In a microwave range measurement system the electromagnetic signal v = A sin 2nft, with f = 100 MHz, is transmitted and its echo vz(1) from the target is recorded. The range is com- puted from t, the time delay between the signal and its echo. (a) Write an expression for v(1) and compute its phase angle for time delays t = 515 ns and t2 = 555 ns. (b) Can the distance be computed unambiguously from the phase angle in vz(1)? _If not, determine the additional 6.2 %3D needed information.
Electromagnetic Power Density
The power density of an electromagnetic (EM) wave is proportional to the square of the frequency of electric field (or magnetic field). The moving charge produces both electric field and magnetic field. Electric and magnetic fields are inseparable and exist in combination in EM waves.
Wave Reflection and Transmission
Reflection is defined as a sudden change in the path of the wave that strikes the boundary between two types of media. The least part of the disturbances coming from the wavelength remains the same within the same area. The reflection of the wave takes place by following everyday thinking, simple rules, at the ends of the plane. The coefficient of reflection is the ratio of the amplitude of the reflected wave to the value of the incident wave.
Skin Depth
The wire-type conductors are used in the transmission of electrical energy or signals using an alternating current. The electrons are driven apart due to the electric field created nearby the copper conductor surface. The magnetic field changes with the change in the current density. This change in the magnetic field produces an electric field that opposes the current density change. This electric field is known as back electromotive force (EMF). This force is stronger inside the copper conductor and is weaker at the surface. An alternating current can be induced due to the magnetic field according to the law of induction. Such a current is produced due to the electromagnetic wave impinging on the copper conductor.
Plane wave propagation
Any of the ways how waves travel is known as wave propagation. One can find the difference between longitudinal wave and transverse wave with respect to the direction of oscillation, which is related to the propagation direction. Propagation may occur in vacuum for electromagnetic waves as well as in material medium. Other types of waves need a transmission medium to exist and cannot propagate through the vacuum. The amplitude of a wave is the maximum distance from the displacement of the particle from its resting position on the medium. The waves that propagate are of two types, Pressure waves or Shear waves.
![In a microwave range measurement system the electromagnetic signal vị = A sin 2rft, with
f = 100 MHz, is transmitted and its echo vý(t) from the target is recorded. The range is com-
puted from t, the time delay between the signal and its echo. (a) Write an expression for v,(1)
and compute its phase angle for time delays t = 515 ns and tz = 555 ns. (b) Can the distance
be computed unambiguously from the phase angle in v,(1)? If not, determine the additional
6.2
needed information.
(a) Let vz(1) = B sin 2tf (t – t) = B sin(2rft – 6).
For f = 100 MHz = 10* Hz, 0 = 2nf t = 2 x 10°rT = 2nk +¢ where 0 < ø < 2n.
For t = 515 x 10-°, 8 = 2710* × 515 × 10-º = 1037 = 51 × 27 + ¢1 or kj = 51 and ø1 = 1.
For 1, = 555 x 10-°, 02 = 2x10* × 555 × 10-° = 1117 = 55 x 27 + ¢2 or kɔ = 55 and ø2 = .
СНAP. 6]
WAVEFORMS AND SIGNALS
117
(b) Since phase angles ø1 and ø2 are equal, the time delays 1, and t, may not be distinguished from each
other based on the corresponding phase angles ø and ø2. For unambiguous determination of the
distance, k and ø are both needed.
6.3
Show that if periods T1 and T2 of two periodic functions v1(1) and v2(1) have a common multiple,
the sum of the two functions, v(1) = v¡(t)+ v½(1), is periodic with a period equal to the smallest
common multiple of T¡ and T2. In such case show that Vavg = V1,avg +V2,avg-
%3D
If two integers n and n, can be found such that T = n¡T¡ = n,T2, then v;(1) = v;(t+n¡T¡) and
v>(t) = v2(t + n2T,). Consequently,
v(1+ T) = vj(1 + T) + vz(t + T) = v1(1) + v>(1) = v(1)
and v(1) is periodic with period T.
The average is
:| [v,(1) + v>()]dt =
+÷ v%(1) dt = V1,avg + V2.avg
V avE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F066e82e8-de44-4e45-ac6f-d99565d02ff7%2F5ce7d88c-edf6-4056-afec-0a64c48cd48e%2Frp0ffnn_processed.png&w=3840&q=75)
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