Brief Lecture for MEL products

★Operation of non-linear element★

＜Linear element and non-linear element＞

The terms linear and non-linear are frequently used in circuit analysis.  Let us explain what they are.
They mean characteristics of elements.  Linear element refers to the output that changes in the form of line, namely, in the direct line according to the input, as the name indicates.  For example, when considering a resistor, the current that flows in the resistor is Ir=Vr/R=G･Vr and it becomes the linear function of Vr that is proportional to the voltage of both ends of the resistor.  Elements with this type of input and output relation are referred to as linear elements.  This is also applicable to the capacitor and it becomes Ic=jωC･Vc and is also the linear function of Vc.
Contrary to this, the output characteristic of non-linear elements is not the linear function of the input.  For example, as for the diode, current changes in the exponential function of voltage and does not change proportionally even if the applied voltage varies linearly.  This type of element is referred to as a non-linear element and semiconductors correspond to it.

＜Operation of non-linear element＞

Now, as for non-linear element, we will examine its operating characteristics.  Distortion is generated in one word.
Distortion is not often desired for the circuits, however, there are many cases in which this distortion is utilized.  A mixer and a multiplication unit utilize distortion.  Talking about distortion, in the digital circuit, it is often said that clock waveform contorts.  However, this contortion is a little different from distortion.  The ratio of harmonic ingredient changes according to the frequency characteristic of the circuit and the waveform becomes out of shape because the digital waveform contains a lot of harmonic ingredients.  Therefore, the distortion of the digital waveform is caused not only by the non-linear elements, but largely by the frequency characteristic of the circuit.
Getting back to our main subject, the operation of the non-linear element distorts the sine wave and it creates a harmonic.

Now, let’s think about this theoretically using S･NAP.  The resistor with the frame shown in the figure below refers to the non-linear resistor created by the macro.  It is set so that the current characteristic is proportional to the square of the terminal voltage.  In other words, it is set so that Ir=Vr²/R holds.  The figure next to it shows current when Vs varies from 0 to 10V in DC analysis.  It can be seen that Y=X² holds.

Now, we connect the AC power supply instead of the DC power supply and try to analyze with the harmonic balance method.  The diagram shown in the middle indicates the spectrum of current.  It can be confirmed that the DC ingredient is 0.5 A, the frequency ingredient is 2 KHz, and the harmonic is 0.5 A, that is twice as large as the input frequency.  The figure shown at right indicates the current waveform and input voltage waveform.  It can be seen that the current waveform is the offset of 0.5 A, the frequency is twice as large as that of input, and the phase delays byπ/2.

The characteristic of the non-linear resistor is simple.  Therefore, current can be calculated easily in terms of analysis.  The current of the circuit is as follows;

I=Vr²/R=sin²(ωt)/1=0.5{1-cos(2ωt)}=0.5+0.5sin(2ωt-π/2)

This matches the result of the simulation.
As you can see from this result, the harmonic ingredient of the input frequency can be obtained in the circuit containing the non-linear element and 2 multiplication units can be easily produced if there is a resistor used here.  n･f1±m･f2 can be obtained as output according to the non-linear characteristic when there are multiple input frequencies and the circuit can also be used as the frequency converter.

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