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Note for Analog Electronic Circuits - AEC By Akash Sharma

  • Analog Electronic Circuits - AEC
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ANALOG ELECTRONICS BİLKENT UNIVERSITY Chapter 1 : SIGNAL S AND COMMUNICATIONS Electronic communications is exchanging signals. While these signals are symbolic in many communication schemes, they are almost exact electrical replicas of original information in analog wireless communications. Sound and vision are all such signals. Signals are converted into a form, by a transmitter, so that they can be transmitted in the air as part of electromagnetic spectrum, and are received by a receiver, where they are converted back to the original form. Two communicating parties can be quite far away from each other, and therefore the term telecommunications is used to describe this form of communications. What follows in this chapter is a descriptive theory of analog signal processing in communications. Transceivers are wireless transmitters (TX) and receivers (RX) combined in a single instrument. This book is structured around building and testing a transceiver, TRC-10, operating in the 10-meter amateur band (28-29.7 MHz). The name is generic: TRC stands for transceiver and 10 indicate that it works in 10-meter band. TRC-10 is an amplitude modulation superheterodyne transceiver. We have to make some definitions in order to understand what these terms mean. 1.1. Frequency The two variables in any electrical circuit is voltage, V, and current, I. In electronics, all signals are in form of a voltage or a current, physically. Both of these variables can be time varying or constant. Voltages and currents that do not change with respect to time are called d.c. voltages or currents, respectively. The acronym d.c. is derived from direct current. Voltages and currents that vary with respect to time can, of course, have arbitrary forms. A branch of applied mathematics called Laplace analysis, or its special form Fourier analysis, investigates the properties of such time variation, and shows that all time varying signals can be represented in terms of linear combination (or weighted sums) of sinusoidal waveforms. A sinusoidal voltage and current can be written as v(t) =V1cos(ωt+θv), and i(t) = I1cos(ωt+θi). V1 and I1 are called the amplitude of voltage and current, and have units of Volts (V) and Amperes (A), respectively. “ω” is the radial frequency with units of radians per second (rps) and ω = 2πf, where “f” is the frequency of the sinusoid with units of Hertz (Hz). “θ” is the phase angle of the waveform. These waveforms are periodic, which means that it is a repetition of a fundamental form in every T seconds, where T=1/f seconds (sec). Quite often, sinusoidal waveforms are referred to by their peak amplitudes or peak-topeak amplitudes. Peak amplitude of v(t) is V1 Volts peak (or Vp) and peak-to-peak amplitude is 2 V1 Volts peak-to-peak (or Vpp). SIGNALS AND COMMUNICATIONS ©Hayrettin Köymen, rev. 3.4/2004, All rights reserved 1-1

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ANALOG ELECTRONICS BİLKENT UNIVERSITY Now we can see that a d.c. voltage is in fact a sinusoid with f = 0 Hz. Sinusoidal voltages and currents with non-zero frequency are commonly referred to as a.c. voltages and currents. The acronym a.c. comes from alternating current. If we know the voltage v(t) across any element and current i(t) through it, we can calculate the power delivered to it as P(t) = v(t)i(t) = V1I1cos(ωt+θv) cos(ωt+θi) or P(t) = V1 I1 VI cos(θ v - θ i ) + 1 1 cos(2ωt + θ v + θ i ) . 2 2 P(t) is measured in watts (W), i.e. (1V)×(1A)=1 W. In case of a resistor, both current and voltage have the same phase and hence we can write the power delivered to a resistor as P(t) = V1 I1 V1 I1 + cos(2ωt + 2θ v ) . 2 2 We shall see that the phase difference between voltage and current in an element or a branch of circuit is a critical matter and must be carefully controlled in many aspects of electronics. P(t) is called the instantaneous power and is a function of time. We are usually interested in the average power, Pa, which is the constant part of P(t): V1 I1 cos(θ v - θ i ) , 2 in general, and Pa = Pa = V1 I1 2 in case of a resistor. We note that if the element is such that the phase difference between the voltage across and current through it is 90o, Pa is zero. Inductors and capacitors are such elements. Radio waves travel at the speed of light, c. The speed of light in air is 3.0 E8 m/sec (through out this book we shall use the scientific notation, i.e. 3.0 E8 for 3×108), to a very good approximation. This speed can be written as c = fλ SIGNALS AND COMMUNICATIONS ©Hayrettin Köymen, rev. 3.4/2004, All rights reserved 1-2

