## 1. Introduction

Hysteresis inverters are used in many low and medium voltage utility applications when the inverter line current is required to track a sinusoidal reference within a specified error margin. Line harmonic generation from those inverters depends principally on the particular switching pattern applied to the valves. The switching pattern of hysteresis inverters is produced through line current feedback and it is not pre-determined unlike the case, for instance, of Sinusoidal Pulse-Width Modulation (SPWM) where the inverter switching function is independent of the instantaneous line current and the inverter harmonics can be obtained from the switching function harmonics.

This chapter derives closed-form analytical approximations of the harmonic output of single-phase half-bridge inverter employing fixed or variable band hysteresis current control. The chapter is organized as follows: the harmonic output of the fixed-band hysteresis current control is derived in Section 2, followed by similar derivations of the harmonic output of the variable-band hysteresis controller in Section 3. The developed models are validated in Section 4 through performing different simulations studies and comparing results obtained from the models to those computed from MATLAB/Simulink. The chapter is summarized and concluded in section 5.

## 2. Fixed-band hysteresis control

### 2.1. System description

Fig.1 shows a single-phase neutral-point inverter. For simplicity, we assume that the dc voltage supplied by the DG source is divided into two constant and balanced dc sources, as in the figure, each of value

In Fig.2, the fundamental frequency voltage at the inverter ac terminals when the line current equals the reference current is the reference voltage,

Referring to Fig.2, when valve

The bang-bang action delivered by the hysteresis-controlled inverter, therefore, drives the instantaneous line current to track the reference within the relay band

### 2.2. Error current mathematical description

The approach described in this section closely approximates the error current produced by the fixed-band hysteresis action, by a frequency-modulated triangular signal whose time-varying characteristics are computed from the system and controller parameters. Subsequently, the harmonic spectrum of the error current is derived by calculating the Fourier transform of the complex envelope of frequency modulated signal.

Results in the literature derived the instantaneous frequency of the triangular error current

and therefore:

where the average switching (carrier) frequency

and

Examining (Eq. 2), the instantaneous frequency _{.}

Now, with the help of Fig.3, we define the instantaneous duty cycle of the error current

Implicit into (Eq. 3) is the reference voltage

As the Fourier series of the triangular signal converges rapidly, the error current spectrum is approximated using the first term of the series in (Eq. 6). Therefore truncating (Eq. 6) to

where

determines the frequency bandwidth

that contains 98% of the spectral energy of the modulated sinusoid in (Eq. 7). To simplify (Eq. 7) further, we use the following convenient approximation (see Appendix-A for the derivation): Given that,

Therefore (Eq. 7) becomes,

Substituting

Next, the cosine term in (Eq. 12) is simplified by using the infinite product identity and truncating to the first term. That is,

Substituting (Eq. 13) into (Eq. 12) and manipulating, the error current approximation becomes:

where

where

The positive frequency half of the spectrum

where

Using the recurrence relation of the Bessel functions,

the positive half of the error current spectrum takes the final form:

where,

The calculation of the non-characteristic harmonic currents using (Eq. 20) is easily executed numerically as it only manipulates a single array of Bessel functions. The spectral energy is distributed symmetrically around the carrier frequency

### 2.3. Model approximation

The harmonic model derived in the previous section describes the exact spectral characteristics of the error current by including the duty cycle

where the carrier (average) frequency

### 2.4. Dc current harmonics

The hysteresis switching action transfers the ac harmonic currents into the inverter dc side through the demodulation process of the inverter. As the switching function is not defined for hysteresis inverters, the harmonic currents transfer can be modeled through balancing the instantaneous input dc and output ac power equations.

With reference to Fig. 1, and assuming a small relay bandwidth (i.e.

