 # Voltage source Inverter

## Square-wave Voltage source Inverter

Voltage source Inverter type Square-wave produces a square-wave-AC voltage output. We never attempt to filter this voltage, only those loads compatible with square-wave voltage can use it.

### Single-phase square-wave Voltage source Inverter

Circuit of single-phase square-wave VSI

𝑄1  to 𝑄4 are IGBTs (Power semiconductor switches)

𝐷1  to 𝐷4 are Fast recovery power diodes

𝑆1  to 𝑆4 are  Binary ON/OFF switching signal inputs

𝐶𝑑  is DC storage capacitor

𝑉𝑑  is input DC voltage

𝐼𝑑  is input current

𝑉 𝑜  is output voltage

Assuming ideal devices: An IGBT acts as short-circuit when its switching signal is HIGH, or open-circuit when switching signal is LOW.

If we need Vo of frequency f, then we should set switching cycle time (Ts)  to 1/f.

Operating wave forms , with inductive load

𝑉 𝑜 =     𝑉𝑑, 𝑖𝑓 𝑄1,𝑄2𝑂𝑁

−𝑉𝑑, 𝑖𝑓 𝑄3,𝑄4𝑂𝑁

In each switch, 𝑄 & 𝐷 are non-separable parts. After 𝑄 is switched on, either 𝑄 or 𝐷 conducts depending on the direction of current.   Indicated on current waveform are the devices conducting over different intervals of time.

𝐼𝑑 =        𝐼𝑜, 𝑖𝑓 𝑄1,𝑄2𝑂𝑁

−𝐼𝑜, 𝑖𝑓 𝑄3,𝑄4𝑂𝑁

During each cycle, inductive load returns some current back to the input.  Capacitor Cd absorbs this current.

In a single-phase square-wave VSI:

• To control output frequency, we should alter the switching cycle time.

• To control output voltage, we should alter the input voltage.

### Example for VSI

A single-phase square-wave VSI is operating on 260V DC input, and delivering 50 Hz square-wave output voltage in the steady state.  Load comprises an inductance 100 mH in series with 50 Ω. Find,

1) RMS value of the fundamental component of output voltage.

2) RMS values of two lowest order harmonics in output voltage.

3) Peak value of the output current.

4) Mean value of input DC voltage

Ans

𝑉𝑑 = 260 V

𝑓 = 50 Hz

𝑅 = 50 Ω

𝐿 = 100 mH

## Three-phase square-wave Voltage source inverter

This inverter is also known as “Six-step inverter”.

𝑄1  to 𝑄6 are IGBTs (Power semiconductor switches)

𝐷1  to 𝐷6 are Fast recovery power diodes

𝑆1  to 𝑆6 are  Binary ON/OFF switching signal inputs

𝐶𝑑  is DC storage capacitor

𝑉𝑑  is input DC voltage

𝐼𝑑  is input current

𝑉𝐴𝐵,𝑉𝐵𝐶,𝑉𝐶𝐴   are output line-voltages

𝑉𝐴𝑁,𝑉𝐵𝑁,𝑉𝐶𝑁   are output phase-voltages

𝑉𝐴𝑂,𝑉𝐵𝑂,𝑉𝐶𝑂   are inverter pole-voltages

𝐼𝐴,𝐼𝐵,𝐼𝐶   are output line-currents

Individual switching signals 𝑆1  to 𝑆6 are derived from 3 main switching signals 𝑆𝑎,𝑆𝑏 & 𝑆𝑐 (for three-legs of inverter) generated by inverter control, according the desired output.  If we need output voltage of frequency f, the switching cycle time (Ts) will be 1/f.

Main switching signals for square wave control of 3-phase VSI (6 intervals have equal time width)

Phase voltage wave forms

Line voltage wave forms

## Equation Derivation of Voltage source Inverter Square-wave

Taking gross rms values,

We can model line-voltages and phase-voltages mathematically, in terms of switching signals Sa, Sb & Sc.

