Lehrstuhl für Elektris he Antriebssysteme und Leistungselektronik

Te hnis he Universität Mün hen Prof. Dr.-Ing. RalphKennel

Ar isstraÿe 21

D80333 Mün hen

Email: ealei.tum.de

Internet: http://www.eal.ei.tum.de

Tel.: +49 (0)89 28928358

Fax: +49 (0)89 28928336

Power Ele troni s

Exer ise: Diode re tiers

2012

1

1 Theory

1.1 General remarks

1.1.1 Diode

In g. 1.1 the ir uit symbol and hara teristi s of an ideal, a real and a power diode are

shown. VS is the threshold voltage, the breakdown voltages is alled VR. In ontrast to onven-

tional diodes, power diodes have a ertain ohmi part whi h has to be onsidered if loss power

al ulations are made.

PSfrag repla ements

+−

pn

Cathode

Anode

I

V

VS

VR

ideal

real

Power diode

Blo king zone

Condu ting zone

Fig. 1.1: Cir uit symbol and hara teristi s of diodes

1.1.2 Average value

The average value of a signal orresponds to the arithmeti average value. For a periodi signal

v(t), e. g. a voltage, the average value an be al ulated as follows:

vM =1

T

t0+T∫

t0

v(t) dt (1.1)

1.1.3 Root mean square (RMS)

The root mean square of a signal is its quadrati mean value. A DC voltage with the same

value as the RMS of a periodi signal produ es exa tly the same thermal power on a resistor

(in average time).

The RMS of a periodi signal v(t) an be al ulated as follows:

Veff =

√

√

√

√

√

1

T

t0+T∫

t0

v2(t) dt (1.2)

For a sinusoidal signal with

v(t) = v sin (ωt)

it an be al ulated to

Veff =1

2

√2v (1.3)

2

1.2 Re tier ir uits

Re tier ir uits are ne essary for the onversion of AC voltages to DC voltages. In general the

ir uits an be divided into one-way ir uits and into two-way ir uits. Another dierentiation

an be made based on the number of ommutations per period. In the following hapters the

• one-way ir uits (M1, M2 and M3) and the

• two-way ir uits (B2 and B6)

are des ribed and explained.

1.2.1 Comparison of one-way ir uits and two-way ir uits

The two most important dieren es between one-way and two-way ir uits are:

1. One-way ir uits need less re tiers or diodes than bridge ir uits.

2. One-way ir uits need more omplex transformers.

In former times, when power semi ondu tors were not yet available, one-way ir uits were used

very often (mer ury vapor re tiers were very expensive). Today mostly two-way re tiers are

used.

3

1.3 M1 ir uit

1.3.1 Cir uit diagram

In g. 1.2 an M1 ir uit is shown.

PSfrag repla ements

V1 V2

Transformer

Vvil

vl

Fig. 1.2: M1 ir uit

1.3.2 Current and voltage traje tories

The ir uits shown in g. 1.3 (R load, RL load and RC load) were simulated with the program

PSIM

R©. The following values were used for the simulations:

V = 230V

f = 50Hz

R = 100Ω

C = 50µF

L = 50mH

The resulting traje tories of urrents and voltages are shown in g. 1.4.

V

A

V

A

V

A

Fig. 1.3: PSIM models of the M1 ir uit

The urrent iR(t) through the resistor R results from Ohm's law:

iR =vR

R

It is proportional to the voltage drop a ross the resistor.

The urrent owing through the apa itor C is given by the equation

iC = C · vC

4

0 0.01 0.02 0.03 0.04 0.05−200

0

200

400Voltages

Time [s]

Voltage [

V]

VRC

VR

VRL

0 0.01 0.02 0.03 0.04 0.05−5

0

5

10Currents

Time [s]

Curr

ent

[A]

IRC

IR

IRL

Fig. 1.4: Current and voltage traje tories (M1 ir uit)

A step in the urrent traje tory results be ause of the salient point in the voltage traje tory,

furthermore a urrent rise an be observed. At the beginning of a half wave the apa itor is

harged. If the peak of the half wave is ex eeded, both the apa itor and the voltage sour e

deliver urrent into the resistor R. The voltage sour e's voltage, however, goes down faster

than the apa itor's one (if the apa itor is dimensioned big enough). As soon as the apa itor

voltage is greater than the voltage of the voltage sour e, only the apa itor is delivering urrent

to the resistor. The diode urrent will stop owing and the diode is now in blo king mode. As

the apa itor is still not ompletely dis harged, it further delivers urrent to the resistor - either

until the omplete harge is drained over R or until the next half wave starts. A apa itor in

parallel to the resistor leads to a smoothing of the voltage.

