2. Harmonics and power factors for different rectifiers

Non-linear loads such as rectifiers can cause harmonics on the aircraft electrical supply systems, this will increase losses and can excite resonance in some circuits resulting in elevated voltages. For a resistive load, the harmonics current are proportional to the voltage harmonics. For a capacitive load the harmonics current will increase the capacitor heating and can cause premature failure.

For an inductive load harmonics increase losses in core components and also rotor losses in an induction motor will increase. Furthermore harmonics current could overheat transformers, therefore transformers, should be derated in the presence of harmonics.

In general rectifiers produce harmonics with the following order [10–14]:

In MEA Power Electronics segment plays a very important part in controlling the energy and improving both generators and actuators energy conversion. Furthermore, in a fixed frequency system (400 Hz) a mechanical constant speed drive set between the engine and the aircraft

The use of Power Electronics helps in reducing weight, is easier to maintain, and provides more controllability and intelligence which includes fault detection and diagnosis [1–6].

A conventional 12 pulse rectifier using Diode Bridge is one of the simplest converter since does not require any control loop, however, this type of converter has a fixed DC output with high Total Harmonic Distortion (THD) on the input current compared with the proposed 12-pulse

The system has the ability to stabilize an output voltage of variable Vdc from a 3 phase 360– 800 Hz, 115 V RMS system. Using a decoupling feed-forward control method by DQ frame technique, the magnitude and the phase of the input current can be controlled and hence the power transfer that occurs between the AC and DC sides can also be controlled. The converter could be suitable to use with an electric actuator (or other) aircraft loads. The system could be used as DC source for DC loads or to feed DC to AC inverter for a fixed 400 Hz supply. The design of this system poses significant challenges due to the nature of the load range and

generator, however this will give extra weight and must be frequently maintained.

supply frequency variation and requires many features such as: 1. Sinusoidal and low harmonics contents on supply current.

active rectifier.

Figure 1. MEA general power distribution system.

212 Flight Physics - Models, Techniques and Technologies

$$h = \frac{f\_h}{f\_1} = K.P \pm 1\tag{1}$$

where h = order of harmonics; fh = frequency of the harmonic current; f1 = fundamental frequency; P = rectifier pulse number; K = 1, 2, 3,….

The amplitude of the harmonic currents caused by rectifier can be calculated as:

$$I\_h = \frac{I\_1}{h} \tag{2}$$

where Ih = amplitude of harmonic current order; I<sup>1</sup> = amplitude of the fundamental current of the rectifier.

In AC power systems with pure sinusoidal voltage and current, the cosine of the phase difference (ϕ) between the voltage and current represents the power factor (PF = cos ϕ). If the voltage or current waveforms contain harmonics, the phase angle between them is no longer represents the power factor. In general, the power factor could be calculated as [11].

$$PF = \frac{\text{mean power}}{V\_{rms}I\_{rms}}\tag{3}$$

Rectifiers draw non-sinusoidal current and have high harmonic components, however, if the input voltage of the rectifier is considered to be a pure sinusoidal, therefore the mean power will be:

$$P\_{\text{mean}} = V\_{rms} I\_{1rms} \cos \varphi\_1 \tag{4}$$

Therefore:

$$PF = \frac{I\_{1rms}}{I\_{rms}} \cos \varphi\_1 \tag{5}$$

Where <sup>I</sup>1rms Irms is defined as the input distortion factor; I1rms is the RMS value of the fundamental current; cos ϕ<sup>1</sup> is the phase angle between the voltage and the fundamental current (input displacement factor).

Electronic devices in MEA technology are increasing, which are usually powered by switched mode power supplies (SMPS). SMPS will properly feed from a diode rectifier which imposes harmonic currents and possibly voltages onto the mains power network on the aircraft systems. This can cause some damage to the cables and equipment within the aircraft electric network. Supply current waveform may be expressed by the Fourier series [11–14]:

$$i\_s(t) = I\_{DC} + \sum\_{n=1}^{\infty} \left( a\_n \cos n\omega t + b\_n \sin n\omega t \right) \tag{6}$$

For three phase 6-pulse diode bridge, the DC output voltage and the RMS input current equal:

$$V\_{DC} = \frac{3\sqrt{3}}{\pi} V\_m \tag{7}$$

Vm is the maximum phase voltage.

