Simplified Analysis of Graetz Circuit

1. Chapter
III:Simplified
Analysis of
Graetz Circuit

2. 3.3.1 Without
Overlap
•At any instant, two valves are conducting in the bridge,
one from the upper and one from the lower
commutation group.
•As the next valve of a commutation group fires, the
preceding valve turns off.
•This assumption, that no overlap between two
valves(meaning no two valves are “on” at the same
instant), is incorrect.
•However, this assumption provides a simpler analysis
into the operation of a converter.

3. 3.3.1 Without
Overlap
•The firing of valves are numbered in sequence, with
60° intervals, and conducts for 120°.
•Consecutive firing pulse is 60° in steady state.

4. Circuit diagram of a
basic Graetz circuit

5. 3.3.1 Without
Overlap
To further simplify analysis, the following assumptions
are taken into consideration
1. The DC current is constant
2. The valves can be modelled as ideal switches
with zero impedance when on or conducting,
and infinite impedance when off or not
conducting.
3. The AC voltages at the converter bus are
sinusoidal and remain constant.

6. 3.3.1 Without
Overlap
•A period of an AC voltage supply can be divided into 6
intervals, corresponding to the firing of a pair of valves.
•The DC voltage waveform repeats for each interval,
thus simplifying the calculation of the average DC
voltage, since we only have to consider one interval.
•Assuming the firing of the 3rd valve is delayed by an
angle α (α° after the crossing of the commutation
voltage for valve 3 – voltage eba ) the instantaneous DC
voltage vd during the interval is given by
vd = eb – ec = ebc, α ≤ ωt ≤ α + 60°

7. 3.3.1 Without
Overlap
Let eba = ELL sin ωt
Then ebc = ELL sin (ωt + 60°)
2
2

8. 3.3.1 Without
Overlap
•Eq. 3.8 that different values of α (range of α is from 0
to 180°), vd is variable. More importantly, when α
crosses 90°, vd is negative value.
•This means that the same converter can act as an
inverter or rectifier depending on the direction of the
DC voltage.

9. 3.3.1 Without
Overlap:
DC Voltage Waveform
•The DC voltage wave contains a ripple whose
fundamental frequency is six times the supply
frequency.
•It contains harmonics of the order h = np, where p is
the pulse number, and n is an integer.
•The rms value of the hth order harmonic is given by:

10. 3.3.1 Without
Overlap:
DC Voltage Waveform
As seen on the figure, 3 voltage
jumps arise due to the
commutation from one valve to
the next. The jumps have the
same magnitude given by eq.
3.10:

11. 3.3.1 Without
Overlap:
DC Voltage Waveform
The full range of 180° cannot be utilized. For all series
thyristors to fire, a minimum limit of alpha is set
greater than zero, likewise, for it to go off, an upper
limit is set less than 180°. Α is not allowed to go beyond
the upper limit (180°-γ), where γ is called the
extinction angle or margin angle. The minimum γ
value is 10°. For normal inverter operation, γ is not
allowed to go below 15° or °.

12. 3.3.1 Without
Overlap:
AC Current Waveform
While it is assumed that dc has no ripple(or harmonics)
due to smoothing reactors in series with the bridge
circuit. The AC currents flowing through the primary
and secondary windings of the transformer has.
Fig. 3.6 shows the current waveform in a valve winding.

13. 3.3.1 Without
Overlap:
AC Current Waveform
• The rms value of the fundamental component of current is
given by:
Where n is an integer, and p is the pulse number.

14. 3.3.1 Without
Overlap:
AC Current Waveform, Power Factor
AC harmonics order(6 pulse bridge converter) : 5, 7, 11,
13 and higher order.
• The first four are filtered out using tuned filters for each.
• The rest requires high pass filters.
The rms value of the hth is given by
The AC power supplied to the converter is
𝐼ℎ =
𝐼𝐼
ℎ

15. 3.3.1 Without
Overlap:
Power Factor
•Ignoring the losses in the converter, the DC power will
be equal to the AC power. Therefore:
Substituting for Vdo and II from eqs. 3.8 and
3.11 gives : cosΦ = cosα
The reactive power requirement increases as α closes
to 90° from 0° or 80°. When α=90°, pf=0 and only
reactive power is consumed.

16. 3.3.2 With Overlap
• Due to the leakage inductance of the converter
transformers and the supply network’s impedance, the
current in the valve will not suddenly change. An
example is when commutation from valve 1 to 3, there
is a finite period of time when both valves are
conducting. That is called overlap and its duration is
measured by u called overlap(commutation) angle.

17. 3.3.2 With Overlap
•The first subinterval shows three valves are conducting and 2nd
subinterval shows 2 valves. This is based on the assumption that the
u<60°. As u goes beyond 60°, there is a finite period where 4 valves are
commutating and 3 valves at the rest of the interval.
• Because of this, there are 3 modes of the converter.