The voltage a ross the indu tor L an be al ulated with the equation

vL = L · iL

the urrent iL owing through the indu tor then results to

iL =1

L

t0+T∫

t0

vL dt

From this results a PT1 traje tory or a smoothing of the urrent. This behavior results from

the fa t that in an indu tor a ux Ψ is reated. The ux always ountera ts against its ause

(the urrent through the indu tor). If the voltage of the sour e has be ome equal to zero,

magneti energy is still stored in the indu tor - the ux is still not zero. This leads to the fa t

that urrent is owing, even if the sour e voltage is already negative (Blo king ondition for

the diode: i = 0). The diode will blo k when the ux has be ome zero and when the urrent

through the diode has stopped owing.

5

1.3.3 Important parameters

For a sinusoidal input voltage

v(t) = v sin (ωt)

the diode's maximum blo king voltage is

Vvmax = v (1.4)

The ideal DC voltage, i. e. the average value of the re tied voltage, an be al ulated as follows:

Vdi =1

T

T∫

0

vL(t) dt =1

2π

π∫

0

v2 sin (ωt) dωt =v2

2π(− cos π + cos 0) =

v2

2π(1 + 1) =

v2

π

With

V2 =1

2

√2v2

follows:

Vdi =

√2

πV2 ≈ 0.4502V2 (1.5)

1.3.4 Transformer power rating

If the voltage for the one-way re tier is provided by a transformer, this part needs to have a

ertain power rating related to the DC part.

The transformer power rating an be al ulated a ording to the equation

PB =1

2

(

∑

i

VPiIPi +∑

i

VSiISi

)

(1.6)

The transformer power rating is the arithmeti average value of the sum of the apparent powers

on the primary and on the se ondary side. For urrents and voltages RMS values have to be

used in equation 1.6.

The power in the DC part is

Pd = VdiId (1.7)

1.3.4.1 Pure R load

The ratio between the transformer power rating and the power in the DC part results to

PB

Pd

≈ 3.09 (1.8)

A derivation shall be omitted at this point. The transformer power rating has to be more than

three times greater than the power in the DC part if a pure resistive load is onsidered!

6

1.4 M2 ir uit

The M2 ir uit uses a transformer with entral tapping.

1.4.1 Cir uit diagram

In g. 1.5 an M2 ir uit is drawn.

PSfrag repla ements

V1

VS1

VS2

Transformer

D1

D2

ilvl

Fig. 1.5: M2 ir uit

1.4.2 General fun tional prin iple

The ir uit shown in g. 1.6 was rst simulated with PSIM

R©with a pure resistive load in order

to show the general fun tional prin iple. Again the values given in hapter 1.3.2 were used. It

has to be noted that here for ea h one of the voltage sour es only

V = 115VRMS

were used. This orresponds to a transformer with entral tapping and a tranformation ratio of

1. In g. 1.7 the transformer voltages, the voltage drop a ross the resistor, the urrents through

A

A

V

V

A V

Fig. 1.6: PSIM model for showing the general fun tional prin iple of the M2 ir uit

both diodes and the urrent through the resistor are shown. As it an be seen, the upper diode

(D1) is ondu ting during the positive half wave, the lower one (D2) during a negative half

wave, i. e. always the diode whi h has a positive potential.

1.4.3 Current and voltage traje tories for R-, RC- and RL load

A further simulation was made using PSIM

R©in order to plot the urrent and voltage traje tories

for a pure R load, an RC load and an RL load. The traje tories an be explained analogously to

hapter 1.3.2. The simulation models are shown in g. 1.8, the urrent and voltage traje tories

in g. 1.9. In ontrast to the M1 ir uit here already a smaller apa itor C is su ient in order

to smooth the load voltage be ause two half waves per period are used.