$$I\_{RMS} = \frac{\sqrt{6}}{3} I\_{DC} \tag{8}$$

Assume losses of the rectifier is zero, therefore the power is

$$P\_{\rm out} = P\_{\rm in} = V\_{\rm DC} I\_{\rm DC} = \frac{3\sqrt{3}}{\pi} V\_{\rm m} I\_{\rm DC} \tag{9}$$

The input apparent power for the rectifier is:

$$\mathcal{S}\_{\text{int}} = \mathfrak{D}V\_{RMS}I\_{RMS} = \sqrt{\mathfrak{J}}\,V\_{m}I\_{D\mathbb{C}} \tag{10}$$

Therefore:

$$PF = \frac{P\_{in}}{S\_{in}} = \frac{3}{\pi} = 0.955\tag{11}$$

Although the power factor is good, the THD value is relatively high and could have a bad effect on the aircraft power systems. The RMS of the input fundamental current for three phase 6-pulse diode rectifier with an inductive load is well known and equals to:

12-Pulse Active Rectifier for More Electric Aircraft Applications http://dx.doi.org/10.5772/intechopen.70882 215

$$I\_{1RMS} = \frac{\sqrt{6}}{3} I\_{DC} \tag{12}$$

$$\text{THD} = \frac{\sqrt{I\_{RMS}^2 - I\_{1RMS}^2}}{I\_{1RMS}} = \frac{\sqrt{\pi^2 - 9}}{3} = 31.08\% \tag{13}$$

The THD could be reduced by using 12-pulse rectifier as shown in Figure 2.

12-pulse diode rectifier is fed from a three phase star connected transformer on the primary side, star and delta transformers on the secondary side. Each transformer on the secondary side feeds a three phase 6-pulse rectifier and they add together to form a 12-pulse rectifier, this configuration gives 30� of phase shift which gave harmonics cancellation. The turn ratio of the delta transformer must be multiplied by ffiffiffi 3 <sup>p</sup> factor in order to get the same voltage level, this illustrated in Figure 3 [14].

Figure 2. 12-pulse diode rectifier.

Pmean ¼ VrmsI1rms cosϕ<sup>1</sup> (4)

cosϕ<sup>1</sup> (5)

<sup>n</sup>¼<sup>1</sup> ð Þ an cos <sup>n</sup>ω<sup>t</sup> <sup>þ</sup> bn sin <sup>n</sup>ω<sup>t</sup> (6)

Vm (7)

<sup>3</sup> IDC (8)

VmIDC (9)

<sup>p</sup> VmIDC (10)

<sup>π</sup> <sup>¼</sup> <sup>0</sup>:<sup>955</sup> (11)

PF <sup>¼</sup> <sup>I</sup>1rms Irms

Irms is defined as the input distortion factor; I1rms is the RMS value of the fundamental

current; cos ϕ<sup>1</sup> is the phase angle between the voltage and the fundamental current (input

Electronic devices in MEA technology are increasing, which are usually powered by switched mode power supplies (SMPS). SMPS will properly feed from a diode rectifier which imposes harmonic currents and possibly voltages onto the mains power network on the aircraft systems. This can cause some damage to the cables and equipment within the aircraft electric

For three phase 6-pulse diode bridge, the DC output voltage and the RMS input current equal:

3 p π

ffiffiffi 6 p

> 3 p π

3

VDC <sup>¼</sup> <sup>3</sup> ffiffiffi

IRMS ¼

Pout <sup>¼</sup> Pin <sup>¼</sup> VDCIDC <sup>¼</sup> <sup>3</sup> ffiffiffi

Sin <sup>¼</sup> <sup>3</sup>VRMSIRMS <sup>¼</sup> ffiffiffiffi

Although the power factor is good, the THD value is relatively high and could have a bad effect on the aircraft power systems. The RMS of the input fundamental current for three phase

PF <sup>¼</sup> Pin Sin ¼ 3

6-pulse diode rectifier with an inductive load is well known and equals to:

network. Supply current waveform may be expressed by the Fourier series [11–14]:

isðÞ¼ <sup>t</sup> IDC <sup>þ</sup>X<sup>∞</sup>

Assume losses of the rectifier is zero, therefore the power is

The input apparent power for the rectifier is:

Therefore:

Where <sup>I</sup>1rms

Therefore:

displacement factor).