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ANALOG ELECTRONICS BİLKENT UNIVERSITY where λ is the wavelength in meters. TRC-10 emits radio waves at approximately 30 MHz (actually between 28 and 29.7 MHz). The wavelength of these waves is approximately 10 meters. The amateur frequency band in which TRC-10 operates is therefore called 10-meter band. 1.2. Oscillators Electronic circuits that generate voltages of sinusoidal waveform are called “sinusoidal oscillators”. There are also oscillators generating periodic signals of other waveforms, among which square wave generators are most popular. Square wave oscillators are predominantly used in digital circuits to produce time references, synchronization, etc. Oscillator symbol is shown in Figure 1.1. Figure 1.1 Oscillator symbol We use oscillators in communication circuits for variety of reasons. There are two oscillators in TRC-10. The first one is an oscillator that generates a signal at 16 MHz fixed frequency. This oscillator is a square wave crystal oscillator module. A square wave of 2 volts peak to peak amplitude is depicted in Figure 1.2. amplitude (V) 2 1.5 1 0.5 time 0 T/2 T 3T/2 2T Figure 1.2 Square wave In fact, such a square wave can be represented in terms of sinusoids as a linear combination: s(t) =1+(4/π)sin(ωt)+(4/3π)sin(3ωt)+(4/5π)sin(5ωt)+(4/7π)sin(7ωt)+….. ∞ = a o + ∑b n sin(nωt) n =1 where ao is the average value of s(t), 1 in this particular case, and bn= (2/nπ)[1-(-1)n] Note here that SIGNALS AND COMMUNICATIONS ©Hayrettin Köymen, rev. 3.4/2004, All rights reserved 1-3

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ANALOG ELECTRONICS BİLKENT UNIVERSITY there are an infinite number of sinusoids in a square wave; the frequencies of these sinusoids are only odd multiples of ω, which is a property of square waves with equal duration of 2’s and 0’s- we call such square waves as 50% duty cycle square waves; the amplitude of sinusoids in the summation decreases as their frequency increases. (i) (ii) (iii) We refer to the sinusoids with frequencies 2ω, 3ω, 4ω,…, nω as harmonics of the fundamental component, sinωt. We can obtain an approximation to a square wave by taking ao, fundamental, and only few harmonics into the summation. As we increase the number of harmonics in the summation, the constructed waveform becomes a better representative of square wave. This successive construction of a square wave is shown in Figure 1.3. a amplitude (V) 2 b 1.5 c 1 d 0.5 time 0 T/2 T 3T/2 2T Figure 1.3 Constructing a square wave from harmonics, (a) only ao+ fundamental, (b) waveform in (a) + 3rd harmonic, (c) waveform in (b) + 5th harmonic, (d) all terms up to 13th harmonic. Even with only 3 terms the square wave is reasonably well delineated, although its shape looks rather corrugated. A common graphical representation of a signal with many sinusoidal components is to plot the line graph of the amplitude of each component versus frequency (either f or ω). This is called the spectrum of the square wave or its frequency domain representation. Spectrum of this square wave is given in Figure 1.4, which clearly illustrates the frequency components of the square wave. Figure 1.4 clearly shows that the square wave, being a periodic signal, has energy only at discrete frequencies. We need sinusoidal voltages in TRC-10, not square waves. Indeed, we must avoid the harmonics of our signals to be emitted from our transceiver, because such harmonics will interfere with other communication systems operating at that frequency. We use this fixed frequency square wave oscillator module, because such modules provide a SIGNALS AND COMMUNICATIONS ©Hayrettin Köymen, rev. 3.4/2004, All rights reserved 1-4

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