The question came up recently about how to model non-linear circuits when performing an EMI Analyst™ radiated susceptibility analysis. The same issue also applies to conducted susceptibility.
At first glance, it seems obvious. Wouldn’t you just model active circuits the same way you model them in PSpice or any other time-domain circuit analysis tool?
The short answer is, no. You can’t.
Active Circuits Are Non-Linear
The problem is, most computer models for non-linear components like diodes and transistors are only valid for low frequencies, the frequency range for which they are intended to operate.
The effects of parasitic elements like junction capacitance, which are significant at high frequencies, are often not included in the models.
Susceptibility Covers a Broad Frequency Range
Circuit immunity is a frequency domain pursuit.
Radio frequency susceptibility requirements are most often defined as a field level or injected signal level over a broad frequency range. For example, the RS103 requirement in MIL-STD-461G specifies 20 V/m from 2 MHz to 18 GHz for spacecraft.
In frequency domain analysis, you have to model the circuit at a single operating point, and then perform the analysis over the specified frequency range. If the circuit has multiple operating points, the analysis has to be done multiple times, once at each operating point.
Diodes are non-linear, right?
Well, yes, but not for all signal levels.
For large signals diodes are non-linear. However, for small signals diodes behave fairly linearly.
Small changes in current through a forward biased diode in saturation result in minute changes in voltage across the diode. Similarly, small voltage changes impressed across a reverse biased diode, or forward biased diode that is not on, produce minuscule current changes.
Diodes are often used for reverse bias protection, over voltage protection, and of course for rectifying AC signals and AC power.
At a given operating point, diodes may be modeled as a resistor in parallel with a capacitor. The model resistor value is obtained by dividing the voltage at the operating point by the current at the operating point. Model capacitance is the junction capacitance, which also varies with operating point.
In reverse bias, diode resistance is the reverse voltage divided by the leakage current. For most applications, the resistance is much greater than the reactance of the junction capacitance and may be ignored.
In forward bias, the small-signal resistance is given by
where ID is the diode forward current, VT is the thermal voltage (kT/q, about 26 mV at normal temperatures), and n is the diode ideality factor (approximately 1 to 2 for silicon diodes.)
At voltages less than the diode turn-on voltage, the diode model resistance is big. In saturation, the diode model resistance is small.
Diode junction capacitance depends on whether the diode is forward biased or reverse biased.
In reverse bias, junction capacitance is a function of charge storage in the depletion region, given by
where CJ0 is the zero-bias value of the junction capacitance, V0 is the barrier potential, mj is a function of the doping profile in the device (typically between 0.2 and 0.5).
In forward bias, diode junction capacitance is approximately
Additionally, in forward bias p-n junction diodes exhibit minority-carrier storage which leads to small-signal diffusion capacitance, given by
where tF is the forward-transit time. The total capacitance of a forward biased diode is Cj + Cd.
If the diode turns on and off during normal circuit operation, the circuit may need to be analyzed at two or more operating points, perhaps one analysis with the diode reverse biased, and one analysis with the diode in saturation.
Fortunately, the calculation time for most EMI Analyst project is quick. Running two or more analyses with different component values is usually just a simple a matter of changing the component values and then clicking Calculate again.
If you need to determine which conditions produce worst case results, simply overlay the graphs from each analysis.
Radiated Susceptibility Analysis Example
When the simple circuit below is exposed to an electric field, current is induced on the pair of wires.
The voltage induced across the resistors at both ends of the cable depends on whether the diode is forward biased or reverse biased.
To capture worst case induced voltage, the analysis is run twice. Once with the diode in saturation and once with the diode off. The two graphs below show the results for the resistor on the left.
Forward Biased Diode
The green plot on the left shows voltage induced when the diode is saturated, in forward bias. When saturated, the impedance of the diode is much lower than the 100-ohm termination impedance, so most of the low frequency induced voltage appears across the termination resistor.
The orange plot on the left shows the voltage induced across the resistor when the diode is reverse biased. When off, diode impedance is larger than the termination impedance, so most of the low frequency induced voltage is dropped across the diode, and less voltage appears across the termination resistor.
At high frequencies, there is only a slight difference between the forward and reverse biased diodes, due mainly to the difference in diode junction capacitance.
The worst-case induced voltage at each frequency is the greater of the two analysis results.