Introduction

Almost every motor drive inverter in the market rely on Pulse-Width Modulation (PWM) to synthesize the output current waveform. Because of that, the actual instantaneous voltage waveform at the inverter terminals will consist of very steep flanks at the beginning and end of each pulse. This is the origin for a multitude of negative effects on the connected equipment.

This post is part one of four:

  1. Reflections causing voltage doubling at the equipment terminals, causing increased stress on winding insulation
  2. Bearing currents inside electrical machines due to capacitive currents and non-uniform voltage and current distribution in windings
  3. Capacitive ground currents
  4. High frequency noise

Voltage reflections

Background

When a PWM inverter is running, it will create a series of voltage pulses with the same amplitude, but different length and separation at the output terminals. Every time a new pulse is created, the rising edge of that voltage pulse will propagate through the cable towards the far end, i.e. to the connected equipment.

Depending on the steepness of this pulse together with the impedance difference between the cable and load, the voltage seen at the cable end might be twice the voltage sent from the inverter. This is caused by the reflective wave phenomenon which is very common in fast switching applications and is extensively investigated.

Its effect on equipment and motor windings in particular was discovered in the early 1990s, when inverter manufacturers seized using voltage sine wave filters at the inverter output because the current was made sinusoidal by the inductance in the machine windings alone. As a result of this, stator windings started to fail prematurely and efforts were put to discover why.

voltage reflection theory

If the time required by the voltage pulse to reach the motor is more than 1βˆ•3 of the voltage rise time π‘‘π‘Ÿ, the pulse will reflect back to the source (e.g. the inverter output terminals). This reflection is maximized when the speed of the propagating pulse is π‘βˆ•2, i.e. half the speed of light.

The propagation speed is dependent on the network of parasitic elements which the cable is built up by: an infinite amount of πœ‹-equivalents in series i.e. inductors in series with a capacitive coupling to earth between each of them. The voltage pulse will charge these capacitances in turn before reaching the connected equipment. Thus the propagation velocity 𝑣 is given by

\begin{equation}
\label{eq:wavespeed}
    v = \frac{1}{\sqrt{L_c C_c}}
\end{equation}

where 𝐿𝑐 and 𝐢𝑐 are the cable inductance and stray capacitance per unit length.

A so-called line equivalent showing a cable or conductor's parasitic elements per unit length, \( \partial x \): DC resistance \( \partial R\), inductance \( \partial L \), capacitance to ground \( \partial C \) and conductivity to ground \( \partial G \)

At very high frequencies, the motor inductance will appear as an infinite impedance/open cir- cuit resulting in reflection of the voltage pulse. Subsequently, the voltage at the motor terminals will see the cable as its only return path to the source. Thus the returning voltage pulse of 1 p.u. will add to the already existing 1 p.u. yielding a total terminal voltage of 2 p.u.
As mentioned, this will only occur if the motor- and the cable impedances are unequal. This impedance mismatch is almost always present in motor drives as the motor impedance tends to be several times larger than the cable impedance.
The resulting terminal voltage caused by the reflection phenomenon is given by: 

\begin{equation}
\label{eq:MotorVoltage}
V_{m} = V_{inv} \cdot (1+\Gamma_L)
\end{equation}

where \( V_m \) is the motor terminal voltage, \( V_{inv} \) is the DC-link voltage at the inverter and \( \Gamma_L \) is the load reflection coefficient which is based upon the relationship between the motor impedance \( Z_{m} \) and the cable impedance \( Z_{c} \). The characteristic equation for a lossy transmission line can be used to calculate the cable impedance:

\begin{equation}
\label{eq:cabel_impedance}
Z_c = \sqrt{\frac{j \omega L_c}{j \omega C_c}}
\end{equation}

Further, the load reflection coefficient is given as: 

\begin{equation}
\label{eq:Gamma_L}
\Gamma_L = \frac{Z_m-Z_c}{Z_m+Z_c} \approx 1
\end{equation}

In almost every installation \( Z_m >> Z_c \) due to the high inductance of the motor winding compared to the cable impedance and thus \( \Gamma_L \) is often approximated to  1. Inserting \eqref{eq:Gamma_L} into \eqref{eq:MotorVoltage} yields a motor terminal voltage which is twice the inverter DC-link voltage:

\begin{equation}
V_{m} = 2V_{inv}
\end{equation}

Figure 1: Lattice diagram explaining voltage reflection theory

However, this will only give the overvoltage caused by the first reflection. If the pulse propagates back to the inverter and reflects back to the motor a second time, an additional pulse will add to the already existing over voltage. This can occur multiple times in a row, depending on the parameters of the source, cable and load. 

