various modeling methods for the analysis of a three … · on a combination of three-phase voltage...

International Journal of Scientific Research and Engineering Studies (IJSRES)

Volume 3 Issue 7, July 2016

ISSN: 2349-8862

www.ijsres.com Page 21

Various Modeling Methods For The Analysis Of A Three Phase

Diode Bridge Rectifier And A Three Phase Inverter

Parvathi M. S

PG Scholar, Dept of EEE, Mar Baselios College of

Engineering and Technology, Trivandrum

Dr. Nisha G. K

Associate Professor, Dept of EEE, Mar Baselios College of

Engineering and Technology, Trivandrum

Abstract: This paper presents the modelling of a three

phase diode rectifier and a three phase inverter in the

Modified Nodal Analysis method and in the Average method.

Virtually all electronic devices require DC, so rectifiers are

used inside the power supplies of all electronic equipment.

Inverters are extensively used in power electronic

applications for converting DC to AC. The applications of

inverters range from uninterrupted power supplies used in

home appliances control to HVDC power transmission. The

average model equations for rectifier and inverter are

formulated using the Forward Euler method.

Keywords: Average Method, Modified Nodal Analysis

method, Three phase diode bridge rectifier and Three phase

inverter.

I. INTRODUCTION

As technology grows every day, the study of power

systems has shifted its direction to power electronics to

produce the most efficient energy conversion. Power

electronics is the study of processing and controlling the flow

of electric energy by supplying voltages and currents in a form

that is suited for user loads [1]. The goals of using power

electronics are to obtain the benefit of lower cost, small power

loss and high energy efficiency. Because of high energy

efficiency, the removal of heat generated due to dissipated

energy is lower.

A rectifier is a “power processor that should give a dc

output voltage with a minimum amount of harmonic contents”

[2]. When a polyphase ac is rectified, the phase-shifted pulses

overlap each other which produce a smoother dc output than

that made by a single-phase ac rectifier. In [3], the author

presents the modeling of single phase, three phase rectifiers

and suitable controller design for the modeled rectifier circuit.

An average inverter model operating in two

complementary modes suitable for micro grid simulation

applications is proposed in [4], by taking into account the

nonlinear behaviour of the switches, delays in the control

loops, and the practical constraints. In [5], the author focuses

on a combination of three-phase Voltage Source Inverter

(VSI) with a predictive current control to provide an optimized

system for three-phase inverter that controls the load current.

With the advent of increased use of inverters, various

switching models are established. The average method is

shown to be an effective method for analysis and controller

design in inverters [6] – [8]. Conventional switching models

and state-space averaging methods which take dead-time

effects into consideration have been applied successfully in

the inverters [9], [10].

II. MODELING OF THREE PHASE DIODE BRIDGE

RECTIFIER

The modeling of a three phase diode bridge rectifier can

be done in two ways – the Modified Nodal Analysis method

and the average model method (using transient simulation

techniques). On comparing the two methods, the Modified

Nodal Analysis method is a bit too long process where, we

need to find the nodal equations and the inverse matrix at each

switching conditions. On the other hand the latter provides

transient equations which enables to obtain the output dc

voltage and current, without the calculation of matrices. The

main advantage of the average model method of modeling of

diode bridge rectifier is that it provides ripple free output.

A. MODIFIED NODAL ANALYSIS METHOD

The power circuit diagram for a three phase diode bridge

rectifier is as shown in the Fig. 1.



ISSN: 2349-8862

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Figure 1: Circuit Diagram of a three phase diode bridge

rectifier

Figure 2: Input Output waveforms and the diode conduction

period of a three phase diode rectifier

The diodes are arranged in three legs. Each leg has two

series-connected diodes. The upper diodes D1, D3, D5

constitute the positive group of diodes. The lower diodes D2,

D4, D6 form the negative group of diodes. The diodes D1, D3,

D5 forming the positive group, would conduct when these

experience the highest positive voltage. Likewise, the diodes

D2, D4, D6 would conduct when these are subjected to most

negative voltage. For a particular interval, a diode from the

positive group and a diode from the negative group conduct.

No two diodes from the same group conduct together at a

time. The waveforms for the three phase diode rectifier and

each diode‟s corresponding conducting period are as shown in

the Fig. 2.

It is seen from the source voltage waveform that, from ωt

= 30 to 150°, voltage Va is more positive than the voltages Vb

and Vc. Therefore, the diode D1 connected to line „a‟ conducts

during this interval [11]. Likewise, from ωt = 150 to 270°,

voltage Vb is more positive therefore, diode D3 conducts.

Similarly, diode D5 from the positive group conducts from ωt

= 270 to 390° and so on. Conduction of the positive group

diodes is shown in Fig. 2 as D5, D1, D3, D5 etc. Similarly, the

negative diodes also conduct accordingly.