214 Flight Physics - Models, Techniques and Technologies

Vm is the maximum phase voltage.

Figure 3. Star-delta configuration.

In a three phase 6-pulse rectifier the dominated harmonics are the 5th and 7th and this why THD is quite high. With a 12 pulse arrangement the 5th and 7th harmonics are canceled as illustrated below [11–14]:

For the star connection, phase (a) current equals:

$$i\_{a\\_starl}(t) = 2\frac{\sqrt{3}}{\pi}I\_d\left(\cos\omega t - \frac{1}{5}\cos 5\omega t + \frac{1}{7}\cos 7\omega t - \frac{1}{11}\cos 11\omega t + \dots\right) \tag{14}$$

For the delta connection phase (a) current equals:

$$i\_{\underline{a}\\_dth}(t) = 2\frac{\sqrt{3}}{\pi}I\_d\left(\cos\omega t + \frac{1}{5}\cos 5\omega t - \frac{1}{7}\cos 7\omega t + \frac{1}{11}\cos 11\omega t + \dots\right) \tag{15}$$

The primary current is equal to the summation of both secondary currents:

$$\dot{q}\_{\text{a\\_imp}}(t) = 4 \frac{\sqrt{3}}{\pi} I\_d \left( \cos \omega t - \frac{1}{11} \cos 11 \omega t + \frac{1}{13} \cos 13 \omega t - \frac{1}{23} \cos 23 \omega t + \dots \right) \tag{16}$$

The series has harmonics of an order of 12k� 1 and the harmonics of orders 6k� 1 circulate between the two converter transformers and do not penetrate the aircraft power system network. Since the magnitude of each harmonic is proportional to the reciprocal of the harmonic number, therefore the 12-pulse rectifier has a lower THD equals to:

$$I\_{1RMS} = \frac{2\sqrt{6}}{3} I\_{\text{DC}} \tag{17}$$

$$THD = \frac{\sqrt{I\_{RMS(12h)}^2 - I\_{1RMS(12h)}^2}}{I\_{1RMS(12h)}} = 2\frac{\sqrt{\pi^2 - 9}}{3} = 15.5\% \tag{18}$$

The THD for the 12-pulse rectifier is reduced by 50% compared with the 6-pulse rectifier.

Figure 4 shows the current waveforms for the supply input current of phase (a) and the currents on the secondary side of each transformer.

Figure 5 shows the harmonics contents for the currents of phase (a) and the currents on the secondary side of each transformer.

For a more THD reduction, 12-pulse active rectifier could be used, this is shown in Figure 6. Many advantages are associated with this type of converter:


Figure 4. Current waveforms for phase (a) at input frequency 400 Hz.

In a three phase 6-pulse rectifier the dominated harmonics are the 5th and 7th and this why THD is quite high. With a 12 pulse arrangement the 5th and 7th harmonics are canceled as

<sup>5</sup> cos 5ω<sup>t</sup> <sup>þ</sup>

<sup>5</sup> cos 5ω<sup>t</sup> � <sup>1</sup>

<sup>11</sup> cos 11ω<sup>t</sup> <sup>þ</sup>

The series has harmonics of an order of 12k� 1 and the harmonics of orders 6k� 1 circulate between the two converter transformers and do not penetrate the aircraft power system network. Since the magnitude of each harmonic is proportional to the reciprocal of the harmonic number,

<sup>I</sup>1RMS <sup>¼</sup> <sup>2</sup> ffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I1RMSð Þ <sup>12</sup><sup>h</sup>

2 1RMSð Þ 12h

The THD for the 12-pulse rectifier is reduced by 50% compared with the 6-pulse rectifier.

Figure 4 shows the current waveforms for the supply input current of phase (a) and the

Figure 5 shows the harmonics contents for the currents of phase (a) and the currents on the

For a more THD reduction, 12-pulse active rectifier could be used, this is shown in Figure 6.