The power balance equation over the switching period when

Using the instantaneous output voltage

in (Eq. 24), the dc current

where

The positive half of the dc current spectrum is thus computed from the application of the Fourier transform and convolution properties on (Eq. 27), resulting in

where

Each spectrum band of the ac harmonic current creates two spectrum bands in the dc side due to the convolution process implicitly applied in (Eq. 28). For instance, the magnitude of the ac spectrum band at

### 2.5. Harmonic generation under distorted system voltages

The harmonic performance of the hysteresis inverter in Fig. 7 under distorted dc and ac system voltages is analyzed. The presence of background harmonics in the ac and dc voltages will affect the instantaneous frequency of the inverter according to (Eq. 30) as

where the dc distortion

Notice that in (Eq. 31),

where

are the frequency noise terms due to the system background distortions. The amplitude modulation indices of the ac and dc harmonic distortions are given by :

Integrating (Eq. 32), the error current

In (Eq. 35): the carrier frequency

where

Applying the Fourier transform and convolution properties on (Eq. 35), the positive half of the frequency spectrum

Where

are the ac and dc modulating spectra. Generally, for any

## 3. Variable-band hysteresis control

### 3.1. Error current mathematical description

The harmonic line generation of the half-bridge inverter of Fig.1 under the variable-band hysteresis current control is derived. The constant switching frequency of the error current in (Eq. 2), i.e.

where the maximum value of the modulating relay bandwidth is

and

Substituting (Eq. 41) in (Eq. 43) and then applying the Fourier transform, the positive half of the frequency spectrum of

The error current spectrum in (Eq. 44) consists of a center band at the switching frequency

### 3.2. Dc current harmonics

The approach developed in 2.2.4 also applies to compute the dc current harmonic spectrum when the variable-band hysteresis control. The positive half of the dc current harmonic spectrum is computed by substituting (Eq. 44) in (Eq. 28).

### 3.3. Harmonic generation under distorted system voltages

The presence of background harmonics in the ac and dc voltages, given in (Eq. 31) will affect the instantaneous frequency of the inverter according to (Eq. 30). Subsequently, to achieve the constant switching frequency

where

where

The new terms introduced by the background distortion appear as amplitude modulations in (Eq. 45). The error current

The harmonic spectrum of the error current

where

Examining (Eq. 49), the presence of the harmonic distortions in the system tends to scatter the spectrum over lower frequencies, more specifically, to

## 4. Simulation

The harmonic performance of the half-bridge inverter under the fixed- and variable-band hysteresis control is analyzed. Results computed from the developed models are compared to those obtained from time-domain simulations using MATLAB/Simulink. Multiple simulation studies are conducted to study the harmonic response of the inverter under line and control parameter variations. The grid-connected inverter of Fig.1 is simulated in Simulink using:

### 4.1. Fixed-band hysteresis current control

The ac outputs of the half-bridge inverter under the fixed-band hysteresis current control are shown in Fig.8. the fundamental component * V*. the inverter line current

The harmonic parameters of the model are computed the system and controller parameters as follows: substituting the reference voltage in (Eq. 4) results in an amplitude modulation index of

The spectrum bands are concentrated around the order of the carrier frequency and are stepped apart by two fundamental frequency orders

To study the effect of line parameter variations on the harmonic performance of the inverter, the DG source voltage is decreased to have the dc voltage

With reference to the results shown in Fig.10, the harmonic spectrum

The total spectral energy of the error current depends on the relay bandwidth

Next, the system and control parameters are set to their original values and the inductance is decreased by 25% to

The width of the relay band is reduced by half while maintaining the rest of the parameters at their base values. As (Eq. 4) indicates,

To study the harmonic performance of the inverter under distorted system voltages, the system and control parameters are set to the original values and the 11^{th} order voltage oscillator

Comparing Fig. 8 and Fig. 13, the reference voltage is distorted due to the presence of the 11^{th} voltage oscillator in the source. The output voltage of the inverter is still bipolar, i.e.

According to (Eq. 32), the carrier frequency

Fig. 15 compares the harmonic spectrum

Similar analysis is performed to study the harmonic performance of the inverter when the dc voltage contains the distortion

The dc distortions impose additional noise component on the instantaneous frequency, see Fig.17, and subsequently, according to (Eq. 38) the harmonic spectrum is drifting to lower order harmonics as shown in Fig.18.