7

0 0.005 0.01 0.015 0.02 0.025−200

0

200Transformer voltages

Time [s]

Vo

lta

ge

[V

]

Voben

Vunten

0 0.005 0.01 0.015 0.02 0.0250

100

200Voltage drop on R

Time [s]

Vo

lta

ge

[V

]

0 0.005 0.01 0.015 0.02 0.025−2

0

2Current in upper branch

Time [s]

Cu

rre

nt [A

]

0 0.005 0.01 0.015 0.02 0.025−2

0

2Current in lower branch

Time [s]

Curr

ent

[A]

0 0.005 0.01 0.015 0.02 0.0250

1

2Current through R

Time [s]

Curr

ent

[A]

Fig. 1.7: Current and voltage traje tories showing the general fun tional prin iple of the M2

ir uit

A A AV V V

Fig. 1.8: PSIM models of the M2 ir uit with dierent loads

8

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−50

0

50

100

150

200Voltages

Time [s]

Vo

lta

ge

[V

]

VR

VRL

VRC

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−1

0

1

2

3

4

5

6Currents

Time [s]

Cu

rre

nt [A

]

IR

IRL

IRC

Fig. 1.9: Current and voltage traje tories (M2 ir uit)

1.4.4 Important parameters

For a sinusoidal input voltage the diode's maximum blo king voltage is

Vvmax = 2vSi (1.9)

The ideal DC voltage is

Vdi =2√2

πVSi ≈ 0.901VSi (1.10)

Important: The voltage loss resulting from ommutation ee ts has to be subtra ted from this

voltage!

1.4.5 Transformer power rating

1.4.5.1 Pure R load

For a pure R load the ratio between transformer power rating and power in the DC part is

ST

Pd

≈ 1.48 (1.11)

A derivation shall be omitted at this point.

1.4.5.2 RL load with L → ∞

The transformer apparent power results to

ST ≈ 1.34Pd (1.12)

9

1.5 B2 ir uit

Nowadays the B2 ir uit is mostly used in power supplies for the re ti ation of single phase

AC voltages.

1.5.1 Cir uit diagram

In g. 1.10 a B2 ir uit is drawn.

PSfrag repla ements

V1 V2vL

iL

D1

D2

D3 D4

Fig. 1.10: B2 ir uit

1.5.2 General fun tional prin iple

The ir uit shown in g. 1.11 was simulated as well, in this ase espe ially in order to show

the general fun tional prin iple. The values given in hapter 1.3.2 were also used for these

simulations. In g. 1.12 the urrent and voltage traje tories are shown. During a positive half

wave the urrent is owing through the diodes D1 and D4, during a negative half wave through

D2 and D3, as these do then have positive potential. In this ase both half waves are used.

A

V

A

A

A

A

Fig. 1.11: PSIM model showing the general fun tional prin iple of the B2 ir uit

1.5.3 Current and voltage traje tories for R, RC and RL load

Another simulation was arried out in order to plot the urrent and voltage traje tories for a

pure R load, an RC load and an RL load. The traje tories an be explained analogously to

10

0 0.005 0.01 0.015 0.02 0.0250

200

400Voltage drop on R

Vo

lta

ge

[V

]

Time [s]

0 0.005 0.01 0.015 0.02 0.025−5

0

5Current through upper left diode

Cu

rre

nt

[A]

Time [s]

0 0.005 0.01 0.015 0.02 0.025−5

0

5Current through lower right diode

Curr

en

t [A

]

Time [s]

0 0.005 0.01 0.015 0.02 0.025−5

0

5Current through upper right diode

Curr

ent

[A]

Time [s]

0 0.005 0.01 0.015 0.02 0.025−5

0

5Current through lower left diode

Curr

ent

[A]

Time [s]

0 0.005 0.01 0.015 0.02 0.0250

2

4Current through R

Curr

ent [A

]

Time [s]

Fig. 1.12: Current and voltage traje tories showing the general fun tional prin iple of the B2

ir uit

11

hapter 1.3.2. The models are shown in g. 1.13, the urrent and voltage traje tories in g. 1.14.

It has to be noted that in this ase, analogously to the M2 ir uit, already a smaller apa itor

C is su ient in order to smooth the load voltage be ause here also two half waves are used

per period.