𝑉𝐴𝑂 ≡ 𝑆𝑎𝑉𝑑

𝑉𝐵𝑂 ≡ 𝑆𝑏𝑉𝑑

𝑉𝐶𝑂 ≡ 𝑆𝑐𝑉𝑑

For a balanced three-phase system,  (𝑉𝐴𝑁 +𝑉𝐵𝑁 +𝑉𝐶𝑁) = 0

𝑉𝐴𝑁 ≡ 𝑉𝐴𝑂 −𝑉𝑁𝑂

𝑉𝐵𝑁 ≡ 𝑉𝐵𝑂 −𝑉𝑁𝑂

𝑉𝐶𝑁 ≡ 𝑉𝐶𝑂 −𝑉𝑁𝑂

∴ 𝑉𝐴𝑁 +𝑉𝐵𝑁 +𝑉𝐶𝑁 = 𝑉𝐴𝑂 +𝑉𝐵𝑂 +𝑉𝐶𝑂 −3𝑉𝑁𝑂 ⇒ 𝑉𝑁𝑂 =1 /3( 𝑉𝐴𝑂 +𝑉𝐵𝑂 +𝑉𝐶𝑂)

Substituting for 𝑉𝑁𝑂 in 𝑉𝐴𝑁, 𝑉𝐵𝑁 and 𝑉𝐶𝑁,

Line currents 𝐼𝐴,𝐼𝐵 & 𝐼𝐶 at inverter output, for an inductive load, can be drawn with  respect to phase-voltages (star connected load),  in terms of step responses.

x, y & z are magnitudes of instantaneous currents at the beginning of intervals, as shown.

Sum of currents  in 3 phases at any instant is zero.  Taking instant t = 0,

(-x) + (-y) + z =0

∴ z = ( x + y )

Load inductance  has changed output currents more towards sinusoidal shape than voltages.

Input current Id is the sum of currents drawn from the positive rail by three legs.  Mathematically, this is expressed as,

𝐼𝑑 ≡ 𝑆𝑎𝐼𝐴 +𝑆𝑏𝐼𝐵 +𝑆𝑐𝐼𝐶 , where 𝑆𝑎, 𝑆𝑏 & 𝑆𝑐 each has value 1 or 0.

Eg:  In interval-2, 𝑆𝑎= 1, 𝑆𝑏= 0  &  𝑆𝑐= 0

∴ 𝐼𝑑 ≡ 1×𝐼𝐴 +0×𝐼𝐵 + 0×𝐼𝐶 = 𝐼𝐴

In the waveform of 𝐼𝑑, we observe  𝐼𝑑 ≡ 𝐼𝐴 in interval-2.

Working through each interval, we see that 𝐼𝑑 is repetitive. Ripple frequency is 6 times fundamental frequency.  Capacitor Cd helps supplying this ripple current

Devices participating in carrying current IA in phase-A during a cycle

## In a three-phase square-wave Voltage source Inverter Square-wave

• To control output frequency, we should alter the switching cycle time.

• To control output voltage, we should alter the input voltage applied to the inverter.

## Example for Voltage source Inverter Square-wave

Ex: Three-phase , IGBT based, square-wave (six-step) VSI is operating on 520 V DC input.  It is operated at a switching frequency 50 Hz.  Load is star-connected and comprises 100 mH in series with 50 Ωin each phase. Determine,

1) Total rmsvalues of phase-voltage and line-voltage at output.

2) Fundamental rmsvalues of phase-voltage and line-voltage at output.

3) RMS values of lowest three harmonics in line-voltage at output.

4) RMS value of the  fundamental component of line-current at output.

5) Ripple frequency of input current.

6) Voltage blocked by each IGBT or diode.

7) Peak current in each IGBT or a diode.

𝑉𝑑 = 520 V

𝑅 = 50 Ω

𝐿 = 100 mH = 0.1 H

𝑓 = 50 Hz

Voltage Source Inverter operation are discussed here with sketches and examples. In first stated before and this PWM is special also because it does not suppress lower-order harmonics in output voltage, which is contrary to the general expectations from a PWM inverter.

## Voltage Source Inverter Square wave PWM

### Single phase Square wave PWM VSI

Single-phase Square-wave PWM VSI has two implementation options.

1. Bipolar PWM implementation

2. Unipolar PWM implementation

### Bipolar Square wave PWM Implementation

In bipolar PWM, voltage pulses of Vo fluctuate between +Vd and –Vd.

#### Square wave Bipolar PWM modulator

Modulator is the circuit that generates PWM switching signals. It has two inputs, carrier-signal input (𝑣𝑐) and reference-signal input (𝑣𝑟).