• The rectifier could be operated with a variable input frequency (usually 360–800 Hz)

RMSð Þ <sup>12</sup><sup>h</sup> � <sup>I</sup>

6 p

¼ 2

1

1

� �

<sup>7</sup> cos 7ω<sup>t</sup> � <sup>1</sup>

<sup>7</sup> cos 7ω<sup>t</sup> <sup>þ</sup>

<sup>13</sup> cos 13ω<sup>t</sup> � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup><sup>2</sup> � <sup>9</sup> <sup>p</sup>

� �

1

� �

1

<sup>11</sup> cos 11ω<sup>t</sup> <sup>þ</sup> …

<sup>11</sup> cos 11ω<sup>t</sup> <sup>þ</sup> …

<sup>23</sup> cos 23ω<sup>t</sup> <sup>þ</sup> …

<sup>3</sup> IDC (17)

<sup>3</sup> <sup>¼</sup> <sup>15</sup>:5% (18)

(14)

(15)

(16)

illustrated below [11–14]:

ia\_starðÞ¼ t 2

216 Flight Physics - Models, Techniques and Technologies

ia\_deltaðÞ¼ t 2

ia\_inpðÞ¼ t 4

For the star connection, phase (a) current equals:

For the delta connection phase (a) current equals:

ffiffiffi 3 p π

ffiffiffi 3 p π

ffiffiffi 3 p π

Id cos <sup>ω</sup><sup>t</sup> � <sup>1</sup>

Id cos ωt þ

Id cos <sup>ω</sup><sup>t</sup> � <sup>1</sup>

therefore the 12-pulse rectifier has a lower THD equals to:

I 2

q

Many advantages are associated with this type of converter:

• The power factor could be controlled by using DQ vectors control.

THD ¼

currents on the secondary side of each transformer.

secondary side of each transformer.

without interrupting its output.

• Bidirectional power flow.

• THD is very low.

The primary current is equal to the summation of both secondary currents:

Figure 5. Harmonics contents for the currents of phase (a) at input frequency 400 Hz.

Figure 6. AC/DC 12 pulse boost converter.

### 3. DQ control circuit

Figure 7 show the configuration of the active rectifier for each secondary side.

The DQ transform is usually called Park transform which is a space vector transformation of the instantaneous 3 phase voltages and currents from a stationary phase coordinate system (ABC) to a rotating coordinate system (DQ) [7, 8].

The general formulas for DQ transformations are given as follows. We assume that the threephase source voltages va, vb and vc are balanced and sinusoidal with an angular frequency ω.

The components of the input voltage phasor along the axes of a stationary orthogonal reference frame (α, β) are given by:

$$V\_{\alpha} = \frac{2}{3}V\_{a} - \frac{1}{3}V\_{b} - \frac{1}{3}V\_{c} \tag{19}$$

$$V\_{\notin} = \frac{1}{\sqrt{3}}V\_c - \frac{1}{\sqrt{3}}V\_b$$

The input voltage can then be transformed to a rotating reference frame DQ chosen with the D axis aligned with the voltage phasor. The voltage components are given by:

$$
\sigma\_d = V\_\alpha \cos \omega t - V\_\beta \sin \omega t \tag{21}
$$

$$v\_q = V\_a \sin \omega t + V\_\beta \cos \omega t \tag{22}$$

The same transformations are applied to the phase currents:

$$i\_d = I\_a \cos\omega t - I\_\beta \sin\omega t \tag{23}$$

$$i\_q = I\_a \sin \omega t + I\_\beta \cos \omega t \tag{24}$$

Referring to Figure 7, let va1, vb<sup>1</sup> and vc<sup>1</sup> be the fundamental voltages per phase at the input of the converter.

$$
\sigma\_a = \text{Ri}\_a + L.\text{di}\_a/\text{d}t + \upsilon\_{a1} \tag{25}
$$

$$
\sigma\_b = \text{Ri}\_b + L.\text{di}\_b/\text{d}t + \sigma\_{b1} \tag{26}
$$

$$
\sigma\_c = \text{Ri}\_c + L.\text{di}\_c/\text{d}t + \upsilon\_{c1} \tag{27}
$$

where L is the value of input line inductance and R is its resistance of the inductor.

Taking the DQ transformation for the inductor, the input voltage to the converter in the DQ reference frame is given by [15–19]:

12-Pulse Active Rectifier for More Electric Aircraft Applications http://dx.doi.org/10.5772/intechopen.70882 219

Figure 7. 6-pulse active rectifier configuration.

$$
\upsilon\_d = \text{Ri}\_d + L.\text{di}\_d/\text{d}t - \omega L i\_q + \upsilon\_{d1} \tag{28}
$$

$$
\sigma\_q = \text{Ri}\_q + L.\text{di}\_q/\text{d}t + \omega L\text{i}\_d + \sigma\_{q1} \tag{29}
$$

Note that vd<sup>1</sup> and vq<sup>1</sup> are the DQ components at the converter terminals.