18. 3.3.2 With Overlap
The three modes are:
◦ 1. Mode 1 – two and three valve conduction u<60°
◦ 2. Mode 2 – Three valve conduction u=60°
◦ 3. Mode 3 – Three and four valve conduction u<60°

19. 3.3.2 With Overlap:
Analysis of Mode 1
For this Mode, the bridge circuit can be simplified to
this:

20. 3.3.2 With Overlap:
Analysis of Mode 1
For that circuit eb – ea = Lc eq. 3.16
The L.H.S is called the commutating emf and its value is:
Which is also the voltage across valve 3 just before it fires.
Since i1 = Id – i3 (eq. 3.18), we get:
𝑑𝑖3
𝑑𝑡
−
𝑑𝑖1
ⅆ𝑡

21. 3.3.2 With Overlap:
Analysis of Mode 1
Where:
The sol’n from 3.20 is from the initial condition i3 (ωt = a) = 0 (eq. 3.22)
At ωt = α + u, i3 = Id,, this gives Id = Is[cos α – cos (α +u)] (eq. 3.23)
Note: during commutation, the
instantaneous dc voltage is:
Instead of (eb – ec)
−
3
2
𝑒 𝑐

22. 3.3.2 With Overlap:
Analysis of Mode 1 : Average Direct
Voltage
The average direct voltage Vd is obtained as:

23. 3.3.2 With Overlap:
Analysis of Mode 1 : Average Direct
Voltage
Where Rc is called equivalent commutation resistance. This is
similar to armature reaction in DC machines as it only
represents voltage drop and not a power loss.

24. 3.3.2 With Overlap:
Analysis of Mode 1 : DC voltage and Valve voltage
waveforms
Notice the valve voltage has now 8 jumps. The two major
jumps occur at the firing and turning off and is given by:

25. 3.3.2 With Overlap:
Analysis of Mode 1 : DC voltage and Valve voltage
waveforms
•The remaining 6 jumps are divided to 2 groups, the first
group composed of jumps that are equal in magnitude
to , and the other equals with .
•The jumps result in extra losses in the damper circuit.
𝑉𝑗1
2
𝑉𝑗2
2

26. 3.3.2 With Overlap:
Analysis of Mode 1 : Inverter equations
To operate as an inverter, an advance angle β is defined
where β = π – α, and use opposite polarity for the dc
voltage with voltage rise opposite to the direction of
current. Thus:

27. 3.3.2 With Overlap:
Analysis of Mode 1 : Inverter equations
The subscript “i” refers to the inverter.

28. 3.3.2 With Overlap:
Analysis of Mode 1 : AC current and DC voltage
harmonics
The waveform of the valve voltage and current are
distorted. And so, the expressions derived for the case
w/out overlap becomes invalid. Through Fourier
Analysis, the actual expression for the current is :
Where Φ is the p.f angle and δ = α + u

29. 3.3.2 With Overlap:
Analysis of Mode 1 : AC current and DC voltage
harmonics
From those expressions Φ can be obtained by:
The harmonic components in the AC current are also
changed. The following is reduced form the values
calculated with no overlap and is called the reduction
factor:

30. 3.3.2 With Overlap:
Analysis of Mode 1 : AC current and DC voltage
harmonics
The DC voltage expression are also affected due to
overlap. It can be shown that

31. 3.3.2 With Overlap:
Analysis of Mode 2
•Mode 1 is the normal mode of operation. Mode 2
occurs during DC line faults or a dip in the AC voltage
occurs.
•The overlap angle u may exceed 60°.
•When u>60°, the minimum number of valves
conducting are three and there are intervals where four
valves.

32. 3.3.2 With Overlap:
Analysis of Mode 2
The equivalent circuit of this mode is shown:

33. 3.3.2 With Overlap:
Analysis of Mode 2
= Is cos(α + u -60°) + B (3.46)

34. 3.3.2 With Overlap:
Analysis of Mode 2

35. 3.3.2 With Overlap:
Analysis of Mode 2 : Average Direct
Voltage

36. 3.3.2 With Overlap:
Analysis of Mode 2 : Average Direct
Voltage
In this case, the comparison between eqs. 3.50 and
3.26 shows that Rc is three times that of the case
where the u<60°. When α=0, u reaches a value of 60°
when Id increases to : Id = Is [1 – cos 60°] = (0.5)Is.
At u=60°, three valves are conducting all the time. For
α≥30, it is possible for the current to be increased
further, which also increases u further. The reason the
overlap angle cannot go beyond 60° for α<30° is shown
in the next figure.

37. 3.3.2 With Overlap:
Analysis of Mode 2 : Average Direct
Voltage
As an example, picture valve 2 and 6 are still conducting, the
anode of valve 3 is at a potential of and the𝑒 𝑏 + 𝑒 𝑐
2
=
−𝑒 𝑎
2

38. 3.3.2 With Overlap:
Analysis of Mode 2 : Average Direct
Voltage
Cathode is at a potential of ea. Hence, unless ea is
negative, valve 3 cannot conduct and this is possible
only for α≥30°. Thus there is an intermediate mode
when the value of overlap angle remains constant at
60°. In this mode, an increase in direct current is
accompanied by automatic increase in the firing angle.

39. 3.3.2 With Overlap:
Analysis of Mode 2 : DC and Valve Voltage
Waveforms
Instantaneous voltage
across the converter
bridge (vd)
Valve voltage of #3 and
#4 valves.

40. 3.3.2 With Overlap:
Analysis of Mode 2 : DC and Valve Voltage
Waveforms
The instantaneous voltage across the bridge can have
both positive and negative excursions, followed by
periods of zero magnitude. The valve voltage has 6
jumps with three of the having the magnitude

41. Reference:
Padiyar, K. R. (1990). HVDC Power Transmission
Systems: Technology and System Interactions. New
Delhi: New Age International