Carrier-signal is a symmetrical triangular signal of amplitude 𝐸𝑐 and frequency 𝑓 𝑐, both of which are fixed for a given implementation. Reference-signal is a square-wave signal of amplitude 𝐸𝑟 and frequency 𝑓 𝑟, both of which are variable.

• 𝑓 𝑐 will be switching frequency.
• 𝑓 𝑟 will be fundamental frequency of output voltage.
• 𝐸𝑟 will determine fundamental amplitude of output voltage.
• 𝐸𝑟 is adjustable between 0 and 𝐸𝑐 only.

𝑚 = 𝐸𝑟/𝐸𝑐 = Depth of modulation

𝑝 = 𝑓𝑐 /𝑓 𝑟 = Carrier ratio

0 ≤ 𝑚 ≤ 1 P is an integer of a larger value and example200.

Output voltage waveform for an example case of p = 9 and m ≈ 0.75. (Rising edge of reference signal is synchronized with positive peak of carrier, by choice)

Bipolar PWM produces p no. of voltage pulses per cycle of Vo and each pulse goes between +Vd and –Vd full span.

For higher values of 𝑝, it can be shown that 𝑉𝑜, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠is virtually independent from 𝑝 then Harmonics in 𝑉 𝑜 contains lower-order harmonics associated with the reference square-wave plus higher order harmonics due to the carrier.

Thus the ,

Magnitude of Vo is controlled by varying depth of modulation. (m)

Frequency of Vo is controlled by varying reference frequency (fr).

#### Uni polar Square wave PWM Implementation

In unipolar PWM and voltage pulses of Vo fluctuate between +Vd and 0 during positive half-cycle and between –Vd and 0 during negative half-cycle.

#### Square wave Uni polar PWM modulator

Unipolar PWM switches two inverter legs A and B by 𝑆𝑎 and 𝑆𝑏, generated by the modulator.  𝑆𝑎is generated by comparing 𝑣𝑟with 𝑣𝑐, and 𝑆𝑏by comparing -𝑣𝑟with 𝑣𝑐. If 𝑆𝑎is HIGH, then the upper-switch 𝑄1will be ON and otherwise the lower-switch 𝑄4 will be ON. After Similar action happens with 𝑆𝑏.

In unipolar PWM too, amplitude 𝐸𝑐 and frequency 𝑓 𝑐 of carrier signal fixed. Amplitude 𝐸𝑟 and frequency 𝑓 𝑟 of reference signal are variable. Here too,

• 𝑓 𝑐 will be switching frequency.
• 𝑓 𝑟 will be fundamental frequency of output voltage.
• 𝐸𝑟 will determine fundamental amplitude of output voltage.
• 𝐸𝑟 is adjustable between 0 and 𝐸𝑐 only.

𝑚 = 𝐸𝑟/𝐸𝑐 = Depth of modulation

𝑝 = 𝑓𝑐 /𝑓 𝑟 = Carrier ratio

0 ≤ 𝑚 ≤ 1 P is an integer of a larger value, and example 200.

for an example case of p = 9 and m ≈ 0.75.

For higher values of 𝑝, 𝑉𝑜, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠is virtually independent from 𝑝. 𝑉 𝑜 contains lower-order harmonics due to the square-wave reference and higher order harmonics due to the carrier signal and  Order of carrier related harmonics are twice as much as those of bipolar PWM, which is an advantage but not effective as the lower order harmonics still dominate.

𝑉𝑜, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠 ≈ 𝑚( (8)^(1/2)𝑉𝑑/ 𝜋 )

Order of Harmonics in 𝑉𝑜 = 3,5,7,9, 11……

Observation on Vo waveform are:

• No. of pulses per half-cycle is p, and that of full cycle is 2p.
• All pulses have same width.
• Pulses distribute symmetrically over the half cycle.
• Pulse-cycle-time is T/2p , and its ON-state duty-factor is  m.
• Adjacent pulses between half-cycles separates by an OFF-state.

#### Example – Voltage Source Inverter operation

Sketch Vo waveform for a single-phase, Square-wave PWM inverter operating on unipolar control mode, with carrier ratio 6 and depth of modulation 0.75, delivering 50 Hz output.  Take DC input voltage as Vd.