Figure 8 shows the phasor diagrams for DQ coordinates.

The instantaneous active and reactive powers are given by:

$$P\_d(t) = \mathfrak{Z}/2\left(\upsilon\_d.i\_d + \upsilon\_q i\_q\right) \tag{30}$$

$$Q\_d(t) = \mathfrak{Z}/2\left(\upsilon\_d.i\_q + \upsilon\_q i\_d\right) \tag{31}$$

During the steady state and by assuming the converter losses are negligible, the DC and AC power are equal, therefore:

$$P\_d = P\_{\rm DC} = V\_{\rm DC} I\_{\rm DC} \tag{32}$$

Therefore

3. DQ control circuit

218 Flight Physics - Models, Techniques and Technologies

ence frame (α, β) are given by:

the converter.

reference frame is given by [15–19]:

(ABC) to a rotating coordinate system (DQ) [7, 8].

Figure 7 show the configuration of the active rectifier for each secondary side.

<sup>V</sup><sup>α</sup> <sup>¼</sup> <sup>2</sup> 3 Va � <sup>1</sup> 3 Vb � <sup>1</sup> 3

axis aligned with the voltage phasor. The voltage components are given by:

The same transformations are applied to the phase currents:

<sup>V</sup><sup>β</sup> <sup>¼</sup> <sup>1</sup> ffiffiffi 3 <sup>p</sup> Vc � <sup>1</sup>

The DQ transform is usually called Park transform which is a space vector transformation of the instantaneous 3 phase voltages and currents from a stationary phase coordinate system

The general formulas for DQ transformations are given as follows. We assume that the threephase source voltages va, vb and vc are balanced and sinusoidal with an angular frequency ω. The components of the input voltage phasor along the axes of a stationary orthogonal refer-

The input voltage can then be transformed to a rotating reference frame DQ chosen with the D

Referring to Figure 7, let va1, vb<sup>1</sup> and vc<sup>1</sup> be the fundamental voltages per phase at the input of

Taking the DQ transformation for the inductor, the input voltage to the converter in the DQ

where L is the value of input line inductance and R is its resistance of the inductor.

ffiffiffi 3 Vc (19)

p Vb (20)

vd ¼ V<sup>α</sup> cos ωt � V<sup>β</sup> sin ωt (21)

vq ¼ V<sup>α</sup> sin ωt þ V<sup>β</sup> cos ωt (22)

id ¼ I<sup>α</sup> cos ωt � I<sup>β</sup> sin ωt (23)

iq ¼ I<sup>α</sup> sin ωt þ I<sup>β</sup> cos ωt (24)

va ¼ Ria þ L:dia=dt þ va<sup>1</sup> (25)

vb ¼ Rib þ L:dib=dt þ vb<sup>1</sup> (26)

vc ¼ Ric þ L:dic=dt þ vc<sup>1</sup> (27)

$$I\_{D\mathbb{C}} = \frac{P\_d}{V\_{D\mathbb{C}}} = \frac{\Im\left(\upsilon\_d.\dot{i}\_d + \upsilon\_q\dot{i}\_q\right)}{2V\_{D\mathbb{C}}}\tag{33}$$

For a power balance, the delivering power should equal to the absorbing power therefore:

$$P\_{A\mathbb{C}} + P\_{D\mathbb{C}} + P\_{\mathbb{C}} = 0 \tag{34}$$

Where PC is the power in the capacitor filter.

If the synchronous frame is aligned to voltage, the quadrature component, vq = 0. Therefore, the power equations reduce to:

Figure 8. DQ phasor diagrams.

$$P\_d = 3/2 \text{ v}\_d \text{.i}\_d \tag{35}$$

$$Q\_d = 3/2 \text{ v}\_d \text{.i}\_q \tag{36}$$

Eq. (32) becomes:

$$I\_{D\mathbb{C}} = \frac{P\_d}{V\_{D\mathbb{C}}} = \frac{\mathfrak{Z}\left(\upsilon\_d, i\_d\right)}{2V\_{D\mathbb{C}}}\tag{37}$$

