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Decibel (Db) – 3Db Point Lab & Checkoff Logistics • Open lab with no scheduled lab sessions –lab can be done any time the lab is open. • Start early –32 students with 17 + 6 lab stations • Staffed lab hours posted. • Tools & Safety •Resonance 2 handouts • Material on cart and round table; additional • Bode Plots lecture notes parts available from parts bins near the center • Wires –theory vs reality pset1 • Amplitude Modulation columns and from EDS 38‐500. • Diodes • Lab reports should be turned in for grading. Acknowledgements: Lecture material adapted from Prof Qing Hu & Prof Jae Lim, 6.003 Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky 6.101 Spring 2020 Lecture 2 1 6.101 Spring 2020 2 Tools & Safety Decibel (dB) –3dB point • Correct tools makes life easier! V P • Understand power ratings. dB 20log o dB 10log o P Vi i • All EECS Instructional Laboratories lab kit voltages are below 50 volts peak or 50 volts DC. See staff if you are working with voltages greater than 50 volts. 100 dB = 100,000 = 105 log10(2)=.301 • All students must read and sign the safety form 80 dB = 10,000 = 104 https : / / eecs‐ug.scripts.mit.edu:444/safety/index.py/6.101 3 dB point = ? 60 dB = 1,000 = 103 half power point 40 dB = 100 = 102 6.101 Spring 2020 3 6.101 Spring 2020 Lecture 2 4 Resonance (Series RLC) –Key points Bandwidth and Q (Series RLC) R • BW (hertz) = h l • Applies to more complex RLC 2 2L + RL circuits V C - •At resonance: power is maximum •Q* (quality factor, radians) resonant frequency bandwidth 1 1 Z R sL R j(L ) •At resonance: phase angle zero, 1 L sC C Q i.e. capacitive reactance = R C inductive reactance, or impedance is real •Higher Q implies more selectivity *Agarwal/Lang Foundation of Analog Digital Elect Circuits equation 14.47, p 794 6.101 Spring 2020 Lecture 2 5 6.101 Spring 2020 Lecture 2 6 Summary – Parallel Series RLC Series Parallel Duality Parallel Series R L 1 1 + f I O fO V C RL CV 2 LC 2 LC I - 2 f L Q 2 f RC Q O o R 1 1 1 1 R V I R jwL I V jwC BW ( fh fl ) BW ( f f ) jwC 2RC h l 2L R jwL 1 Series Parallel 2 f 0 LC 0 VI 1 R1/R f0 LC 2 LC CL 6.101 Spring 2020 Lecture 2 7 6.101 Spring 2020 8 Bode Plot ‐ Review • A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. • Magnitude plot on log‐log scale – Slope: 20dB/decade, same as 6dB/octave • Bode plot provides insight into impact of RLC in frequency response. • Stable networks must always have poles and zeroes in the left‐half plane. https://web.njit.edu/~levyr/Physics_121/chapter32.ppt 6.101 Spring 2020 9 6.101 Spring 2020 Lecture 2 10 MATLAB MATLAB commands • % comment delimiter • Matlab windows: • MATLAB arrays starts with index=1 current folder, – command, work a = [4,5,6] is a row vector a(2)=5 space (workspace), – b = [7;8;9] is a column vector command history • “;” don’t print values (commandhistory) • Variables are case sensitive • Set folder to your • Variables must start with a letter Aa favorite folder • who/whos: list the current variables in short/long form • Built in help in • shg – show recent graph, pop to the front command window • use apostrophes for FILENAME • docking/undocking • format shortENG – display engineering notation Command window Workspace Current folder Command history MATLAB MATLAB Matrix Operation >> a=[2,3,4] pi 3.14159265 a = 2 3 4 i,j sqrt(-1) imaginary unit >> b=[1,0,0] b = 1 0 0 zeros(n,m) an n x m matrix of zeros >> c=a+b ones(n,m) an n x m matrix of ones c = 3 3 4 >> d=a*b % dot product operation ??? Error using ==> mtimes + - addition, subtraction Inner matrix dimensions must agree. */ ^ multiplication, division, power >> d=a.*b d = 2 0 0 sqrt square root >> e=a*b' % ' transpose e = 2 MATLAB Flow Control MATLAB example sin(x) >> t=[0:1/100:1-1/100]; % create t from 0 to .99, 100 values if a == 0 >> x=sin(2*pi*t); • if else statement b = a; >> plot(t, x); else >> stem(t,x); b = 1/a; >> shg end • for loop n = 100 for m = 1:n a(m) = a(m) + 1; end • while loop n = 10 while n > 0 n = n – 1 end Bode vs Freqs Plots MATLAB Functions bode, freqs format shortENG num=[1/L 0] denom=[1 R/L 1/(L*C)] 1 f=1/(2*pi*sqrt(L*C)) w=2*pi*f 1 • BODE(SYS,W) uses the vector W of frequencies (in radians/TimeUnit) to w_range = [.8*w:20:1.2*w]; evaluate the frequency response h=bode(num,denom,w_range); magh=abs(h); plot(w_range,magh) R=1 L=47uh C=1.8nf f=540khz • [MAG,PHASE] = BODE(SYS,W) and [MAG,PHASE,W] = BODE(SYS) return the shg freqs(num,denom,w_range) response magnitudes and phases in degrees (along with the frequency vector W if unspecified). • SYS is the transfer function expressed