For the three-phase PWM inverter, there are three commonly used coefficients to indicate the DC voltage utilization level, i.e.

## Modulation Level (调制度)

`M = \cfrac{V_R}{V_T}`

where `V_R`

is the magnitude (note, not fundamental amplitude) of modulation wave, `V_T`

is the magnitude of carrier wave.

When `M \leq 1`

, there is no distortion in the output. While `M > 1`

means the output is distorted from the modulation wave.

## DC Voltage Utilization Ratio (直流电压利用率)

`M_u = \cfrac{V_{\rm 1max}} {V_{\rm dc}}`

where `V_{\rm dc}`

is the DC voltage. For single phase output application, `V_{\rm 1max}`

is the maximum amplitude of phase fundamental voltage. For three phase output application, `V_{\rm 1max}`

is the maximum amplitude of line-line fundamental voltage.

For single phase SPWM, the maximum `M`

before distortion is `M=1`

. At `M=1`

, the amplitude of fundamental voltage is `V_{\rm dc}/2`

, thus the DC voltage utilization ratio `M_{u \rm\ 1-Phase\ SPWM}=1/2`

.

For three phase SPWM, the maximum `M`

before distortion is `M=1`

. At `M=1`

, the amplitude of fundamental voltage is `\sqrt{3}V_{\rm dc}/2`

, thus the DC voltage utilization ratio `M_{u \rm\ 3-Phase\ SPWM}=\sqrt{3}/2`

.

In order to Increase the `M_u`

of three-phase Y-connected motor, the third-order harmonic can be added to the SPWM, named as THIPWM. When the amplitude of third-order harmonic is `1/6`

of the SPWM, the method is named as THIPWM1/6, as shown below.

The modulation wave magnitude of THIPWM1/6 is `\sqrt{3}/2`

of the SPWM. Thus, when `M=1`

, the `M_u`

of THIPWM1/6 is `2/\sqrt{3}`

of SPWM. Consequently, the DC voltage utilization ratio of THIPWM1/6 `M_{u\rm\ 3-Phase\ THIPWM1/6} = 1`

.

Also, for SVPWM the DC voltage utilization ratio is the same as THIPWM1/6, thus `M_{u\rm\ 3-Phase\ SVPWM} = 1`

## Modulation Index (调制系数) or Voltage Utilization Level

`M_i = \cfrac{V_{1\rm max}}{V_{1\rm max\ 6-step}}`

where `{V_{1\rm max\ 6-step}}`

is the amplitude of the fundamental voltage (the green line below) of square form modulation wave (the blue line below) with `M=1`

.

The fundamental wave of the square wave has the formula `4\sin{\theta}/\pi`

, thus

`V_{\rm 1max\ 6-step}=\cfrac{4}{\pi}\cfrac{V_{\rm dc}}{2}=\cfrac{2}{\pi}V_{\rm dc}`

.

Then,

`M_i = \cfrac{\pi}{2}\cfrac{V_{\rm 1max}}{V_{\rm dc}}`

or,

`V_{\rm 1max}=\cfrac{2V_{\rm dc}}{\pi}M_i`

Consequently, for three-phase SPWM with `M=1`

, the `M_i`

is

`M_{i\ \rm 3-Phase\ SPWM}=\cfrac{\pi}{2}\cfrac{1}{V_{\rm dc}}\cdot \cfrac{V_{\rm dc}}{2}=\cfrac{\pi}{4}\approx0.785`

For three-phase THIPWM1/6 and SVPWM with `M=1`

, the `M_i`

is

`M_{i\ \rm 3-Phase\ THIPWM1/6}=M_{i\ \rm 3-Phase\ SVPWM}=\cfrac{\pi}{4}\cfrac{2}{\sqrt{3}}=\cfrac{\pi}{2\sqrt{3}}\approx0.907`

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