Automobiles present a very harsh environment for the electrical and electronic devices incorporated into them. This includes the automobile’s electrical system that powers them, which is by no means ideal. For this reason, there are a variety of electrical disturbance tests that have been defined that these devices must be tested under to validate they will survive and function properly in use. Two well-established industry standards that define a wide range of electrical disturbance tests are ISO-7637-2 and ISO-16750-2. There are several other similar standards, both industry-based and manufacturer-specific, that are basically derivations of these two ISO standards.

Around the 2010 to 2011 timeframe, the crank start profile test pulse and the unsuppressed and suppressed load dump test pulses were removed from the ISO-7637-2 standard. At the same time, ISO-16750-2 gained the unsuppressed and suppressed load dump test pulses, and it already had its own version of crank start profile test pulse. The net result is ISO-7637-2 incorporates test pulses for disturbances having mostly very high speed rise or fall times of nanoseconds to microseconds, while ISO-16750-2 incorporates test pulses for disturbances having relatively slow rise and fall times, of around a 1 millisecond or slower.

The ISO-7637-2 disturbances having microsecond to nanosecond-fast rise and fall times are primarily a result of voltage spikes on the DC voltage bus created by switching inductive devices on and off, or electrical devices creating a stream of voltage spikes while active. One exception to all the very high speed disturbances in ISO-7637-2 is test pulse 2b of section 5.6.2. This test pulse addresses the electrical disturbance created by an electrical motor, like that of a blower motor within the heating and air conditioning system. When the motor is running and then the ignition is switched off, the motor will change over from consuming power to generating power. The result is it creates a relatively slow voltage pulse back onto the electrical system, until all the energy from its spinning mass is dissipated. Test pulse 2b is depicted in Figure 1.

**Figure 1: Test pulse 2b of ISO-7637 Section 5.6.2**

The first part of the test pulse is a step drop due to the ignition being switched off, while the second part is a pulse from the motor regeneration energy. This motor regeneration pulse is referred to as a double exponential waveform. This is a characteristic shape of many types of pulses, where stored energy is dumped onto, and then subsequently dissipated by a load. Other test pulses in ISO-7637-2 have this characteristic shape, but are of much shorter rise and fall times.

Most often, a double exponential test pulse is produced by a test pulse generator specifically designed to this purpose. The basis of many double exponential test pulse generators is depicted in Figure 2. It makes use of reactive components to store the energy and shape the waveform. For these types of pulse generators, the loading impedance of the device under test (DUT) itself is a factor that must be taken into consideration in determining the pulse waveform. As such, it can be an iterative process to set up this kind of pulse generator to get the desired pulse amplitude, rise time, and duration. Appendix E of ISO 7637-2 describes how to calculate the energy delivered to DUT by a double exponential pulse.

**Figure 2: Simplified diagram of a double exponential test pulse generator**

Unlike other tests that call out for a generator as depicted in Figure 2, the ISO 7637-2 standard instead recommends using an arbitrary waveform generator driving a DC power supply/amplifier with an analog control input for generating test pulse 2b of section 5.6.2. This is sensible given its complex shape, consisting of a step drop followed by the double exponential pulse, and having relatively slow rise and fall times. Another advantage of using an arbitrary waveform generator and DC power supply/amplifier is that its output is independent of the DUT’s load impedance. This eliminates the iterative trial and error approach to get the desired pulse, that can be associated with pulse generators that make use of reactive components.

While the step portion of test pulse 2b is easy to define, how does one define the double exponential motor regeneration pulse? There are a few possible approaches, including:

- Construct a piecewise linear model to approximate the double exponential shape;
- Use software tools to generate a data file of points from a graphical image of the waveform; or
- Make direct use of a double exponential mathematical expression that defines this waveform, which can be utilized once it is understood how to do so, which will be explained here.

The double exponential expression for test pulse 2b is basically a fast exponential being subtracted from a slow exponential, as shown by the following:

*U _{DE}(t)= U_{S }*

***

*G*

***

*(e*

^{-K1t}– e^{-K2t})Where U_{DE}(t) is the double exponential voltage, U_{A} is the electrical system voltage, G is a gain correction factor for the double exponential, K_{1} is the slow time constant related to the duration of the test pulse, t_{d}, and K_{2} is the fast time constant related to the rise time of the test pulse, t_{r}.

The trick of making use of this double exponential expression is to understand how to relate the constants in the equation to the test pulse parameters shown in Figure 1. It turns out that this is relatively straight forward for this application, once understood. In good part, this is simplified due to the large relative difference between test pulse 2b’s rise time compared to its duration. This reduces the overlapping interaction between the two exponentials by a considerable amount, keeping the mathematical expression relatively straight forward to apply.