That gives:

$$P\_{A\mathcal{C}} + P\_{D\mathcal{C}} + P\_{\mathcal{C}} = 3/2 \text{ v}\_d.i\_d + V\_{D\mathcal{C}}.I\_{D\mathcal{C}} + V\_{D\mathcal{C}}.i\_{\mathcal{C}} = 0 \tag{38}$$

Therefore the capacitor current becomes:

$$\dot{\mathbf{u}}\_{\mathcal{C}} = -\left(\frac{\mathbf{3} \left(\mathbf{v}\_d \,\dot{\mathbf{u}}\_d\right)}{2V\_{D\mathcal{C}}} + I\_{D\mathcal{C}}\right) \tag{39}$$

But:

$$\dot{a}\_{\mathbb{C}} = \mathbb{C} \frac{dV\_{D\mathbb{C}}}{dt} \tag{40}$$

From Eqs. (38)–(40):

$$\frac{dV\_{D\mathbb{C}}}{dt} = \frac{\dot{\mathbf{i}}\_{\mathbb{C}}}{\mathbb{C}} = \frac{-1}{\mathbb{C}} - \left(\frac{\Im\left(V\_{d}\dot{\mathbf{i}}\_{d}\right)}{2V\_{D\mathbb{C}}} + I\_{D\mathbb{C}}\right) \tag{41}$$

From Eq. (41) by controlling the active Current id the DC output voltage of the rectifier could be controlled.

Inverse DQ transformations then need to be applied to provide the three phase modulating waves (varef, vbref and vcref) for the PWM generator. DQ vector control has several benefits such as reactive and active power will be easy to control and the dynamic response on the current loop will be very fast.

The PWM generator employs a 20 kHz carrier and is based on a regular asymmetric PWM strategy. The line inductor has a value of 100 μH per phase which limits the Total Harmonic Distortion (THD) to the required value.

Figure 9 shows the schematic of the DQ control scheme implemented in the input converter.

The proposed control scheme consists of two parts [13–15]:


Pd ¼ 3=2 vd:id (35)

Qd ¼ 3=2 vd:iq (36)

dt (40)

(37)

(39)

(41)

IDC <sup>¼</sup> Pd VDC

iC ¼ � <sup>3</sup> ð Þ vd:id 2VDC

<sup>C</sup> <sup>¼</sup> �<sup>1</sup>

dVDC dt <sup>¼</sup> iC iC <sup>¼</sup> <sup>C</sup>dVDC

From Eq. (41) by controlling the active Current id the DC output voltage of the rectifier could

Inverse DQ transformations then need to be applied to provide the three phase modulating waves (varef, vbref and vcref) for the PWM generator. DQ vector control has several benefits such as reactive and active power will be easy to control and the dynamic response on the current

The PWM generator employs a 20 kHz carrier and is based on a regular asymmetric PWM strategy. The line inductor has a value of 100 μH per phase which limits the Total Harmonic

<sup>C</sup> � <sup>3</sup> ð Þ Vd:id 2VDC

þ IDC

<sup>¼</sup> <sup>3</sup> ð Þ vd:id 2VDC

PAC þ PDC þ PC ¼ 3=2 vd:id þ VDC:IDC þ VDC:iC ¼ 0 (38)

þ IDC 

Eq. (32) becomes:

Figure 8. DQ phasor diagrams.

220 Flight Physics - Models, Techniques and Technologies

Therefore the capacitor current becomes:

That gives:

But:

From Eqs. (38)–(40):

be controlled.

loop will be very fast.

Distortion (THD) to the required value.

The outer voltage controller regulates the DC link voltage. The error signal is used as an input for the PI voltage controller this provides a reference to the D current of the inner current controller. Figure 10 shows the DC link model and Figure 11 shows the close loop control of outer voltage control.

Figure 9. DQ control for the input converter.

Figure 10. DC link model.

Figure 11. Close loop control of outer DC voltage control. PI TF: Kp + Ki 1 <sup>S</sup> = KP <sup>S</sup>þai S � �; converter TF: <sup>3</sup> <sup>2</sup> ffiffi 2 <sup>p</sup> M; plant TF: <sup>1</sup> CS.

The relationship between the DC voltage and the D axis input voltage is given by:

$$V\_{DC} = \frac{2\sqrt{2}V\_d}{M} \text{ or } M = \frac{2\sqrt{2}V\_d}{V\_{DC}}\tag{42}$$

$$I\_{DC} = \frac{3}{2\sqrt{2}} \, MI\_d \tag{43}$$

Where: M is the modulation index.

