COMMUNICATIONS Formulas and Concepts

AMPLITUDE MODULATION •

AM Wave

Vm V cos(ω c − ω m )t − m cos(ω c + ω m )t 2 2 where: Vc = maximum voltage of the carrier signal Vm = maximum voltage of the original modulating signal ωc = 2πfc = frequency of the carrier signal ωm = 2πfm = frequency of the modulating signal V (t ) = Vc sin ω c t +

mt = m12 + m 22 + m32 ...... where: mt = total modulation index m1, m2, m3 = modulation index of signal having index 1, 2, 3 respectively Power Savings a. Single Sideband (SSB) P + PC PS = LSB / USB Pt b. Single Sideband full carrier (SSBFC) P PS = LSB / USB Pt

V m V max −V min = V c V max +V min where: m = modulation index Vmax = maximum peak-to-peak voltage swing of AM wave Vmin = minimum peak-to-peak voltage swing of AM wave m=

AM wave equation in terms of modulation index V (t ) = Vc sin ω c t +

mVc mVc cos(ω c − ω m )t − cos(ω c + ω m )t 2 2

•

AM Bandwidth

•

AM Power and Current V2 V2 V2 V2 Pt = carr + LSB + USB Pc = c R R R 2R 2 2 2 2 m Vc V m PLSB = PUSB = = m = Pc 8R 8R 4

BW = 2 f m where: BW = bandwidth fm = modulating signal frequency

c. Two independent Sidebands P PS = C Pt • Tuned Radio-Frequency (TRF)AM Receiver TRF Design formulas 1 fr = 2π LC f Q= r BW where: Q = quality factor fr = frequency BW = Bandwidth •

Superheterodyne Receiver f si = f s + 2 f i where: fsi = image frequency fs = signal frequency fi = intermediate frequency

2

 It  Pt m2 m2   = 1 + =1+ 2 Pc 2  Ic  where: Pt = total transmitted power Pc = unmodulated carrier power It = total transmitted current Ic = unmodulated carrier current m = modulation index

α = 1+ Q2ρ 2 f image f f f ρ = si − s = − RF fs f si f RF f image where: α = image-frequency rejection ratio(IFRR) Q = quality factor of the circuit

Note: The voltage should be in rms Amplitude Modulation with Multiple Signals  m2 Pt = Pc 1 + t  2   PDF created with pdfFactory trial version www.pdffactory.com

TELEVISION Details of Horizontal Blanking Period Time, μsec Total line (H) 63.5 H blanking 0.15H-0.18H or 9.5-11.5 H sync pulse 0.08H, or 4.75 ± 0.5 Front porch 0.02H, or 1.27 Back porch 0.06H, 3.81 Visible line time 52-54 Details of Vertical Blanking Period Time Total field (V) 1/60s 0.0167s V blanking 0.05V-0.08V or 9.5-11.5 Each V sync pulse 27.35 μs Total of 6 V sync pulse 3H = 190.5 μs Each equalizing pulse 0.04H = 2.54 μs Each serration 0.07H = 4.4 μs Visible field time 0.92V-0.95V, 0.015-0.016s

PAV = PPEAK × DR

NAVIGATIONAL AIDS Directional Gain

4π λ G dir = θ= θφ L where: θ = horizontal beam-width (radians) λ = the wavelength of the radar L = the dimension of the antenna in the direction of interest (i.e. width or height) φ = vertical beam-width (radians)

RADAR Pulse (Waveform) PRT = PW + RT 1 PW PRF = DR = PRT PRT

PAV PRT PW

where: PRT = Pulse Repetition Time PW = Pulse Width (μs) RT = Rest Time (μs) PRF = Pulse Repetition Frequency DR = Duty Cycle or Duty Ratio PAV = Average Power PPEAK = Peak Power Maximum unambiguous range PRT Runamb = c 2 Minimum displayed range PW 2 where: c = speed of light (3×108 m/s) Rmin = c

Radar Range R=4

Picture Information Encoding Y = 0.30 R + 0.59G + 0.11B I = 0.60 R − 0.28G − 0.32 B Q = 0.21R − 0.52G + 0.31B Relative amplitude for the AM RF picture signal Tip of sync = 100% Blanking level = 75% Black setup = 67.5% Maximum white = 10 to 15% or 12.5% (typical)

PPEAK =

PT AP SAO (4π ) 2 PR min

AP =

4πAO λ2 2

P A S R =4 T 2O 4πλ PR min where: R = Radar Range PT = Transmitted Power AP = antenna gain S = cross-sectional area of the target A0 = captured area of an antenna PRmin = detected signal level in W Doppler Effect

2v cos θ λ where: FD = frequency change between transmitter and reflected signal v = relative velocity between RADAR and target λ = wavelength of the transmitted wave θ = angle between target direction and RADAR system