From inspection, the time for the fast exponential to decay from 90% to 10%, or decay from 0.9 to 0.1, is also the rise time of the test pulse, t_{r}. Remember, the fast exponential is being subtracted from the slow exponential so the decay of the fast exponential corresponds to the rise time of the test pulse. Taking the natural log of these two amplitude values gives the following:

*Ln (0.9) = -0.1054 and Ln (0.1) = -2.3026 *

Taking the difference of these two log values and then dividing it by the rise time, t_{r}, normalizes the 90% to 10% exponential decay to rise time, t_{r}. Thus, the time constant K_{2} for the fast exponential in the double exponential expression can be expressed as:

*K _{2} = (2.197/t_{r})*

*Where t*

_{r}is the rise time of the test pulseFrom inspection, the slow exponential decays from 100%, or 1, at t = 0, to 10%, or 0.1 at the end of the test pulse duration, t_{d}. Just as before, taking the natural log of these two values gives:

*Ln (1) = 0 and Ln (0.1) = -2.3026*

Note, however, the test pulse duration, t_{d}, starts at the 10% point of the rise time of the test pulse, not at t = 0, which is the starting point of the slow exponential’s decay time. This additional time for the test pulse to rise from zero to the starting point for the pulse duration, t_{d}, is given by:

*t _{(0 to 10%)} = [Ln(0.9) / (Ln(0.9)-Ln(0.1))] *

***

*t*

_{r}= 0.048***

*t*

_{r}Thus, the total time it takes for slow exponential to decay from 100% to 10% is (t_{d} + 0.0148 * t_{r}). In practice the additional time, 0.048 * t_{r}, can be neglected, but has been included here for completeness. The slow exponential decay time is primarily associated with the test pulse duration.

Taking the difference of the two log values and then dividing it by the value (t_{d} + 0.0148*t_{r}) normalizes the 100% to 10% exponential decay to the test pulse’s rise time and duration. Thus, the time constant K_{1} for the slow exponential in double exponential expression can be expressed as:

*K _{1} = (2.303/(t_{d} + 0.0148 *

***

*t*

_{r})) ≈ (2.303/t_{d}) for t_{d}>>t_{r}Where t_{d} is the duration and t_{r} is the rise time of the test pulse.

Except for the limiting case with a rise time of zero, the base double exponential waveform will always have an amplitude of less than one due to the overlap of the two separate exponentials. Thus, some gain correction is required, which is provided by the gain correction factor, G. As the ratio of the duration time, t_{d}, to rise time, t_{r}, lessens, there is more overlap between the two exponentials, requiring additional compensation be made, to adjust for increasing loss in amplitude. In the example illustrated in Figures 3 and 4, the ratio of the duration to the rise time was 200 to 1. A gain correction factor of G = 1.03 was used to compensate for a 3% amplitude loss.

**Figure 3: Single and double exponential waveforms implementing ISO 7637-2 test pulse 2b**

The resulting exponential and double exponential waveforms are shown in Figure 3, using the double exponential expression just defined. They are based on typical rise and duration times, and amplitude value given in Figure 1 for test pulse 2b.

To generate the actual test pulse 2b, a Keysight N7951A 20V, 50A, 1KW Advance Power System DC power supply was used, as it has suitable bandwidth and power for the application, and already has arbitrary waveform generation capabilities built in. Companion Keysight 14585A software was used to create the arbitrary waveform sequence consisting of a voltage step drop followed by the mathematical expression of the double exponential just defined, and load it into the N7951A. The resulting test pulse waveform was run, captured, and in turn displayed by the 14585A software, as shown in Figure 4.

**Figure 4: Test pulse 2b of ISO 7637-2 generated and captured using Keysight 14585A software**

In closing, test pulse 2b of section 5.6.2 of the ISO 7637-2 automotive electrical test standard can be easily generated based on the mathematical expression for a double exponential waveform. It just a matter of understanding the relation between the test pulse’s amplitude, rise time and duration, and the double exponential waveform expression’s amplitude and time constants, as demonstrated here. As the double exponential waveform is a common characteristic shape of many kinds of electrical disturbances, understanding how to mathematically define in terms of the test pulse parameters has utility for many other disturbances beyond test pulse 2b of the ISO 7637-2 standard as well!

*Edward Brorein** is an application specialist at Keysight Technologies, Inc. He received his BSEE from Villanova University in 1979 and his MSEE from the New Jersey Institute of Technology in 1987. Ed joined Keysight Technologies (at the time Hewlett Packard) in 1979 and worked as an R&D engineer, manufacturing engineer, and presently as a marketing engineer, helping customers using power products, both in manufacturing and R&D. Ed has been actively and deeply involved with the design, engineering, and application of DC power products for testing a variety of electronic devices. *

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