A three phase diode bridge rectifier consists of six diodes

arranged in three legs. Each diode has an on-resistance value

Ron, when the diode conducts and an off-resistance value Roff,

when the diode is not conducting. For the analysis purpose,

the three phase diode bridge rectifier is replaced by its

equivalent circuit diagram with its on-state and off-state

resistance values. The equivalent circuit diagram is obtained

for each conduction period, where two diodes conduct, one

diode from the positive group and the other from the negative

group.

Consider the interval ωt = 0 to 30° and ωt = 330 to

360°, where the diodes D5 and D6 conduct. The conducting

diodes D5 and D6 are replaced by their on-resistance values

Ron and the rest of the diodes with their off-resistance values

Roff. The equivalent circuit diagram will be as in Fig. 3.

Figure 3: Equivalent circuit diagram of a three phase diode

bridge rectifier for the interval ωt = 0 to 30° and ωt = 330

to 360° (D5 and D6 conduct)

Now, applying KCL at the nodes a, b, c and d, we get the

following equations.

At node „a‟;

At node „b‟;

At node „c‟;

At node „d‟;

Converting the above nodal equations into matrix form

which amounts to solve a linear system AX = Z, where, A

denotes the admittance matrix or the modified nodal analysis

matrix, X denotes the unknown matrix (unknown node

voltages and currents) and Z is the input matrix (source

voltages). We can obtain the unknown matrix X as X = A-1

Z.



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Assuming suitable values for the resistances as Ron =

1mΩ, Roff = 1MΩ, RL = 10kΩ and substituting in the above

matrix equation, the inverse of A matrix (A-1

) can be obtained

using an online inverse matrix calculator. The A-1

matrix (after

substituting the resistance values) for the interval taken, ωt =

0 to 30° and ωt = 330 to 360°, can be evaluated as below.

During the interval, ωt = 30 to 90°, the diodes D1 and D6

will be conducting and the inverse matrix can be obtained as

below.

Similarly, A

-1 matrix for each conducting interval can be

obtained. The A-1

matrix and the Z matrix is given as input to

the program and the corresponding outputs waveforms are

obtained.

B. AVERAGE METHOD FOR RECTIFIER MODELING

USING TRANSIENT SIMULATION TECHNIQUES

In order to make prime decisions in the design

procedures, power electronic modeling methods are widely

used. The modeled circuit helps to determine the performance

of the circuit along with the component values as well. State

space average modeling method is a common technique that is

utilized to obtain models of power electronic circuits for ac

and dc power conversions. The main advantage of this

particular method is that the accuracy can be increased and

ripple free waveforms can be obtained.

A simplified three phase diode bridge rectifier is shown in

the Fig. 4. The circuit consists of three ac input sources (VA,

VB and VC), input source reactor (Ls), diodes (D1 to D6),

output inductance (Lout), output filter capacitance (C) and

resistance (R) which represents the load on the rectifier [12].

Figure 4: Simplified circuit model of a three phase diode

bridge rectifier

Utilizing the general state space averaging method, the

switching functions sa, sb and sc for a three phase diode bridge

rectifier can be expressed by a Fourier series using the first

harmonic terms of the series as shown in the equation (5).

The average dc output voltage Vpn can be expressed as the

equation (6).

Also, the dc current equation can be obtained as shown in

equation (7).

From the equation (6), the left hand side of the equation

can be expanded as:

Now, the right hand side of the equation (6) can be solved

using the Transient simulation method. In this method, the

differential current equation can be converted to its equivalent

Forward Euler form and solved to obtain the value for the

(n+1)th

value of the dc current.

From the equation (9), the (n+1)

th value of the dc current

can be calculated as:

where, „h‟ is the delta constant value or the time step.

Similarly, from the equation (7), the differential voltage

equation can be converted to its equivalent Forward Euler

form and solved to obtain the value for the (n+1)th

value of the

dc voltage.

Thus, from the equation (11), the (n+1)

th value of the dc

voltage can be calculated as:

The modeling equation for a three phase diode bridge

rectifier is obtained using the average state space model and



ISSN: 2349-8862


the transient simulation techniques. The equations (8), (10)

and (12) represent the average model mathematical equations

for the rectifier output dc voltage and dc current.

III. MODELING OF THREE PHASE INVERTER

Similar to the modeling of a three phase diode bridge

rectifier, a three phase inverter can also be done in two ways –

the Modified Nodal Analysis method and the average model

method (using transient simulation techniques). The main

advantage of the ripple free output in the case of the average

model method holds the same for the modeling of a three

phase inverter as well.

A. MODIFIED NODAL ANALYSIS METHOD

A basic three phase inverter is a six-step bridge inverter.

