# An Analytical Time–Domain Expression for the Net Ripple Produced by Parallel Interleaved Converters

## Abstract

We apply modular arithmetic and Fourier series to analyze the superposition of N interleaved triangular waveforms with identical amplitudes and duty-ratios. Here, interleaving refers to the condition when a collection of periodic waveforms with identical periods are each uniformly phase-shifted across one period. The main result is a time-domain expression which provides an exact representation of the summed and interleaved triangular waveforms, where the peak amplitude and parameters of the time-periodic component are all specified in closed-form. Analysis is general and can be used to study various applications in multi-converter systems. This model is unique not only in that it reveals a simple and intuitive expression for the net ripple, but its derivation via modular arithmetic and Fourier series is distinct from prior approaches. The analytical framework is experimentally validated with a system of three parallel converters under time-varying operating conditions.

- Authors:

- Publication Date:

- Research Org.:
- National Renewable Energy Lab. (NREL), Golden, CO (United States)

- Sponsoring Org.:
- NREL Laboratory Directed Research and Development (LDRD)

- OSTI Identifier:
- 1346338

- Report Number(s):
- NREL/JA-5D00-66257

Journal ID: ISSN 1549-7747

- DOE Contract Number:
- AC36-08GO28308

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: IEEE Transactions on Circuits and Systems. II, Express Briefs; Journal Volume: 64; Journal Issue: 3

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 24 POWER TRANSMISSION AND DISTRIBUTION; interleaving; parallel converters; dc-dc converters; inverters; Fourier analysis; modular arithmetic

### Citation Formats

```
Johnson, Brian B., and Krein, Philip T.
```*An Analytical Time–Domain Expression for the Net Ripple Produced by Parallel Interleaved Converters*. United States: N. p., 2017.
Web. doi:10.1109/TCSII.2016.2557620.

```
Johnson, Brian B., & Krein, Philip T.
```*An Analytical Time–Domain Expression for the Net Ripple Produced by Parallel Interleaved Converters*. United States. doi:10.1109/TCSII.2016.2557620.

```
Johnson, Brian B., and Krein, Philip T. Wed .
"An Analytical Time–Domain Expression for the Net Ripple Produced by Parallel Interleaved Converters". United States.
doi:10.1109/TCSII.2016.2557620.
```

```
@article{osti_1346338,
```

title = {An Analytical Time–Domain Expression for the Net Ripple Produced by Parallel Interleaved Converters},

author = {Johnson, Brian B. and Krein, Philip T.},

abstractNote = {We apply modular arithmetic and Fourier series to analyze the superposition of N interleaved triangular waveforms with identical amplitudes and duty-ratios. Here, interleaving refers to the condition when a collection of periodic waveforms with identical periods are each uniformly phase-shifted across one period. The main result is a time-domain expression which provides an exact representation of the summed and interleaved triangular waveforms, where the peak amplitude and parameters of the time-periodic component are all specified in closed-form. Analysis is general and can be used to study various applications in multi-converter systems. This model is unique not only in that it reveals a simple and intuitive expression for the net ripple, but its derivation via modular arithmetic and Fourier series is distinct from prior approaches. The analytical framework is experimentally validated with a system of three parallel converters under time-varying operating conditions.},

doi = {10.1109/TCSII.2016.2557620},

journal = {IEEE Transactions on Circuits and Systems. II, Express Briefs},

number = 3,

volume = 64,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2017},

month = {Wed Mar 01 00:00:00 EST 2017}

}