i_i_A, i_i_B, i_i_C [A]')
legend('i_i_A', 'i_i_B', 'i_i_C')
grid
22
subplot(4,1,3)
plot(t,VLAB,'-', t,VLBC,'-.',t,VLCA,':')
axis([0.9 1 -400 400])
ylabel('V_L_A_B, V_L_B_C, V_L_C_A [V]')
legend('V_L_A_B', 'V_L_B_C', 'V_L_C_A')
grid
subplot(4,1,4)
plot(t,iLA,'-', t,iLB,'-.',t,iLC,':')
axis([0.9 1 -50 50])
ylabel('i_L_A, i_L_B, i_L_C [A]')
legend('i_L_A', 'i_L_B', 'i_L_C')
xlabel('Time [Sec]')
grid
%For only Sine PWM
figure(6)
subplot(4,1,1)
plot(t,Vtri,'-', t,Vsin,'-.')
axis([0.9 0.917 -1.5 1.5])
ylabel('V_t_r_i, V_s_i_n [V]')
legend('V_t_r_i', 'V_s_i_n')
title('V_t_r_i and V_s_i_n')
grid
subplot(4,1,2)
plot(t,ViAn)
axis([0.9 0.917 -500 500])
ylabel('V_i_A_n [V]')
grid
23
subplot(4,1,3)
plot(t,Vtri,'-', t,Vsin,'-.')
axis([0.9 0.909 -1.5 1.5])
ylabel('V_t_r_i, V_s_i_n [V]')
legend('V_t_r_i', 'V_s_i_n')
grid
subplot(4,1,4)
plot(t,ViAn)
axis([0.9 0.909 -500 500])
ylabel('V_i_A_n [V]')
xlabel('Time [Sec]')
grid
24
A.3 Simulink Code
Simulink Model for Overall System
25
Simulink Model for “Sine-PWM Generator”
26