It uses a minimum of six thyristors. In inverter terminology, a

step is defined as a change in the firing from one thyristor to

the next thyristor in the proper sequence. For one cycle of

360°, each step would be of 60° interval for a six-step inverter.

This means that the thyristors would be gated at regular

intervals of 60° in proper sequence so that a three phase ac

voltage is synthesized at the output terminals of a six-step

inverter.

The Fig. 5 shows the power circuit diagram of a three

phase bridge inverter using six thyristors and six diodes.

Presently, the use of IGBTs in single phase and three phase

inverters is on the rise. The basic circuit configuration of

inverter, however, remains unaltered with just a small change

of replacing the thyristors with IGBTs. A large capacitor

connected at the input terminals tends to make the input dc

voltage constant. This capacitor also suppresses the harmonics

fed back to the dc source. In the Fig. 5, commutation and

snubber circuits are omitted for simplicity. The thyristors are

numbered in the sequence in which they are triggered to

obtain voltages vab, vbc, vca at the output terminals a, b, c of the

inverter. There are two possible patterns of gating the

thyristors. In one pattern, each thyristor conducts for 180° and

in the other, each thyristor conducts for 120°. But in both

these patterns, gating signals are applied and removed at 60°

intervals of the output voltage waveform.

Figure 5: Circuit Diagram of a three phase bridge inverter

using thyristors

In three phase inverter, each thyristor conducts for 180° of

a cycle. The thyristor pair in each arm, i.e. T1, T4; T3, T6 and

T5, T2 are turned on with a time interval of 180. The thyristors

in the upper group, i.e. T1, T3, T5 conduct at an interval of

120°. The conduction time periods for 180° mode 3 phase

inverter is as shown in the Fig. 6. It can be understood from

the table that in every step of 60° duration, only three

thyristors are conducting – one from the upper group and two

from the lower group or vice versa.

Figure 6: Conduction period for a 180° mode 3-phase VSI

Similar to the analysis of a three phase diode bridge

rectifier, a three phase bridge inverter consists of six thyristors

arranged in three legs. Each thyristor has an on-resistance

value Ron, when the thyristor conducts and an off-resistance

value Roff, when it is not conducting. For the analysis purpose,

the three phase bridge inverter is replaced by its equivalent

circuit diagram with its on-state and off-state resistance

values. The equivalent circuit diagram is obtained for each

conduction period, where three thyristors conduct, one

thyristor from the upper group and two from the lower group

or vice versa.

Consider the interval ωt = 0 to 60°, where the thyristors

T1, T5 and T6 conduct. The conducting thyristors are replaced

by their on-resistance values Ron and the rest of the thyristors

with their off-resistance values Roff. The equivalent circuit

diagram will be as in Fig. 7.

Figure 7: Equivalent circuit diagram of a three phase bridge

inverter for the interval ωt = 0 to 60° (T1, T5 and T6 conduct)

Now, applying KCL at the nodes a, b, c and d, we get the

following equations.

At node „a‟;

At node „b‟;

At node „c‟;



ISSN: 2349-8862


At node „d‟;

At node „o‟

Converting the above nodal equations into matrix form

which amounts to solve a linear system AX = Z, where, A

denotes the admittance matrix or the modified nodal analysis

matrix, X denotes the unknown matrix (unknown node

voltages and currents) and Z is the input matrix (source

voltages). We can obtain the unknown matrix X as X = A-1

Z.

Assuming suitable values for the resistances as Ron =

1mΩ, Roff = 1MΩ, RL = 10kΩ and substituting in the above

matrix equation, the inverse of A matrix (A-1

) can be obtained

using an online inverse matrix calculator. The A-1

matrix (after

substituting the resistance values) for the interval taken, ωt =

0 to 60° can be evaluated as below.

During the interval, ωt = 60 to 120°, the diodes T1, T2and

T6 will be conducting and the inverse matrix can be obtained

as below.

Similarly, A

-1 matrix for each conducting interval can be

obtained. The A-1

matrix and the Z matrix is given as input to

the program and the corresponding outputs waveforms are

obtained.

B. AVERAGE METHOD FOR 3 PHASE INVERTER

MODELING USING TRANSIENT SIMULATION

TECHNIQUES

In the study of inverters and their dynamic performance,

various switching models are obtained based on the switching

conditions, as in the Modified Nodal Analysis method, which

is feasible for a single inverter or a small scale inverter

system. Therefore, in case of large scale inverter systems,

switching models becomes complicated. Hence, a state space

averaging method is formulated which is effective for the

analysis and controller design purpose in inverters.

A generalized state space averaging model is considered

which enhances the fundamental AC voltage and current

calculation with desired accuracy and precision. At the same

time, this method enables the steady state as well as transient

analysis processes [13]. The circuit diagram of a 3 phase 3

wire Voltage source inverter is as shown in the Fig. 8.