A

V

A

V

A

V

Fig. 1.13: PSIM models of the B2 ir uit with dierent loads

0 0.005 0.01 0.015 0.02 0.025−100

0

100

200

300

400Voltages

Voltage [V

]

Time [s]

VR

VRL

VRC

0 0.005 0.01 0.015 0.02 0.025−2

0

2

4

6

8Currents

Curr

ent [A

]

Time [s]

IR

IRL

IRC

Fig. 1.14: Current and voltage traje tories (B2 ir uit)

1.5.4 Important parameters

The ideal DC voltage is

Vdi =2√2

πVS ≈ 0.901VS (1.13)

The diode's maximum blo king voltage is

Vvmax = vS (1.14)

12

1.5.5 Transformer power rating

1.5.5.1 Pure R load

The transformer apparent power results to

ST =≈ 1.23Pd (1.15)

1.5.5.2 RL load with L → ∞

In this ase the transformer apparent power results to

SS ≈ 1.11Pd (1.16)

1.5.5.3 RC load with C → ∞

The transformer apparent power an be al ulated to

S = 1.21 (Vdi + 2VS) Id (1.17)

A derivation shall be omitted at this point.

13

1.6 Three-phase ir uits

The following ir uits are used for the re ti ation of three-phase alternating urrents.

1.6.1 M3 ir uit

1.6.1.1 Cir uit diagram

In g. 1.15 an M3 ir uit is drawn, the transformer is onne ted on both sides via a star

onne tion.

PSfrag repla ements

3 ∼

vL

iL

D1

D2

D3

Fig. 1.15: M3 ir uit

1.6.1.2 General fun tional prin iple

The ir uit shown in g. 1.16 was also simulated, in this ase espe ially in order to show the

general fun tional prin iple. The values given in hapter 1.3.2 were used, in this ase, however,

a three-phase voltage sour e with VS = 230V. In g. 1.17 the urrent and voltage traje tories

are shown. It an be seen that only the diode with positive potential is ondu ting. This means

that, if, e. g., V1 is the highest voltage, D1 is ondu ting, if V2 is the highest one, then D2 is

ondu ting. The same is valid for U3 and D3.

1.6.1.3 Current and voltage traje tories for R, RC and RL loads

A further simulation was arried out in order to plot the urrent and voltage traje tories for

a pure R, an RC and an RL load. The traje tories an be explained analogously to hapter

1.3.2. The models are shown in g. 1.18, the urrent and voltage traje tories in g. 1.19.

A

A

A

A

V

Fig. 1.16: PSIM model for showing the general fun tional prin iple of the M3 ir uit

14

0 0.005 0.01 0.015 0.02 0.025−500

0

500Transformer voltages

Time [s]

Vo

lta

ge

[V

]

V1

V2

V3

0 0.005 0.01 0.015 0.02 0.0250

200

400Voltage drop on R

Time [s]

Vo

lta

ge

[V

]

0 0.005 0.01 0.015 0.02 0.025−2

0

2

4Current through branch 1

Time [s]

Curr

ent

[A]

0 0.005 0.01 0.015 0.02 0.025−2

0

2

4Current throuhg branch 2

Time [s]

Curr

ent

[A]

0 0.005 0.01 0.015 0.02 0.025−2

0

2

4Current through branch 3

Time [s]

Curr

ent

[A]

0 0.005 0.01 0.015 0.02 0.0250

2

4Current through R

Time [s]

Curr

ent

[A]

Fig. 1.17: Current and voltage traje tories for showing the general fun tional prin iple of the

M3 ir uit

15

A

V

A

V

A

V

Fig. 1.18: PSIM models of the M3 ir uit with dierent loads

1.6.1.4 Important parameters

As it has already been shown in hapter 1.6.1.2, always only one diode is ondu ting, while

the other two ones are blo king. Hen e, the blo king diodes are stressed with the line to line

voltage of the se ondary transformer side.

In a three-phase system the line to line voltages (mesh voltages) an be al ulated from the

line to neutral voltages as follows:

VV =√3VS (1.18)

The following an be obtained for the maximum blo king voltage of the diodes:

Vvmax =√2VV =

√2√3VS =

√6VS ≈ 2.45VS (1.19)

For the al ulation of the ideal DC voltage it is better to integrate only over a third of the period,

2π3. In this ase the integration is done for the voltage V1. The lower integration boundary is

obtained from the interse tion of V1 with V2, the upper one from the interse tion of V1 with V3.

The boundaries an be obtained if the equations for the voltages are set equal to ea h other.

In g. 1.20 the voltages are shown. Hen e, the ideal DC voltage an be al ulated to

Vdi =3

2π

5

6π∫

π

6

√2VS sin (ωt) dωt =

3

2π

√2VS [− cos (ωt)]

5

6π

π

6

= . . . =3√6

2πVS ≈ 1.17VS (1.20)

If, instead of this, the line to line voltage is used,

Vdi =3√2

2πVV ≈ 0.68VV (1.21)

an be obtained.

