As mentioned before, in some situations the voltage overshoot at the machine terminals can exceed the theoretical maximum value of twice the DC bus voltage. Some modes in a PWM modulation cycle can lead to these increased voltage stress levels when long cable lengths are used. If the inverter dwell time (the time the line-to-line voltage is zero) is shorter than the time it takes for the last cable transient to decay (T ), a residual charge is trapped in the cable and may lead to increased overvoltages. This phenomenon is known as double pulsing. Due to the spacing of inverter PWM pulses, the carrier switching frequency and modulation technique have a predominant effect on voltage overshoot in this mode. The rise time of the inverter output PWM pulses has a lesser effect on how often the double pulsing phenomenon occurs [18].
The amount of residual charge trapped in the cable depends on the cable AC damping resistance. The natural oscillation frequency of the cable has a large influence on this damping resistance. As was discussed in Chapter 2.1 and illustrated in Figure 2, during a voltage reflection cycle the voltage pulse travels the length of the cable four times, i.e.
Tcycle = 4 ·tt. The oscillation frequency is thereby given by
c
This means that short cable lengths lead to high oscillation frequencies. The oscillation frequency affects the skin [Kskin(fo)] and proximity (Kp) effects, which in turn increase the AC resistance of the conductor above the DC value. The series conductor AC resistance rs ( / unit length) and the cable parallel insulation resistance rp ( / unit length) cause power losses as dissipated heat during both the forward-travelling and the reflected wave pulses.
This attenuates the initial pulse amplitude u0 to its final value u depending on cable
whererDC is the cable DC resistance. Distancex is given by
p t. v
x= ⋅ (18)
The reflected waves dampen quickly, since distance lc is travelled four times during each oscillation cycle. Furthermore, smaller machines have greater damping than more powerful ones, since the DC resistance values in thers term of smaller gauge wires are higher.
Skin effect results from the inductance in a conductor being unevenly distributed. The inductance is highest in the centre of the conductor and least near the edges. This means that high frequency current does not penetrate the conductor deeply and only travels near the surface, which decreases the apparent conductor area and increases AC resistance. The skin effect factorKskin(fo) in (17) is a function of frequency [23].
Proximity effect, as the name suggests, is a result of two neighbouring conductors. The magnetic field of an adjacent neighbouring conductor distorts and reduces the current flow area in the primary conductor. In (17), the proximity effect increases AC resistance by a factor of two (Kp = 2) for tightly bundled round cables [24].
Skin depth is defined when the conductor current density in the radial dimension is -1. The total amount of high frequency power in the conductor can be described using the Poynting vector equation
where is the depth of conductor current density penetration. The radial depth after which only minimal high frequency power is carried inside the conductor is defined when
2
γ =δ . (20)
Skin depth is given by
δ =α1 , (21)
where is the attenuation coefficient. For a good conductor, skin depth can be defined as
where is conductivity and resistivity. With the condition set in (20), the expression for the skin effect becomes
2
wheredo is the wire diameter.
The expression for the AC resistance of a solid wire conductor which takes into account the skin and proximity effects can now be written as
2
Predicting thers value for wires with large diameter d0 is difficult, because the number and size of the conductor strands are critical in the calculation.
The voltage overshoot caused by reflected waves is damped out in a
α =3⋅τ
T (25)
time interval, where the time constant is given by
s
2 c
r
⋅L
τ = . (26)
The time needed for the reflected pulse to damp to less than five per cent of the initial peak value can be estimated by substituting (18), (8) and (6) into (17), which gives
ε τ
The skin effect AC resistance significantly affects the damping time of the reflected transients. With shorter cable lengths, the reflected transients are damped out faster because the oscillation frequency is higher.
In Figure 4, the inverter dwell time is short compared withT , which leads to a voltage overshoot of over two times the magnitude of DC bus voltage. Initially the cable is in a fully charged condition. When the cable is discharged for approximately 4 s, the voltage at the machine terminals is forced to approximately negative DC bus value.
When the third resulting voltage pulse reflection reaches the machine terminals, the pulse from the inverter arrives at the same time and both are reflected. This double pulsing event increases the voltage at the machine terminals to 1670 V.
1670 V
<Tα 0 1
0 2 500 V/div 500 V/div
T
Inverter
Motor
0 5 10 15 20 25 30 35 40 45 50
Time (µ sec)
Figure 4. Motor and inverter line-to-line voltages showing the effect of double pulsing. A 7,5 kW unloaded induction motor at 60 Hz, a third-harmonic PWM (TPWM) modulator with a 4 kHz carrier frequency and 152 metres of 3,31 mm2 cable were used. The DC bus voltage is about 650 V. Modified from [19].
