Non-Linearity. Frequency-doubling, degree variation, amplitudemodulation and demodulation

NON-LINEARITY -

FREQUENCY-DOUBLING, DEGREE VARIATION, AMPLITUDE MODULATION AND DEMODULATION

LAURA IMPERATORI

MURRAY EDWARDS COLLEGE LSI22

PRACTICAL PARTNER: GOODWIN GIBBINS

EXPERIMENT PERFORMED THURSDAY, OCTOBER 25, 2012

Abstract. In order to test non-linearity, the effects of different transfer func- tions of an AD633 multiplier in a given electrical circuit were investigated and compared with the theoretical expectations. First of all, the phenomenon of frequency doubling was found to occur when squaring the input voltage. Sec- ondly, the multiplier was reconfigured to give a square-root response. This allowed us to vary the degree of non-linearity by choosing the parameters of input voltage and DC offset such that we could determine which terms in the Taylor expansion of the transfer function were relevant and hence to what de- gree the circuit behaved non-linearly. For a small, sinusoidal variation about a large DC level, the system was found to be weakly non-linear. For high amplitude and a low DV offset we observed strong non-linearity. Compared to weak non-linearity, we were able to detect the third harmonic as well as the first and the second one. The existence of harmonics was investigated on the PicoScope screen and verified by plotting output amplitude (dBV) versus input amplitude (dBV) and finding the gradient of the slope corresponding to the respective harmonic. Finally, frequency mixing was explored in its broader context by investigating amplitude modulation and demodulation on the same circuit board.

1. Introduction

The aim of the experiment is to investigate the behaviour of non-linearity in electrical circuits. Most common circuit elements such as capacitors, inductors and resistances as well as operational amplifiers behave very linearly and thus enable an intuitive understanding of the qualitative behavior of the circuit, but most physical systems are inherently nonlinear in nature such as neural circuits [?] and climate dymamics [?]. Therefore, nonlinear problems are of interest to engineers, physicists and mathematicians. The most common consequence of non-linearity is frequency- doubling which is also known as Second Harmonic Generation. It is a nonlinear optical process, in which photons interacting with a nonlinear material are effec- tively “combined” to form new photons with twice the energy, and therefore twice the frequency and half the wavelength of the initial photons. It was first showed by Peter Franken et al. at the University of Michigan in 1961.[?]

2. Theoretical Background

2.1. Non-linear circuits in constrast to linear circuits. In linear circuits, the output voltage is a function of the input voltage: V out = LV in, where

illustration not visible in this excerpt

Date: September 18, 2013.

2 LAURA IMPERATORI

Hence, linear circuits obey the superposition principle. For a sinusoidal input volt- age of frequency f, any steady-state output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency f. Examples of linear circuits are circuits composed exclusively of ideal resistors, ca- pacitors, inductors, and operational amplifiers in the non-saturated regime, such as small-signal amplifiers, differentiators, and integrators. Some examples of nonlinear electronic components are: diodes, saturated transistors, iron core inductors and transformers. These non-linear components obey a transfer function, which speci- fies how the input voltage is modified. In this experiment, we introduced the AD633 multiplier, which takes two input voltages and multiplies them together giving an output voltage. Hence, its transfer function is as follows: V out = V x × V y / 10.

2.2. Frequency doubling. In the given experimental setup, frequency doubling can be explained by looking at the effect of the multiplier. For an input voltage V in = A cos ω t, the output of the AD633 multiplier is

illustration not visible in this excerpt

Hence, there are two different frequencies in the output: a DC-offset and twice the frequency of the input voltage. The transfer function of a general non-linear system, V out = f (V in) can be approximated by a Taylor series, which with sinusoidal input A cos ω t, gives

illustration not visible in this excerpt

Using power-reduction formula, the (cos ω t) n terms can be written as sums of terms with frequencies that are multiples of ω t. Thus, a non-linear system generates an output with frequency components of ω , 2 ω , 3 ω , 4 ω,etc.

2.3. Varying the degree of non-linearity. The square-root circuit transforms the output voltage into the square-root of the input voltage: V out =√ √ − 10 V in. This equals 10 B + 10 A cos(ω t) on defining − V in as B + A cos(ω t). Hence, the Taylor

series of V out can be obtained by multiplying the Taylor expansion of f (x) = √ √ 1 + x

for x = A 10 B:

B cos ω t withthefactor

illustration not visible in this excerpt

This can be simplified to obtain the harmonic coefficients of the series cos(ω t) , cos(2 ω t) , cos(3 ω t):

illustration not visible in this excerpt

Figure 1. The main circuitboard with the non-linear component

- the AD633 multiplier. Special features of this circuit are the two inverting amplifiers (-1 buffers) before as well as the two +1 buffers after the AD366.

3.1. Frequency doubling. In order to investigate a non-linear system with a response that depends on the square of the input, the AD366 multiplier was set to X ². This then simplified the given circuit diagram in Figure ??.

illustration not visible in this excerpt

Figure 2. The key features are the -1 and +1 buffer.

The BNC X input was connected to the signal generator at the 50Ω output. The input signal was then observed at the test point IN X and the output was available at the test point MULT. The different frequency values were measured using the built-in function of the PicoScope Software. The different amplitudes were measured with the ruler function to get more precise values and to estimate errors better. At 100 kHz there was no more a visible amplitude of the output signal. The gain of operational amplifiers decreases at high frequencies due to saftey reasons.¹ Considering the two operational amplifiers in the circuit, we decided to conduct the amplitude measurements at a frequency of 1 . 066 kHz. Based on our theoretical prediction, we expected a straight line with gradient 2 for the plot of the output ⁴

Here we can see that cos(ω t)³ contains a term cos(ω t) as well as cos(ω t)⁴ includes a term cos(2 ω t). These coefficients show that large values of the DC-offset (B) and small amplitudes of the input voltage (A) imply a small number of relevant terms in the Taylor series, which corresponds to weak non-linearity, whereas small values in B and large amplitude values correspond to big changes in the function of the output voltage, hence to strong non-linearity. In general, the smaller the gradient, the smaller amout of terms in the corresponding Taylor series.

2.4. Amplitude modulation and demodulation. Electromagnetic energy can only be transferred in an efficient way if the dipole length fits well to the wavelength of the corresponding frequency. An acoustic signal of frequency f = 1 kHz would have to be emitted by an antenna of length l = c = 150 km. As this is not 2 f

very practical, signals are transformed to be emitted at a higher frequency via amplitude, frequency or phase modulation: A high carrier frequency is multiplied by the signal frequency. In the time domain, this gives rise to an envelope of the carrier frequency, which contains the information of the signal frequency. In the frequency domain, amplitude modulation produces a signal with power concentrated at the carrier frequency and two adjacent sidebands, each one equal in bandwidth to that of the modulating signal. The data is shown in Figure ?? and ?? respectively. This phenomenon can be explained mathematically by frequency mixing. If two frequencies ω 1 and ω 2 are fed into the multiplier, the output will be:

illustration not visible in this excerpt

The original signal can then be recovered from the modulated waveform by a further multiplication with the carrier.

illustration not visible in this excerpt

3. Methods and Results

The following measurements of non-linear behaviour in electrical circuits due to a non-linear compononent (AD633 multiplier) were conducted based on the Printed Circuit Board in Figure ??.

[...]

¹ Over a big enough frequency range, the feedback resistor which has inductive and capacitive components will lead to a phase shift of π. This will change negative feedback to posive feedback, which implies oscillation rather than amplification.