16

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350Voltages

Time [s]

Voltage [V

]

VR

VRL

VRC

0 0.005 0.01 0.015 0.02 0.025−2

0

2

4

6

8

Time [s]

Curr

ent [A

]

Currents

IR

IRL

IRC

Fig. 1.19: Current and voltage traje tories (M3 ir uit)

0 0.5 1 1.5 2−400

−300

−200

−100

0

100

200

300

400

ω t / π

Voltage [V

]

V1

V2

V3

Fig. 1.20: Voltages in three-phase systems

17

1.6.2 B6 ir uit

Nowadays the B6 ir uit is mostly used for the re ti ation of three-phase AC voltages.

1.6.2.1 Cir uit diagram

In g. 1.21 a B6 ir uit is drawn.

PSfrag repla ements

3 ∼ vL

iL

D1 D2 D3

D4 D5 D6

Fig. 1.21: B6 ir uit

1.6.2.2 General fun tional prin iple

The ir uit shown in g. 1.22 was also simulated, in this ase espe ially in order to show the

general fun tional prin iple. The values given in hapter 1.3.2 were used as well. In g. 1.23

the urrent and voltage traje tories are shown. From the upper diodes (D1, D2 and D3) the

one with the highest potential is ondu ting, from the lower diodes (D4, D5 and D6) the one

whose phase urrently has the most negative potential.

A

V

V

V

V

AAA

A A A

Fig. 1.22: PSIM model for showing the general fun tional prin iple of the B6 ir uit

18

0 0.005 0.01 0.015 0.02 0.025−500

0

500Transformer voltages

Time [s]

Voltage [V

]

V1

V2

V3

0 0.005 0.01 0.015 0.02 0.0250

200

Voltage drop on R

Time [s]

Voltage [V

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through upper left diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through upper middle diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through upper right diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through lower left diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through lower middle diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.025−10

0

10Current through lower right diode

Time [s]

Curr

ent [A

]

0 0.005 0.01 0.015 0.02 0.0250

5

Current through R

Time [s]

Curr

ent [A

]

Fig. 1.23: Current and voltage traje tories showing the general fun tional prin iple of the B6

ir uit

19

1.6.2.3 Current and voltage traje tories for R, RC and RL load

A further simulation was made in order to plot the urrent and voltage traje tories for a pure

R, an RC and an RL load. The traje tories an be explained analogously to hapter 1.3.2. The

models are shown in g. 1.24, the urrent and voltage traje tories in g. 1.25. In order to make

the voltage smoothing visible, in this ase a apa itor with C = 500µF was sele ted.

A

VV

A

V

A

Fig. 1.24: PSIM models of the B6 ir uit with dierent loads

1.6.2.4 Important parameters

The B6 ir uit an be interpreted as a series onne tion of two M3 ir uits. Hen e, the ideal

DC voltage is twi e the one of the M3 ir uit:

Vdi = 2 ·3

2π

√6VS =

3√6

πVS ≈ 2.34VS (1.22)

Related to the line to line voltage VV the following relation an be obtained:

Vdi =3√2

πVV ≈ 1.35VV (1.23)

the diode's maximum blo king voltage is (without derivation for sake of brevity)

VVmax = 1.05Vdi (1.24)

The B6 ir uit is quite often also used in industrial abinets to provide a 24V DC voltage: First

the voltage is transformed via a three-phase transformer to VV = 18V (line to line voltage),

the ideal DC voltage then results to a. 24V a ording to equation 1.23. If the spe i ation

for the voltage ripple is sele ted well, no smoothing apa itor is ne essary. This makes su h a

setup very robust against power u tuations.

20

0 0.005 0.01 0.015 0.02 0.025150

200

250

300

350Voltages

Voltage [V

]

Time [s]

VR

VRL

VRC

0 0.005 0.01 0.015 0.02 0.0250

2

4

6Currents

Curr

ent [A

]

Time [s]

IR

IRL

0 0.005 0.01 0.015 0.02 0.025−50

0

50

100

150Current through RC element

Curr

ent [A

]

Time [s]

Fig. 1.25: Current and voltage traje tories (B6 ir uit)

21