The magnitude of the voltage overshoot resulting from double pulsing depends on the damping characteristics of the cable, DC bus voltage, inverter dwell time, modulation technique, duty cycle and carrier frequency. Double pulsing occurs more often if carrier frequency is increased.
To prevent the charge trapped in the cable from causing possible overvoltage transients of three times the magnitude of DC bus voltage with short PWM dwell times, or when the carrier frequency is increased to reduce allowable decay times, the reflected wave transients must decay before the next PWM pulse is sent to the cable. This is achieved if the inverterTon time is greater than 3 · .
The reflection coefficients of low power AC drives are approximately one, so the primary mechanism affecting transient overvoltage decay is skin effect AC resistance damping. Cables with larger diameters are used with high power drives and generators, so overvoltage transient decay as a result of skin effect resistance damping is minimal.
In this case, the damping of the reflected waves is a result of the reduced reflection coefficients. The voltage pulse is damped after each oscillation cycle and eventually decays [19].
The operating frequency after which the inverter dwell time is too short and overvoltages in excess of twice the magnitude of DC bus voltage start to appear is referred to as the double pulsing inception frequency. Together with the cable’s damping ratio, this inception frequency defines the minimum dwell time of the line to line voltage. If the condition in (25) is met, voltage at the machine terminals will decay before the arrival of the next PWM pulse. The double pulsing inception frequency is modulator dependant [19].
Modulators may be characterised by their respective double pulsing inception modulation indexMi ). This is achieved by relatingT to the modulation index, which eliminates the DC bus voltage and operating frequency as parameters. By describing double pulsing withMi( ), the induced overvoltages can be reduced utilising minimum time, pulse elimination and different modulator techniques.
All modulators – sine wave (SPWM), third harmonic injection (TPWM), space vector (SVPWM) and two phase discontinuous (TPPWM) – can be viewed as duty cycle comparisons. Figure 5 further illustrates this method. By denoting the modulating voltage withU* and the carrier period withTc, the on timeTon of an upper power device of one inverter pole can be calculated using equation
U .
2 DC
* c c on
T U
T =T + ⋅ (28)
Ton equalsTc / 2 (50 % duty cycle) whenU* equals zero [Figure 5 (a)]. IfU* = -UDC / 2, Ton = 0 for that carrier cycle [Figure 5 (b)]. The upper device is on for the complete carrier cycle only ifU* =UDC / 2 [Figure 5 (c)].
Tc
Ton(n)= 0
Ton(n-1) = 0,5 Ton(n+1)= 1
U(n)* U(n+1)*
U(n-1)*
0 1
½UDC
-½UDC 0
½UDC
-½UDC 0
(a) (b) (c)
Figure 5. Pulse time generation. The sawtooth line represents the trigger voltage, U* is the modulating voltage (sometimes also referred to as the control voltage) and Tc is the carrier period. Ton
changes state every time the modulating voltage and the trigger voltage intersect. Modified from [25].
The effects ofT can be investigated by selectingTon equal to Tc -T and solving the per unit modulation voltage from the double pulsing inception modulation index
2 T 2 1 ) 2
(
DC c
*
⋅
−
⋅ ≥
= T
U
Miα U . (29)
It can be seen from equation (29) that Mi( ) decreases as the carrier frequency increases. WhenT and cable damping characteristics are known, a minimum allowable dwell time can be calculated and Mi( ) determined as a function of carrier frequency.
The shape of the modulating signal also affects the Mi( ) characteristic of each modulator type [26]. Only one overvoltage region exists for the continuous modulators (SPWM, TPWM and SVPWM), whereas the discontinuous modulators – for example TPPWM – have multiple overvoltage regions.
Figure 6 on the next page shows the double pulsing inception modulation index as a function of carrier frequency for different modulator types. In the case depicted in Figure 6, there are two overvoltage regions for TPPWM. The operating regions where overvoltages are possible are separated by the Mi( ) curves so that the regions where Ton <T are below and the regions whereTon >T are above the curves.
0 2000 4000 6000 8000 10000 12000 14000 16000 0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
fc[Hz]
Mi(α)
TPPWM TPPWM
SPWM TPWM&SVPWM
Figure 6. Double pulsing inception index withT = 12 s. TheMi( ) curves separate the regions where Ton <T (above) from the regions whereTon >T (below). Modified from [25].