### 4.2. Variable-band hysteresis control

The harmonic performance of the same half-bridge inverter used in section 2.4.1 is analyzed when the variable-band hysteresis current control is employed. Similar harmonic studies to those in the previous section are performed to compute the spectral characteristics of the inverter harmonic outputs using the developed models in section 2.3 and compare them with results obtained from time-domain simulations using Simulink.

The instantaneous line outputs of the single-phase inverter operating under variable hysteresis control are shown in Fig.19. With the maximum relay band

The dc voltage * V*while all other parameters remain unchanged from Study 1. Decreasing

The new values are shown in Fig.21. Consequently, the spectrum

when

The value of the inductance is decreased to

Lower inductance results in higher switching frequency. The harmonic spectrum

The harmonic performance of the inverter under distorted system voltages is studied by simulating the system with the distorted 8^{th} order dc voltage ^{th} order ac voltage

### 4.3. Comparison and discussion

The spectral characteristics of the line current under the fixed- and variable-band hysteresis control are compared in this section. For identical system configurations and controller settings, i.e.

The THD of the line current is directly proportional to relay bandwidth. For similar controllers setting

## 5. Conclusion

A closed-form numerically efficient approximation for the error current harmonic spectrum of single-phase two-level inverters employing either fixed- or variable-band hysteresis current control is derived. The models are based on the amplitude and frequency modulation theorems.

The instantaneous frequency of the inverter is first derived. Then it is used to closely approximate the error current by a modulated sinusoid. The error current harmonic spectrum is basically the Fourier transform of error current complex envelop. In the case of the fixed-band control, the spectrum reduces to a series of Bessel functions of the first kind whose argument is implicitly expressed in terms of the system and controller parameters, where as in the variable-band mode, the spectrum reduces to a 3-element array.

The spectral characteristics such as the carrier frequency and frequency bandwidth are derived analytically and related to line parameters; it is a development useful in inverter-network harmonic interactions. Unlike time-domain simulators, the developed models provide fast numerical solution of the harmonic spectrum as they only involve numerical computation of single arrays. Simulation results agree closely with the developed frequency-domain models in terms of frequency order, magnitude and angle.

In addition to the single-phase two-level inverter, the proposed approximations apply also to the harmonic output of certain three-phase two-level inverters where independent phase control is applicable, such as the neutral point inverter, and the full-bridge inverter in bipolar operation.

## 6. Future directions of research

The models detailed in this chapter can be extended in a number of ways, both in terms of improving the proposed models as well as in the application of the models in other PWM applications.

The developed models neglected the dynamics of the Phase-Locked Loop (PLL) and assumed that the inverter line current tracks a pure sinusoidal reference current. Possible extensions of the models include the effect of the harmonic current propagation through the ac network and the deterioration of the terminal voltage at the interface level and its effect on the reference current generation. As the PLL synchronizes the reference current with the terminal voltage, the propagation of harmonic currents might affect the detection of the zeros-crossings of the terminal voltage resulting in generating a distorted reference current. The hysteresis controller consequently will force the line current to track a non-sinusoidal reference which, in turn, modifies the harmonic output of the inverter.

The implementation of an LC filter at the inverter ac terminals could trigger a parallel-resonance which tends to amplify the harmonic voltages and currents in the ac network leading, in some cases, to potential harmonic instabilities. The improvement of the developed models to include the effect the filter capacitance on the harmonic performance of the inverter is an interesting improvement.

Reviews of the developed models show that hysteresis current controlled inverters can have a ‘switching function’ notation similar to those inherit with the Sinusoidal PWM inverters. The switching function is based on the error current characteristics which implicitly depend on the system and controller parameters. Such development will enable the various time- and frequency-domain algorithms developed for the harmonic assessment of linear PWM inverters to be applied to hysteresis controlled inverters.

Harmonic load flow studies of systems incorporating inverters with hysteresis current control can be formulated based on the developed models. The iterative solution of the harmonic load flow shall incorporate the harmonic magnitudes and angles obtained from the developed models for a faster convergence to the steady state solution.

## 7. Appendix - A

Function

satisfies the same properties in