Figure 8: Circuit diagram of a three phase three wire voltage

source inverter

In the Fig. 8, assuming that the loads are 3 phase

symmetrical resistive loads in delta connection, whose value

are R, we obtain the equations from (7.6) to (7.11).

Where, are virtual line currents which can be

calculated as mentioned in the equations from (24) to (26).

Similar to the equation (6) in the modeling of three phase

diode bridge rectifier by the average model, it can be written

that,

Thus, the conventional state equations of three phase

inverter can easily be constructed.



ISSN: 2349-8862


The differential voltage and current equations can be

converted to its equivalent Forward Euler form and

substituting the equations (27) to (29) in the Forward Euler

equations we obtain the value for the (n+1)th

value of the

corresponding ac voltage and ac current.

The switching function is given by:

where k = a, b, c ; m is the modulation ratio taken as 0.5

; where is the

initial phase angle.

Thus, the modelling equations are formed for a three

phase inverter in average model using the Transient simulation

techniques.

IV. RESULTS AND DISCUSSIONS

The output waveforms obtained by simulating the MNA

model of rectifier with the parameter specification as Ron=1

mΩ, Roff=1 MΩ and RL=10 kΩ is as shown in the Fig. 9. It

depicts rectifier output voltage with amplitude 100 V.

Figure 9: Rectifier output voltage waveform in MNA method

(X axis: 1 unit = 0.002s & Yaxis: 1 unit = 10V)

In the case of the average model of the three phase diode

bridge rectifier, the parameter specifications used for the

simulation are as in the Table 1.

SYMBOL PARAMETERS VALUE UNIT

Vabc Input Voltage 415 V

f Fundamental

frequency

50 Hz

L Filtering Inductor 0.05 H

C Filter Capacitor 0.05 F

R Load Resistance 2 Ω

h Time step 100 µs

Table 1: Parameters for rectifier simulation in average model

The dc voltage and dc current waveforms at the output of

the rectifier for the specified parameters are as shown in the

Fig. 10.

Figure 10: Output dc voltage and dc current value of a 3

phase rectifier ,Vdc = 687 V and Idc = 344 A (X axis: 1 unit =

0.5s and Yaxis: 1 unit = 100 units of voltage and current)

The output waveform obtained in the MNA method of

modelling of inverter with parameter specifications as Ron =

1mΩ, Roff = 1MΩ and RL = 10kΩ is shown in the Fig. 11. The

figure depicts the phase voltage in the MNA method with

input amplitude of 70 V.

Figure 11: Inverter output phase voltage waveform in MNA

method (X axis: 1 unit = 0.005s & Yaxis: 1 unit = 20V)



ISSN: 2349-8862


In the case of the average model of a three phase inverter,

the parameter specifications used for simulation are as in the

Table 2.

SYMBOL PARAMETERS VALUE UNIT

Vdc Input Voltage 100 V

m Modulation Ratio 0.6 -

f Fundamental

frequency

50 Hz

φ0 Initial phase angle 0 rad/s

L Filtering Inductor 1.0 mH

rL ESR of filtering

inductor

1.245 Ω

C Filter Capacitor 1.0 mF

R Load Resistance 10 Ω

h Time step 100 µs

Table 2: Parameters for inverter simulation in average

method

The simulation waveform for the input dc voltage to the

inverter is as shown in the Fig. 12.

Figure 12: Input dc voltage of magnitude 100 V to the inverter

(X axis: 1 unit = 0.1s and Y axis: 1 unit = 50V)

The simulation waveform for phase currents at the output

of the inverter is as shown in the Fig. 13. The figure shows the

ac phase currents with a magnitude of 20 A for the given input

dc of 100 V

Figure 13: Output phase current waveform for the dc input of

100 V (X axis: 1 unit = 0.1S and Y axis: 1 unit = 10 A)

The simulation waveform for phase voltages at the output

of the inverter is as shown in the Fig. 14. The figure shows the

ac phase voltages with a magnitude of (2Vdc/3 = 66.67 V) for

the given input dc of 100 V.

Figure 14: Output phase voltage waveform for the dc input of

100 V (X axis: 1 unit = 0.1s and Y axis: 1 unit = 20 V

V. CONCLUSION

In this paper, the modeling of three-phase diode bridge

rectifier and a three-wire inverter is done based on the

Modified Nodal Analysis method and in the state space

average method. The simulation waveforms obtained from the

two methods are discussed. It is observed from the paper that

the average model of the rectifier and inverter is more

effective and efficient as it gives ripple free waveforms, thus

the use of filters for smoothening the ripples can be

eliminated.

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various modeling methods for the analysis of a three … · on a combination of three-phase voltage...

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