For PI controller:

$$\text{PI TF} = K\_p + K\_i \frac{1}{S} = K\_P \left(\frac{S + a\_i}{S}\right) \tag{44}$$

Where ai <sup>¼</sup> Ki Kp

For the converter the system has the following TF:

$$\text{ConvertTF} = \frac{I\_{D\mathbb{C}}}{I\_d} = \frac{3}{2\sqrt{2}}M \tag{45}$$

For the plant TF, The dc link may be modeled by a capacitor:

$$\text{Plant TF} = \frac{1}{\text{CS}}\tag{46}$$

Therefore the characteristic equation for the DC link voltage control is given by:

$$S^2 + \frac{3MK\_p}{2\sqrt{2}\mathcal{C}}S + \frac{3MK\_p a\_i}{2\sqrt{2}\mathcal{C}} = 0\tag{47}$$

General equation for second order characteristic equation is given by:

$$\mathcal{S}^2 + 2\xi\omega\_n \mathcal{S} + \omega\_n^2 = 0 \tag{48}$$

Therefore the controller parameters are given by:

$$K\_p = \frac{4\sqrt{2}\text{ C\xi\omega}\_n}{3M} \tag{49}$$

$$a\_i = \frac{2\sqrt{2}\,\text{C}\omega\_n^2}{\text{3MK}\_p} \tag{50}$$

where ω<sup>n</sup> and ξ are the closed loop natural frequency and damping ratio, therefore the controller parameters can be easy calculated by choosing the value of the modulation index, ω<sup>n</sup> and ξ.

A PI inner DQ current control refers the phase current measurements to a rotating coordinate frame DQ fixed to the supply voltage. Figure 12 shows the DQ model of input stage and Figure 13 shows the close loop control of the inner current control. If the phase currents are in phase with the supply voltages, the current referred to the direct D axis becomes the DC link current and the current referred to the quadrature Q axis is equal to zero. The coordinate transformation is done using phase angle information derived from the measurement of the supply voltages. However, if the system is needed to operate with a leading or a lagging power factor, the Q axis reference value could be changed to define the displacement angle of the rectifier. The D axis and Q axis currents are compared to their respective demands values and the error is applied to individual PI controllers give voltage demands referred to D axis and Q axis. In the rotating co-ordinate frame the D axis and Q axis currents are inter-related due to their rotation. The rotation introduces an orthogonal component into time derivative of each current which, when applied to an inductive load, gives a voltage components along the axis orthogonal to that of the current. The DQ scheme studied uses two feed forward terms:


The relationship between the DC voltage and the D axis input voltage is given by:

2 <sup>p</sup> Vd

IDC <sup>¼</sup> <sup>3</sup> 2 ffiffiffi 2

> 1 <sup>S</sup> <sup>¼</sup> KP

> > Id

3MKpai 2 ffiffiffi 2

Plant TF <sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>3</sup> 2 ffiffiffi 2

<sup>M</sup> or <sup>M</sup> <sup>¼</sup> <sup>2</sup> ffiffiffi

2 <sup>p</sup> Vd VDC

S þ ai S � �

p MId (43)

p M (45)

CS (46)

<sup>p</sup> <sup>C</sup> <sup>¼</sup> <sup>0</sup> (47)

<sup>n</sup> ¼ 0 (48)

<sup>3</sup><sup>M</sup> (49)

(42)

(44)

(50)

VDC <sup>¼</sup> <sup>2</sup> ffiffiffi

PI TF ¼ Kp þ Ki

Converter TF <sup>¼</sup> IDC

Therefore the characteristic equation for the DC link voltage control is given by:

3MKp 2 ffiffiffi 2 <sup>p</sup> <sup>C</sup> S þ

<sup>S</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ξω<sup>n</sup> <sup>S</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

Kp <sup>¼</sup> <sup>4</sup> ffiffiffi 2 <sup>p</sup> <sup>C</sup>ξω<sup>n</sup>

ai <sup>¼</sup> <sup>2</sup> ffiffiffi 2 <sup>p</sup> <sup>C</sup>ω<sup>2</sup> n 3MKp

where ω<sup>n</sup> and ξ are the closed loop natural frequency and damping ratio, therefore the controller parameters can be easy calculated by choosing the value of the modulation index,

<sup>S</sup><sup>2</sup> <sup>þ</sup>

General equation for second order characteristic equation is given by:

Where: M is the modulation index.

