Electric and Magnetic Fields

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11/24/2020

COLLEGE OF ENGINEERING School of Electrical Engineering and Computer Science

ECE 390 – Electric & Magnetic Fields L18 – Friis Equation and the Radar Range Equation

Fall 2020 T. Weller

School of EECS – ECE 390

Today’s Topics

• Communications system and link calculations • Friis equation • Radar range equation

11/24/2020

School of EECS – ECE 390

Communications System

~ R

Source (High Power) Amplifier

(Low Noise) Amplifier

Receiver

(Transmit) Antenna

(Receive) Antenna

GHPA

GLNAGTXA GRXA

PTX PRX

Gain

Gain Gain Gain

School of EECS – ECE 390

Source Power

PSRC P Watts = 1000P milliWatts

10log10(P) power in dBW (dB Watts)

30 + 10log10(P) power in dBm (dB milliWatts) 10log10(1000) 30

Recall that log10(AB) log10(A) + log10(B) 10log10(1000P) 30+ 10log10(P)

P Watts = 10PdBW/10 0 dBW = 1 W, 10 dBW = 10 W, ‐10 dBW = 0.1 W

11/24/2020

School of EECS – ECE 390

Amplifier Power

Pout PinGHPA

10log10(Pout) 10log10(PinGHPA)

Pout dBW = Pin dBW + GHPA dB

1000Pout (1000Pin)GHPA

Pout dBm = Pin dBm + GHPA dB

If the input power is 1 W, and the gain is 1000, then the output power is 1000 W. If the input power is 1 mW, and the gain is 1000, then the output power is 1000 mW. If the input power is 0 dBW, and the gain is 30 dB, then the output power is 30 dBW. (Identical to the first statement.)

School of EECS – ECE 390

Antenna • A very small antenna,

infinitesimally small, will radiate power equally in all directions – it is a point source

• Recall the Poynting vector – the power continues to spread out (lowering the power density) as the wave travels farther from the source

Surface area of sphere is 4R2

11/24/2020

School of EECS – ECE 390

Antenna • A real antenna has

some non-zero physical size and will always radiate more power in some directions than in others

• In fact, many antennas are specifically designed to focus most of the power in one direction

School of EECS – ECE 390

Antenna

• Radiation Pattern: graphical representation of the spatial variation of the antenna radiation properties, typically either the far-field electric field (field pattern) or power (power pattern)

• Directivity, D: ratio of radiation intensity in a given direction to the radiation intensity averaged over all directions – Do is the maximum directivity – For a half-wavelength dipole

Radiation pattern of an x‐ directed dipole antenna

67.10 D

11/24/2020

School of EECS – ECE 390

Antenna

• Efficiency: How effectively available incident power is converted to power delivered to the antenna load

• Gain, G: Equal to the product of Directivity and Efficiency

• Assuming high efficiency – Cell phone antenna: G = 1.6 = 2 dB – Cell tower antenna: G = 1000 = 30 dB – Large satellite antenna: G = 10000 = 40 dB

School of EECS – ECE 390 10

Propagation Channel

• Friis Transmission Equation: maximum possible received power for a transmitter-receiver separation distance of R at frequency f ( = c/f)

• Assumptions: – Impedance-matched antennas – Polarization alignment between

antennas – The absence of atmospheric loss (e.g.

rain, fog) – The absence of multipath effects – The antennas are at or beyond the far-

field distance of each other

  WattsP

GG P t

rt r 2

4 



Gt Gr

Pr R

Free Space Path Loss (FSPL)

At 2.45 GHz, a 100 W transmitter with a 20 dBi gain antenna delivers at most 10 uW to a receiver 500 m away that has a 2 dB gain antenna

 = c/f100 W = 20 dBW

11/24/2020

School of EECS – ECE 390

Link Budget Between Two Antennas

~ R

Source (High Power) Amplifier

(Low Noise) Amplifier

Receiver

(Transmit) Antenna

(Receive) Antenna

GHPA

GLNAGTXA GRXA

PTX PRX

Gain

Gain Gain Gain

Friis in dB form for this problem: PRX = PTX + GTXA + GRXA + FSPL

School of EECS – ECE 390

Radar Range Equation

11/24/2020

School of EECS – ECE 390

Radar Range Equation

~ RSource

GHPA GTXA

GRXA PTX

GLNA

PRX

Some object, e.g. Earth, with permittivity that varies with location

Variation in permittivity lead to variations in the amount of power reflected back to the radar

School of EECS – ECE 390

Radar Range Equation

• t is the radar cross section (m2) – Bird: 0.01 m2 – Human: 1 m2 – Cargo aircraft: 100 m2 – Stealth aircraft: 0.1 m2

• For a very large target (e.g. Earth) the entire incoming wave can be reflected and

• Typically, Gt = Gr (i.e. the same antenna is used for transmit and receive)

  WattsP

GG P t

trt r 22

44  



222 mRt  