Electronic Design

Predict ESD response using first-order differential equations

Tests for electrostatic-discharge (ESD) susceptibility sometimes call for charging a test capacitor to a specified voltage (example: 100 pF charged to 15,000 V) and then connecting that capacitor to discharge into the circuit being tested.

Voltage transients in the circuit under test can be estimated with LaPlace transforms or with several different software packages. However, it may be possible to use first-order differential equations and just a few simple lines of code, as demonstrated in the following example.

A circuit under test is taken as resistor R2, for which some kind of input filtering has been provided (Fig. 1). That filtering can be taken as an RLC low-pass filter. As a result, we can write our descriptive set of first-order differential equations for the filter.

When defining the initial conditions, the inductor’s current is zero and the voltage across capacitor C2 is zero. As a result, we enter the loop of differential equations and trace each voltage and current over very small time intervals (dT).

In this example, when we plot the values of voltages e1 and e2, we find that e2 will peak at almost 40 V in response to an initial 15,000 V on C1 (Fig. 2).

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