Electromagnetic pulse (EMP) is a kind of transient electromagnetic phenomenon with short rise time of the leading edge and wide spectrum, which usually disrupts communications and damages electronic equipment and system. It is challenging for an EMP sensor to measure a wideband electromagnetic pulse without distortion for the whole spectrum. Therefore, analyzing the distortion of EMP measurement is crucial to evaluating the sensor distortion characteristics and correcting the measurement results. Waveform fidelity is usually employed to evaluate the distortion of an antenna. However, this metric depends on specific signal waveforms, thus is unsuitable for evaluating and analyzing the distortion of EMP sensors. In this paper, an associated-hermite-function based distortion analysis method including system transfer matrices and distortion rates is proposed, which is general and independent from individual waveforms. The system transfer matrix and distortion rate can be straightforwardly calculated by the signal orthogonal transformation coefficients using associated-hermite functions. Distortion of a sensor

Electromagnetic pulse (EMP) is a kind of transient electromagnetic phenomenon. In the time domain, the rise time of the leading edge is short, and its spectrum is wide. Lightning, electrostatic discharge and high-power switch can produce electromagnetic pulse. EMP usually disrupts communications and damages electronic equipment and system by means of electromagnetic radiation and conduction.

EMP measurement is one of the important research topics in the damage and protection mechanism to the electronic equipment [

For ultra-wideband (UWB) antenna, waveform fidelity is an important index to measure antenna performance [

The AH basis function is [

Hermite polynomial

Therefore, the AH basis function is the orthonormal basis

As the AH basis function is the characteristic function of Fourier transform [

It can be seen from the above formula that the shape of

The differential form of the AH basis function has the following relationship with the original function [

When the AH basis function is used to analyze the signal, this property makes it convenient to analyze the characteristic relationship between the input and output of a signal system.

The scaling factor

The relationship between time domain support interval

When the time period of the time domain signal

Substituting formulas

The time-domain support range of AH functions extends to both positive and negative sides with

The new definition of distortion rate is derived from the system transfer matrix represented by orthogonal basis functions. A system transfer function can be obtained by a time-domain measurement, and then the basis vector is taken as the excitation function and substituted into the transfer function to obtain N responses. Finally, the system transfer matrix can be obtained by calculating the coefficients of the N responses. The distortion rate can be obtained by comparing the system transfer matrix with the identity matrix.

An EMP signal with limited energy and certain pulse width belongs to the typical compactly support signal [

The coefficient of the basis vector can be expressed by the inner product as

The input and output signals can be represented by the below vectors

According to formula

Then, from

Since the Fourier transform form of the Associated-Hermite function is its characteristic function in the time domain, whose expressions in frequency domain and time domain have the same coefficients. In order to obtain the

The coefficient

Then, the system transfer matrix is

For an ideal electromagnetic field sensor, the energy is normalized by the system transfer matrix as below

The absolute value graph of the corresponding system transition matrix is shown in

The absolute value graph of the normalized system transfer matrix is shown in

When analyzing the distortion of a sensor, in order to do quantitative analysis, a distortion rate

When calculating the distortion rate of differential sensor, the output waveform is usually integrated first, then the system transfer matrix is obtained by the

The distortion rate is different from the fidelity, and the calculation result no longer depends on the specific input waveform. Three-dimensional electromagnetic field computer simulation technology (CST) is employed to verity it by taking the inverted cone sensor as an example. The inverted cone sensor is placed in an electromagnetic environment formed by plane wave irradiation, and the load is a resistance and a capacitance in parallel. The simulation model is shown in

Excitation waveforms | fidelity |
Distortion rate |
---|---|---|

IEC61000-2-9 | 0.8981 | 0.2230 |

square wave | 0.9337 | 0.2226 |

DOD-STD-2169 | 0.8885 | 0.2235 |

CS116 | 0.9996 | 0.2228 |

Generally speaking, a signal is considered to be distorted when its output and input are nonlinear. For the differential sensor, its output is the differential form of the input, that is, obvious nonlinear distortion appears. In practical application, the output of the differential sensor is usually integrated, and the integrated waveform and input waveform will present a good linear relationship.

In order to verify the effect of distortion analysis based on the system transfer matrix and distortion rate based on AH functions, the waveforms measured by a current probe, a coaxial pulse voltage probe and a magnetic field sensor (also known as B sensor) are analyzed respectively.

The measured waveforms of the current probe, the coaxial pulse voltage probe and the B sensor are shown in

For the current probe, the frequency band width of the input waveform is

Sensors and probes | |||||
---|---|---|---|---|---|

current probe | 250 | 0.15 | 60 | 12 | 125 |

coaxial pulse voltage probe | 160 | 0.2 | 60 | 8 | 80 |

B sensor | 150 | 0.2 | 60 | 7 | 75 |

According to the calculation, the absolute value graph of the system transfer matrix of the current probe is shown in

The absolute value graph of the system transfer matrix of the coaxial pulse voltage probe is shown in

The absolute value of the system transfer matrix of the B sensor is shown in

The distortion rates of the current probe, the coaxial pulse voltage probe and the B sensor are shown in

Sensors and probes | Distortion rate |
---|---|

current probe | 0.1768 |

B sensor | 0.4255 |

coaxial pulse voltage probe (output waveform) | 0.7602 |

coaxial pulse voltage probe (integrated output waveform) | 0.1718 |

It is challenging for an EMP sensor to measure a wideband electromagnetic pulse without distortion for the whole spectrum. Therefore, analyzing the distortion of EMP measurement is crucial to evaluating the sensor distortion characteristics and correcting the measurement results. Waveform fidelity is usually employed to evaluate the distortion of an antenna. However, this metric depends on specific signal waveforms, thus is unsuitable for evaluating and analyzing the distortion of EMP sensors. In view of the limitation of the fidelity, an AH-function based distortion analysis method including system transfer matrices and distortion rates is proposed in this paper, which is general and independent from individual waveforms. The system transfer matrix and distortion rate can be straightforwardly calculated by the signal orthogonal decomposition coefficients using AH functions. According to a set of measured time-domain signal waveforms, the system transfer function is obtained by the orthogonal decomposition with AH basis function. Then, by exciting the transfer function using each order AH basis vector, there will be different outputs and the system transfer matrix can be established from orthogonal decomposition coefficients of those outputs. The distortion rate can be calculated by comparing the system transfer matrix with the identity matrix. Distortion of a sensor