As the first overvoltage will have discharged slightly before the second pulse arrives, the resulting terminal voltage will be the sum of several decaying terms from each of the reflections. This is illustrated in figure 1. 

\begin{equation}
\begin{aligned}
V_m = & \underbrace{V_{inv}(t) \cdot e^{-\tau_t t} + V_{inv}(t) \Gamma_L \cdot e^{-\tau_t \cdot t}}_\text{First reflection} 
+ \underbrace{V_{inv}(t) \cdot \Gamma_L \Gamma_S \cdot e^{-3\tau_t t} + V_{inv}(t) \cdot \Gamma_L^2 \Gamma_S \cdot e^{-3\tau_t \cdot t}}_\text{Second reflection} \\
& + \underbrace{V_{inv}(t) \cdot \Gamma_L^2 \Gamma_S^2 \cdot e^{-5\tau_t t} + V_{inv}(t) \cdot \Gamma_L^3 \Gamma_S^2 \cdot e^{-5\tau_t \cdot t}}_\text{Third reflection} \\
& + \underbrace{V_{inv}(t) \cdot \Gamma_L^3 \Gamma_S^3 \cdot e^{-7\tau_t t} + V_{inv}(t) \cdot \Gamma_L^4 \Gamma_S^3 \cdot e^{-7\tau_t \cdot t}}_\text{Forth reflection}
+ \dots
\end{aligned}
\label{eq:voltage_reflections}
\end{equation}

where \( \tau_t \) equals the propagation time given by \( \ell_c \cdot \sqrt{L_{c} C_{c}} \) and \( L_C \) and \( C_C \) is the per-unit length cable inductance and capacitance respectively and \( \ell_c \) is the cable length. 
\( \Gamma_S \) equals the source reflection coefficient:

\begin{equation}
\Gamma_S = \frac{Z_s-Z_c}{Z_s+Z_c} \approx 1
\end{equation}

As in \eqref{eq:Gamma_L}, also the source impedance is generally much larger than the cable impedance, hence the source reflection coefficients \( \Gamma_S \) can be approximated to 1. That will yield the following simplification: 

\begin{equation}
\begin{aligned}
V_m = & V_{inv}(t) \cdot \left[2e^{-\tau_t \cdot t}  + 2e^{-3\tau_t \cdot t} + 2 e^{-5\tau_t \cdot t} + 2 e^{-7\tau_t \cdot t} + \dots\right]...\\
= &  V_{inv}(t) \cdot  \sum_{i=0}^{n} 2e^{-(2i+1)\tau_t \cdot t}  
\end{aligned}
\end{equation}

where \( n \) is the number of decaying reflections desired for inclusion in the final answer.

The propagation time for the traveling wave from the source to the load is given as:

    \begin{equation}
    \label{eq:prop_time}
    \tau_t = \frac{\ell_c}{v}  = \ell_c \cdot \sqrt{L_c C_c}
    \end{equation}

where \( \tau_t \) is the traveling wave propagation time and \( v \) is given by \eqref{eq:wavespeed}. 

After time \( \tau_t \), the wave will travel back to the inverter. The amplitude of this backward traveling wave, \( V_t \), is:

\begin{equation}
        \label{eq:prop_back_amplitude}
        V_t(\tau_t) = \frac{\tau_t\cdot  V_{inv}\cdot  \Gamma_L}{t_r}  \quad\quad\text{for  } \left(\tau_t < t_r\right)
        \end{equation}

and

 \begin{equation}
        \label{eq:prop_back_amplitude2}
        V_t(\tau_t) =  V_{inv} \cdot  \Gamma_L  \quad\quad\text{for  } \left(\tau_t \geq t_r\right)
        \end{equation}

In \eqref{eq:prop_back_amplitude} and \eqref{eq:prop_back_amplitude2} it is seen that when the travel time \( \tau_t \) is larger than the switching device's rise time \( t_r \), both the travel time and rise time is excluded from the equation. 

