NONLINEAR RESPONSE AND FREQUENCY MODULATION Eric Prebys P

NONLINEAR RESPONSE AND FREQUENCY MODULATION Eric Prebys

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 2 Linear Elements • So far, most of our formalism has been built on the assumption that our circuits (or digital processing operations) are linear • If we operate on two signals, we assume that it’s some linear combination of the two • If this is true, then all of our assumptions regarding Fourier Transforms, etc still hold • But what if it’s not?

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 3 Linear Operations in the Frequency Domain • In the frequency domain, all linear operations result in simple multiplications • If the response of a circuit is characterized by Spectrum of Input Response of circuit • In other words, a linear circuit can eliminate frequencies, but it can never change them or introduce frequencies that are not present in the input.

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 4 Nonlinear Operations on a Single Input • We can write a general non-linear operation on a single input as • Assume • Now we have introduced new frequencies

P 116 A - Fall 2019 5 Nonlinear Response and Frequency Modulation Product of Two Terms difference sum

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 6 Nonlinear Mixing • We showed an example of a multiplication circuit early in the course but that’s kind of of complicated • In general, we just need an nonlinear operation that has a quadratic term to get what we’re looking for • “mixer”

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation Example: Exponential Adder 7

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 8 Diode Mixer in Frequency Space • Assume • Look at first two terms multiplication term • So the frequency content is

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 9 Mixer Output Example • Example f 1=10 MHz, f 2=100 MHz. • Output frequencies Filter around this component

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 10 Application of Frequency Mixing • Only really meaningful if one frequency is a delta function (pure sine wave) “modulator” “carrier”

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 11 Ring Modulator Mixer • Our “multiplication” circuit isn’t very practical, because the diodes will only behave properly over a small range of amplitudes. • The most common type of mixer is the “ring Modulator” • This is designed to keep one pair of diodes always forward biased.

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 12 Understanding the Ring Modulator* • It’s easier to understand the ring modulator if you draw it like this • The role of the transformers is to isolate the input and output voltages and allow the carrier wave (vc) to drive their central position. • If |vc|>|vm|, then • On the half of the cycle where VA>VB, D 1 and D 2 will conduct. • On the half of the cycle where VA<VB, D 3 and D 4 will conduct. *Explanation and figures by Jarmo Lähdevaara, published at guitarscience. net

P 116 A - Fall 2019 First half of cycle The voltage across the first diode is: The voltage across the 2 nd diode is: The current in diodes is The current through the diodes is then The net current through the primary of T 2 is Nonlinear Response and Frequency Modulation 13

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 14 Second Half of the Cycle • If we analyze the second half of the cycle we get vc has switched sign, but remember this is a (positive) amplitude. The sign of the carrier wave has also switched sign during this half of the cycle, so this is consistent with a simple multiplication by the carrier wave

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation Example: Audio Frequency • Raw Audio • Mixed with 2500 Hz through a ring modulator • Hearing mostly the subtracted frequency • If you listen carefully, you can hear the high frequency, too. • Popular effect for synthesizers and guitars Audio processors can now do this entirely digitally. I don’t know if this has a real ring modulator in it or not 15

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation Application: Frequency Modulation • Amplitude Modulation vs. Frequency Modulation 16

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation 17 Frequency Modulation • Frequency modulation is usually effected with some sort of Voltage Controlled Oscillator (VCO) • Here’s a simple example with a 555 (which would not work for high frequencies) Changing the voltage at the “control” input, will change the voltage were the 555 resets, and therefore the frequency

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation Use of Mixers in RF Circuits LO=“Local Oscillator” IF=“Intermediate Frequency” 18

P 116 A - Fall 2019 Nonlinear Response and Frequency Modulation Processing FM Signals 19