## Three phase Square wave PWM Voltage source Inverter

Modulator for three-phase, square-wave PWM VSI (Reference signals are balanced three-phase, square-wave signals)

Amplitude 𝐸𝑐 and frequency 𝑓 𝑐 of carrier signal are fixed. Amplitude 𝐸𝑟 and frequency 𝑓 𝑟 of reference signal are variable.

• 𝑓 𝑐 will be switching frequency.
• 𝑓 𝑟 will be fundamental frequency of output voltage.
• 𝐸𝑟 will determine fundamental amplitude of output voltage.
• 𝐸𝑟 is adjustable between 0 and 𝐸𝑐 only.

𝑚 = 𝐸𝑟/𝐸𝑐 = Depth of modulation

𝑝 = 𝑓𝑐 /𝑓 𝑟 = Carrier ratio

0 ≤ 𝑚 ≤ 1 P should be an ODD integer of a higher value. It should be a multiple of 3 to enable synchronizing between carrier and three phase reference signals.

### line voltage waveform has Characterstics

• Symmetric half-cycles
• 2𝑝 3 pulses in each half-cycle, packed over 120° span
• Pulse cycle-angle of 120° 2𝑝 3, which is 180° 𝑝
• ON-state duty-factor of m for pulse-cycles
• Gap between half-cycles of (60°+ OFF angle)

For higher values of 𝑝, 𝑉𝐿𝑖𝑛𝑒, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠is virtually independent from 𝑝.  Lower-order harmonics associated with the reference three-phase square-waves continue to appear in 𝑉𝐿𝑖𝑛𝑒. High frequency switching harmonics, too, are present due to the carrier.  It can be shown,

𝑉𝐿𝑖𝑛𝑒, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠 ≈ 𝑚 ( 𝑉𝑑/ 𝜋)

Order of Harmonics in 𝑉𝐿𝑖𝑛𝑒 = 5,7,11, 13……

### Example for Three phase Square wave PWM Voltage source Inverter

Sketch line voltage waveform for three-phase, square-wave PWM VSI, for carrier-ratio 12 and depth of modulation 0.73.  Take input DC voltage as Vd.  Use ωt for horizontal axis, where ω is the fundamental frequency of output voltage.

Phase-voltage (𝑉𝐴𝑁) follows the six-step-profile of square-wave inverter but with pulses.  The number of pulses per half-cycle is p, and all pulses have equal cycle-length and an ON-state duty-factor of m. It can be shown that,

𝑉𝑃ℎ𝑎𝑠𝑒, 𝐹𝑢𝑛𝑑 𝑟𝑚𝑠 ≈ 𝑚( (8)^1/2𝑉𝑑/𝜋)

Order of Harmonics in 𝑉𝑃ℎ𝑎𝑠𝑒 = 5,7,11, 13……

Phase voltage waveform for an example case of p = 9 and m = 0.75

Software implementation of three-phase square-wave PWM can be done indirectly, by modifying mid one-third portions of each half-cycle of square wave switching signals 𝑆𝑎, 𝑆𝑏, 𝑆𝑐, according to the values of 𝑚, 𝑝 𝑎𝑛𝑑 𝑇.

Introduce 𝑝 3   number of pulses of duty-factor m and cycle-time 𝑇 2𝑝   to the mid one-third portions of each half-cycle of switching signals.

### Illustration of criteria for the case

criteria for the case of p = 9 and m = 0.75

If the mid-portion is originally HIGH, the modification will begin with a HIGH pulse of duration 𝑚(𝑇/2𝑝) followed by a LOW pulse of duration (1−𝑚) (𝑇/2𝑝), repeating for (𝑝/3) cycles.

If the mid-portion is originally LOW, the modification will begin with a LOW pulse of duration 𝑚(𝑇/2𝑝) followed by a HIGH pulse of duration (1−𝑚) (𝑇/2𝑝), repeating for (𝑝/3) cycles.

Line voltage 𝑉𝐴𝐵 produced by modified switching signals 𝑆𝑎, 𝑆𝑏, 𝑆𝑐, is same as that produced by the normal modulator.  From real time processing point of view, only one switching signal is to be modified at a time, which is an ease.