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FD =

TRANSMISSION LINES • Electrical Characteristics Characteristic Impedance Z ZO = Y where: Z = R +jωL Ω/m Y = G +jωC S/m R G L ZO = C ZO =

R=

Ω

Also, Z O = Z SC Z OC where: ZSC = short circuit impedance ZOC = open circuit impedance

Ω

For Parallel-wire line: µ 2D H L = ln π d m πε F C= 2D m ln d where: L = Inductance C = Capacitance D = Separation between center to center d = diameter of the wire

εr ZO

× 10 −3

Characteristic Impedance, Z0 L ZO = C 120 2 D ZO = ln d εr 276 2D log d εr Note: 150Ω ≤ Z0 ≤ 600Ω ZO =

f 5d

where: a = radius (m) f = frequency (MHz) d = diameter (inches)

at high frequency, Ω

C = 1.016

f Ω a m Ω 100 − ft

R = 8.34 × 10 −8

at low frequency, Ω

Alternate formulas: L = 1.016 Z O ε r × 10 −3

Resistance, R

For coaxial line:

µ D H ln 2π d m F 2πε C= D m ln d where: D = diameter of the outer conductor d = diameter of the inner conductor L=

Alternate formulas: L = 1.016 Z O ε r × 10 −3 C = 1.016

εr ZO

× 10 −3

μH/ft

Characteristic Impedance, Z0 60 D ZO = ln d εr 138 D ZO = log d εr Note: 40Ω ≤ Z0 ≤ 150Ω

μF/ft

Resistance, R

μH/ft μF/ft

 1 1 Ω f +  D d m where: D = diameter of the outer conductor (m) d = diameter of the inner conductor (m) f = frequency (MHz) R = 8.34 × 10 −8

Ω  1 1 R = 0.1 f  +   D d  100 − ft where: D = diameter of the outer conductor (inches) d = diameter of the inner conductor (inches) f = frequency (MHz)

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Complex Propagation constant, γ γ = α + jβ = ZY where: α = attenuation constant or coefficient (Nepers/length) β = phase constant or coefficient (Radians/length)  R   dB/length α = 4.343 Z  O ω 2π β = ω LC = = radians/length λ VP 1 VP = m/s LC where: Vp = propagation velocity • Loading Conditions Note: The zero reference is at the load not on the generator. 1. ZL = Z0 (match load) with ZL = Z0 then Zin = Z0 I R = I S e − γL V R = V S e − γL

V (d ) = V + e jγd + V − e − jγd 1 I (d ) = V + e jγd − V − e − jγd ZO Loss-less transmission line where: V(d) = line voltage at point d I(d) = line current at point d Z0 = characteristic impedance of the line V+ = incident voltage V– = reflected voltage γ = complex propagation constant for lossyline β = complex propagation constant for lossless line d = distance from the load

(

Four Cases (loss-less transmission line) 1. ZL → 0 (short circuit) V (d ) = 2 jV + sin( β d ) 2V + cos(β d ) I (d ) = Z0 V (d ) Z (d ) = = jZ 0 tan( β d ) I (d ) ΓR = −1

I S = I R e γL V S = V R e γL

PR = PS e −2γL PS = PR e 2γL where: IR, IS, VR, VS = receiving and sending end current and voltages respectively PR, PS = power at the receiving and sending end γ = complex propagation constant L = length of the transmission lone Zin = input impedance ZL = load impedance 2. ZL ≠ Z0 (Mismatch) Z2 Z in = O ZL

 Z + Z O tanh γL   for L > λ/4 Z in = Z O  L  Z O + Z L tanh γL  where: Zin = the equivalent impedance representing the entire line terminated by the load

(

2. ZL → ∞ (open circuit) V (d ) = 2V + cos( βd ) 2 jV + sin( β d ) Z0 V (d ) Z (d ) = = − jZ 0 cot(β d ) I (d ) ΓR = 1 I (d ) =

3. ZL = Z0 (matched load) V (d ) = V + e jβd

for λ/4 line

Load boundary characteristics V (d ) = V + e jβd + V − e − jβd 1 I (d ) = V + e jβd − V − e − jβd ZO Loss-less transmission line

)

)

V + e jβd Z0 Z (d ) = Z 0 ΓR = 0 I (d ) =

4. ZL = jX (pure reactance) - Reactive impedance can be realized with transmission lines terminated by a short or by an open circuit.

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Z in = jZ 0 tan( β L)

-

Reflection coefficient has a unitary magnitude, as in the case of short and open circuit load.

Shorted Transmission Line – Fixed Frequency Series L=0 Z in = 0 Resonance Inductance λ Im(Z in ) > 0 0