The p-n junctions of modern diodes and bipolar transistors can be approximated as an exponential nonlinearity in the form introduced by Shockley. This permits analytic prediction of the harmonics produced by the junction, even in circuits that include series resistance. Although the circuit equation is transcendental and in implicit form, usefully simple approximations are developed. This permits circuit design tradeoffs of other requirements with distortion and can give a more nearly optimum result. Harmonic structures produced by p-n junctions depend on the input voltage, the bias current, the series resistance, and the junction properties, but can be characterized by only two parameters. Circuit resistance and bias current are analogous in their effect on harmonic structure. When the circuit resistance is zero, the harmonic amplitudes are given by the modified Bessel function whose order is the harmonic number and whose argument depends on the input voltage and diode. For greatest usefulness, the modified Bessel function is normalized to the fundamental amplitude and presented in logarithmic graphical form. For large resistance or bias current, the distortion is small and is directly proportional to the input voltage and inversely proportional to the square of the series resistance. For intermediate resistance values, complex harmonic structures are found that can be expressed in series form involving modified Bessel functions of variable argument for small input voltages. Useful Taylor series expansions are found for current and voltage in the implicit circuit equation. An opposing series pair diode circuit is considered and the distortion evaluated for large series resistance circuits. This circuit is useful in consideration of different amplifiers, diode modulators and signal keyers, and similar circuits.
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