as numerator and denominator in the form d eS SYS aS2 bS c • bode(num,denom,range) num=[d e], denom=[a b c], range= desired freqencies in radians • freqs(num,denom,range) plots frequency response and phase angle bode freqs not same scale 6.101 Spring 2020 17 6.101 Spring 2020 Lecture 2 18 Bode in Hz Selectivity and Q 1 R=1 L=47uh C=1.8nf 1 f=540khz L=47uh C=1.8nf [Mag, Phase, W] = bode(num, denom, w_range); f=540khz Freq_Hz = W/2/pi; Mag_dB = 20*log10(Mag); subplot(2,1,1) semilogx(Freq_Hz, Mag_dB) RQ title('Bode Diagram') ylabel('Magnitude (dB)') 1 160 subplot(2,1,2) semilogx(Freq_Hz,Phase) 532 xlabel('Frequency (Hz)') ylabel('Phase (deg)') 10 16 shg 6.101 Spring 2020 Lecture 2 19 6.101 Spring 2020 Lecture 2 20 r=1 l=4.7e-5 bw1=1/(2*pi*l) Selectivity and Q c=1.8482e-009 q1=f/bw1 num=[1/l 0] bw5=5/(2*pi*l) denom=[1 r/l 1/(l*c)] q5=f/bw5 f=1/(2*pi*sqrt(l*c)) bw10=10/(2*pi*l) w=2*pi*f q10=f/bw10 w_range = [.8*w:20:1.2*w] L=47uh [w,mag1]=bode_gh(1,l,c) >> bw1=1/(2*pi*l) [w,mag5]=bode_gh(5,l,c) bw1 = 3.3863e+003 C=1.8nf [w,mag10]=bode_gh(10,l,c) >> f f=540khz hold on f = 5.4000e+005 plot(w,mag1) >> q1=f/bw1 plot(w,mag5/max(mag5),'r-') q1 = 159.4683 RQ shg >> bw5=5/(2*pi*l) plot(w,mag10/max(mag10),'g-') bw5 = 1.6931e+004 1 160 shg >> q5=f/bw5 xlabel('w(rad/sec)') q5 = 31.8937 532 shg >> bw10=10/(2*pi*l) ylabel('Amplitude') bw10 = 3.3863e+004 10 16 shg >> q10=f/bw10 title(' Response for RLC series circuit') q10 = 15.9468 shg 6.101 Spring 2020 21 6.101 Spring 2020 Lecture 2 22 Lab 1 Topics Proper External Grounding for Lab 1 IF Transformer • Resonance, Q, bandwidth • Transformers and impact on load and PC Board pri sec bandwidth NC • Diode detector, demodulation • Simple AM transmitter and receiver Can 6.101 Spring 2020 23 6.101 Spring 2020 Lecture 2 24 Schematics & Wiring Wire Gauge • IC power supply connections generally not drawn. • Wire gauge: diameter is inversely proportional to the All integrated circuits need power! wire gauge number. Diameter increases as the wire gauge decreases. 2, 1, 0, 00, 000(3/0) up to 7/0. • Use standard color coded wires to avoid confusion. – red: positive • Resistance – black: ground or common reference point – 22 gauge .0254 in 16 ohm/1000 feet – 12 gauge .08 in 1.5 ohm/1000 feet – Other colors: signals – High voltage AC used to reduce loss • Circuit flow, signal flow left to right • Higher voltage on top, ground negative voltage on • 1cm cube of copper has a resistance of 1.68 micro ohm (resistance of copper wire scales linearly : bottom length/area) • Neat wiring helps in debugging! 6.101 Spring 2020 25 6.101 Spring 2020 26 Wires Theory vs Reality ‐ Lab 1 Bypass (Decoupling) Capacitors Electrolytic Bypass capacitor Capacitor 10uf 0.1uf typical • Provides additional filtering from main power supply • Used as local energy source – provides peak current during 30-50mv voltage drop in chip transitions Wires have inductance and • Provided decoupling of resistance noise spikes during power transitions supply • Placed as close to the IC noise as possible. noise during transitions • Use small capacitors for Voltage drop across wires high frequency response. • Use large capacitors to LC ringing after transitions Through hole PCB (ancient) shown for clarity. localize bulk energy storage 6.101 Spring 2020 Lecture 2 27 6.101 Spring 2020 Lecture 2 28 Two of Many Methods of Modulation The Concept of Modulation (modulating a carrier) x(t) Transmitted Signal Carrier Signal Why? • More efficient to transmit E&M signals at higher frequencies. • Transmitting multiple signals through the same medium using different carriers. • Increase signal/noise ratio in lock‐in measurements. • others... How? • Many methods Focus is on Amplitude Modulation (AM) 6.101 Spring 2020 Lecture 2 29 6.101 Spring 2020 30 Fourier Series Time Domain Analysis 1(n1) n ? v (Ac KAm cosmt)*cosct T= 1/f1 KA v A cos t m [cos( )t cos( )t] c c 2 c m c m ? V=|Asin(ω0t)| ω0 = 2π f1 ? 6.101 Spring 2020 31 6.101 Spring 2020 Lecture 2 32 Amplitude Modulation (AM) of a Complex Asynchronous Demodulation Exponential Carrier • Assume c >> M, so signal envelope looks like x(t) j c t • Add same carrier ct e , c — carrier frequency j t y(t) x(t) e c 1 Yj Xj Cj 2 1 Xj 2 Time Domain 2 c Frequency Domain A + x(t) > 0 why? Xj c 6.101 Spring 2020 Lecture 2 33 6.101 Spring 2020 34 Lecture 2 AM with Carrier (for different Amplitudes of A) Asynchronous Demodulation (continued) xm (t) A c(t) D1 Envelop Detector xm (t) t c(t) In order for it to function properly, the envelop function must be positive definite, i.e.
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