222 Flight Physics - Models, Techniques and Technologies

For the converter the system has the following TF:

Therefore the controller parameters are given by:

For the plant TF, The dc link may be modeled by a capacitor:

For PI controller:

Where ai <sup>¼</sup> Ki

ω<sup>n</sup> and ξ.

Kp


These feed forward terms de-couple the two currents. In addition the supply voltage is referred to D axis and this added to the Vd demand to avoid the integrator having to compensate for it. The final demand voltages are transferred back into stationary co-ordinates and the resulting sinusoids are used to generate the PWM.

Figure 13 shows the close loop control of inner current control. The following transfer functions are applied to the control block.

The plant is represent the line from the generator to the input of the converter which has RL network with following TF:

Figure 12. Equivalent circuit for DQ input supply.

Figure 13. Close loop control of inner current control. PI TF: Kp + Ki 1 <sup>S</sup> = KP ( Sþai <sup>S</sup> <sup>Þ</sup>; converter TF: <sup>1</sup> <sup>1</sup>þTS; plant TF: <sup>1</sup> <sup>R</sup>þLS.

$$\text{Plant TF} = \frac{1}{R + LS} \tag{51}$$

The converter may be modeled as a first order lag. = <sup>1</sup> <sup>1</sup>þTS where <sup>T</sup> <sup>=</sup> <sup>1</sup> 2Fs

Fs is the switching frequency. The same procedure can be used to calculate the parameters of the controller. Simulink or other tools can be easily used to tune the PI controller by using Zeigler-Nichol's method.

#### 4. Simulation results for the 12-pulse active rectifier

In order to optimize the power quality and transient behavior of the power distribution system, a well-designed simulation model of the 12-pulse active rectifier based on detailed component models will be necessary.

For high voltage demand, the two rectifiers are connected in series, and for high current demand, the rectifiers may connect in parallel. The converter has been simulated for various operating conditions with the following parameters.


Simulation results show that comparing with a conventional 12-pulse diode rectifier, the low order harmonics are totally eliminated and only very low harmonics around the switching frequency at frequencies f = mfs where m = 1, 2, …, ∞

Figures 14–19 show different simulation results.

Figure 14. Waveforms results for 360 Hz input frequency.

Figure 15. THD for phase a current—input frequency 360 Hz.

Plant TF <sup>¼</sup> <sup>1</sup>

Fs is the switching frequency. The same procedure can be used to calculate the parameters of the controller. Simulink or other tools can be easily used to tune the PI controller by using

In order to optimize the power quality and transient behavior of the power distribution system, a well-designed simulation model of the 12-pulse active rectifier based on detailed

For high voltage demand, the two rectifiers are connected in series, and for high current demand, the rectifiers may connect in parallel. The converter has been simulated for various

Simulation results show that comparing with a conventional 12-pulse diode rectifier, the low order harmonics are totally eliminated and only very low harmonics around the switching

• Input inductance for L = 100 μH, input resistance 0.2 Ω. For each converter

The converter may be modeled as a first order lag. = <sup>1</sup>

4. Simulation results for the 12-pulse active rectifier

Zeigler-Nichol's method.

component models will be necessary.

224 Flight Physics - Models, Techniques and Technologies

operating conditions with the following parameters.

• Switching frequency 20 kHz. For each converter

frequency at frequencies f = mfs where m = 1, 2, …, ∞

Figures 14–19 show different simulation results.

Figure 14. Waveforms results for 360 Hz input frequency.

• The DC voltage reference for each converter is set to 320 V.

• DC filter C = 200 μF. For each converter

• Input frequency 360–800 Hz.

• Resistive load = 15 Ω.

• AC input voltage = 115 V RMS.

<sup>R</sup> <sup>þ</sup> LS (51)

2Fs

<sup>1</sup>þTS where <sup>T</sup> <sup>=</sup> <sup>1</sup>

Figure 16. The 30 shift for the delta current and the input current for phase a—input frequency 360 Hz.

Figure 17. DC voltage level for each converter and the overall DC voltage.

Figure 18. Waveforms results for 600 Hz input frequency.

Figure 19. Waveforms results for 800 Hz input frequency.