The peak line to line motor terminal voltage, \( \hat{V}_{LL} \) can be expressed as:

\begin{equation}
        \label{peak_term_volt}
        \hat{V}_{LL} = 3\cdot \frac{\ell_c \cdot V_{inv} \cdot \Gamma_L} {v \cdot t_r}  + V_{inv}    \quad\quad\text{for  } \left(\tau_t < \frac{t_r}{3}\right)
        \end{equation}

 and

\begin{equation}
        \label{peak_term_volt2}
            \hat{V}_{LL} =  V_{inv} \cdot \Gamma_L + V_{inv}  \quad\quad\text{for  } \left(\tau_t \geq \frac{t_r}{3}\right)
        \end{equation}

showing that if the cable length is such that propagation time is longer than  1/3  of the voltage rise time, voltage doubling occurs at the motor terminal, given that \( \Gamma_L \approx 1 \).

\eqref{peak_term_volt2} can be rewritten to express the line to line voltage in per unit by dividing by the inverter DC-link voltage, \( V_{inv} \). This can be used to express the per unit motor terminal voltage in terms of \( t_r \):

\begin{equation}
    \label{eq:peak_term_volt_normalized}
    \frac{\hat{V}_{LL}}{V_{inv}} = 3\cdot \frac{\ell_c \cdot \Gamma_L} {v \cdot t_r}  + 1
    \end{equation}

From \eqref{eq:peak_term_volt_normalized} it is apparent that the first term alone defines the actual overvoltage and that this should be kept as small as possible.

In a given system, $ t_r $ is based by the inverter switching device and cannot be adjusted to any significant degree without adding components to the system. New \ac{SiC} devices with shorter rise time $ t_r $ will according to \eqref{eq:peak_term_volt_normalized} increase the motor terminal overvoltage if the cable length and load reflection coefficient is unchanged.

The load reflection coefficient \( \Gamma_L \) can be lowered by introducing a filter with a resistor on the inverter output terminals which increases the cable impedance to match the motor impedance. This filter will also have a positive impact on the voltage rise time of dimensioned properly.

Both will be discussed in a later blog post.

The cable length as function of rise time and load reflection coefficient is given by:

\begin{equation}
\label{eq:ell_c}
\ell_{c} = V_{max} \cdot \frac{ v \cdot t_r}{3 \cdot \Gamma_L}
\end{equation}

where \( V_{max} \) is the maximum allowable per-unit voltage overshoot at the motor terminals, e.g. 0.2 for 20 %

Consequences

This voltage doubling occurs at the rising edge of every pulse from the inverter. The frequency of these overvoltages are dictated by the inverter's switching frequency and modulation scheme. It can vary from a few hundred Hz to several tens of  kHz. This depends on the application, switching device parameters and the manufacturer's design. For induction motor applications, the frequency range is typically a few kHz.

These transients put a significant amount of stress on the motor windings closest to the terminal. Depending on the rise time, up to 85% of the overvoltage can be dropped across the first turn of the first winding. This can lead to premature winding insulation failure and subsequent earth fault in the machine. 

Remedies

Because this phenomena has been known for a long time, motor manufacturers has taken measures by applying additional insulation on the most affected windings. One important measure is to increase the insulation system's withstand voltage.

Industry organizations such as NEMA and IEC has also issued recommendations and design guides for drive- and motor manufactures. Two important ones are recommendations for minimum voltage rise times (NEMA ICS 61800-2-2005) and drive design guidelines (IEC 60034-18-42).

For applications where additional insulation is impractical, a proven solution is to add an output filter to delay the rise time and \( dV/dt \). Such filter design will be discussed in a different blog post.

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