For 𝑝 = 9 and 𝑚 = 0.75, assuming 50 Hz fundamental frequency:

Number of pulse-cycles for mid one-third portion = 𝑝/3 = 3

Fundamental cycle-time = 𝑇 = 1000/50 ms = 20 ms

Pulse cycle-time = 𝑇/2𝑝 = 20 2×9 ms = 1.11 ms

Active pulse time = 𝑚 (𝑇/2𝑝) = 0.75×1.11 ms = 0.83 ms

Opposite pulse time =   1−m𝑇 2𝑝 = 1−0.75 ×1.11 ms = 0.28 ms

## Voltage and Frequency control

• To control output frequency we should alter reference signal frequency  𝑓 𝑟 .
• To control output voltage we should alter depth of modulation  𝑚 .

Constant  (𝑉/𝑓) control & Gear changing

Some applications (eg: AC motor drives) need inverter deliver constant  (𝑉/𝑓)  type output, where the ratio between   𝑉𝐿𝑖𝑛𝑒,𝐹𝑢𝑛𝑑 𝑟𝑚𝑠and  𝑓 𝑟 is constant.  This requires the ratio  (𝑚/𝑓 𝑟 ) be held constant.

We can show that, when (𝑚/𝑓 𝑟 ) is constant, the width of line-voltage-pulses is constant irrespective of 𝑓 𝑟, for a given p.

Width of line voltage pulses = 𝑚(𝑇/2𝑝) = 𝑚 (1 /𝑓𝑟 2𝑝) = 1 /2𝑝 (𝑚 /𝑓𝑟)

Illustration of the effect’s online voltage waveform at low frequency in V/f control with fixed p

Line voltage waveform is poor and unacceptable at low frequency, as the load tends to respond to individual pulses, causing significant distortions in its response.

To address this issue and make the output voltage acceptable, we increase carrier ratio p when fr is bring lower.  This acts to fill the occupying-span of the half-cycle with far more greater number of pulses of lesser width.  Increasing of carrier ratio can be do in steps, according to a chosen plan.

Raising of carrier ratio p in steps when the reference frequency fr is lower, in order to improve the output voltage of an inverter on constant V/f control, is call Gear-changing.

An example Gear-changing plan. (Values for information only)

## Over modulation with Square wave PWM – Voltage source Inverter

Over-modulation means taking m beyond 1. Over modulation transfers inverter operation from square wave PWM to ordinary square-wave control.

### Case 1: Single-phase bipolar square-wave PWM

Vo waveform with over-modulation of single-phase bipolar PWM

Ideal over-modulation makes a smooth transition from square-wave PWM to square-wave VSI but, practically, there is a step jump in the output voltage owing to the “dead-band” adopt in switching times.

Dead band is the minimum time  that an IGBT is set to stay in the OFF state.  This is require to avoid two IGBTs in the same leg getting short-circuit during switchovers.  This is implement by introducing a short delay-time Δ t B (call blanking time) for each switch-ON but not for switch-OFF.  Thus, once an IGBT is switch-off, it must stay there for Δ t B on the minimum before responding to next switch-ON.  Δ t B is choose conservatively, typically few microseconds for IGBTs .

ideal switching case

Real switching case with  Δ t B

After Er exceeds the threshold corresponding to OFF-state time of Δ t B , the output voltage will not change until E r exceed E c at which point the output voltage rises up stepwise to the level of square wave VSI output.

Also Size of this voltage step depends on Δt B , f c and V d . If any of these 3 parameters goes up the size of voltage step will go up. While some applications may tolerate this kind of voltage step some others may not.

Δ V o is the step jump of rms fundamental output voltage

## Case 2: Single-phase Unipolar square-wave PWM

With unipolar PWM too, over-modulation can be apply as a means of transitioning from square-wave PWM to just square-wave VSI.  Here too, as in bipolar PWM, transition occurs with s step jump of output voltage, due to the blanking time adopted in switching as a measure of safety.

This Δ V o jump of the rms fundamental output voltage is same as that in the bipolar PWM.

In addition Vd is DC input voltage, fc is carrier frequency and Δ t B is blanking time adopt in the inverter.

## Case 3: Three-phase square-wave PWM – Voltage source Inverter

With three-phase square-wave PWM too, over-modulation can be apply as a means of transitioning from square-wave PWM to just square-wave VSI.  Here too, as in single-phase PWM, transition occurs with a step jump of output voltage, due to the blanking time adopted in switching.

Step jump Δ V o of the rms fundamental output is find as,