# Transmitter And Receiver Antenna Basics

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Applications Note
Antenna Basics

6
An RF signal propagates through space at a velocity of about 300x10 meters/second, so it’s
wavelength:

300
λ=          where f is in MHz
f
At a distance greater than about 1 wavelength from the transmitter the RF signal is composed of
E and H fields which are at right angles to each other and propagate in a direction at right angles
to the E and H fields. The E and H field are related to the characteristic impedance of space just
as V and I are related to resistance.

E         µ
=       ε   = 120 π ≈ 377
H
µ = 4 π × 10 7                   ε = 8.854 ×10 −12

The RF power density and any point is given by the product of E and H:

1       1 E2
S = E× H =
2       2 377
Where E and H are the peak field values

When working with H, or E, care must be taken to know whether the peak or RMS value (often
called average) is being used. Most textbooks on antennas use peak values and most
regulations and technical articles on antennas use RMS values. In order to be consistent here,
RMS values will be used unless otherwise noted

Therefore, if the electric or magnetic field strength at any point is known, the power density can
be calculated. This will be convenient when we want to know how much power the transmitter
can put out to meet the maximum field strength limits imposed by a government agency.

A basic property of passive antennas is that they are reciprocal: They have the same gain and
pattern if they are being used as a transmitting or receiving antenna. If you had a transmitter and
receiver matched to the antenna, the results would be the same if they were interchanged.
However, the transmitted field strength may be a lot greater in one case, and it might exceed the
allowed limits. Therefore, it is common practice to put the good antenna on the receiver and the
poor one on the transmitter because it is usually much easier to increase the transmitter power
than to make the receiver more sensitive.

Antennas are often characterized in different ways, and this does not make understanding them
easier. Years ago when frequencies were low, and receivers were usually not matched to the
antenna, a term effective height was used. It was defined as:

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Voc
Heff =
E
Voc = open circuit antenna voltage         E = RMS electric field strength in volts/meter

Another term commonly used for antennas used for measuring field strength is Antenna Factor.
1
It is defined as:
E
AF =
Vout
E = electric field strength   Vout = antenna output voltage into the receiver
AF is usually given in dB to make it easier to add in factors like cable loss.

AfdB = 20log f –10 log Zo – 10log Gr – 12.79

Zo = antenna/load impedance assuming they are matched
Gr = receiving antenna gain (not in dB)
If a dipole antenna is used, Gr = 1.62, and usually Zo = 50 Ohms, so

AfdB = 20log f – 31.4

If the standard dipole antennas are constructed (ANSI standard C63.5 –1998) they can be used
with a calibrated receiver or spectrum analyzer to measure field strengths. You can also pay a lot
and get a broadband calibrated antenna to do the same thing. The advantage of this is that the
antenna does not have to be tuned for every frequency being measured.

An antenna which radiates uniformly in all directions is called an isotropic antenna. They don’t
exist, but the concept is used for calculating the gain of practical antennas. The term is also used
to set maximum signal levels for low power radios. The term ERP is used by European agencies
to set limits and is the same as the term EIRP used by radio stations. It is the equivalent
isotropic radiated power of the transmitter which would give the same field strength as the
maximum from the directional antenna. In the U.S., the limits are given in microvolts/meter, and
they can be converted to ERP for comparison.
G t Pt
S=
At the location of maximum signal from a transmitter:              4 πD 2
Where Gt = transmitter antenna gain
Pt = transmitter output power
D = distance from the transmitter

In the U.S. the FCC allows from 260 to 470MHz:

E = 41.67 × f − 7084.2          up to a maximum of 12,500 microvolts/meter at 3 meters

1
Frank Egenstafer, “Path Loss and Antenna Gain Elementary Calculations”, RF Design, Feb 1995

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Where E is the average field strength. The average term is important because the FCC
regulations allow the peak of an ASK signal to be as much as 20dB higher than the average. The
calculation is detailed in part 15.35 of the FCC regulations.

From 902 to 928 MHZ, the limit is 50mV/M at 3Meters

Since Gt for an isotropic antenna is 1.0, this translates to:
2
Pti = 0.3 * E max

Using the FCC limits, we get for ERP:

315MHz = 10.95 x10 −6 W
418MHz = 32.04 x10 − 6 W
433.92 MHz = 36.28 x10 − 6 W
915MHz = 750x10 −6 W
The proposed European standard is 5 to 25 mW depending on the application.

A practical antenna has some directivity D which is the ratio to maximum power density to the
power density which would be received from an isotropic antenna with the same power applied.
Directivity does not count losses in the antenna which become very important with small
antennas and loops. The only directivity we really need to be concerned with here are:
short dipole and small loop: D = 1.5
short monopole: D = 3.0
1/2 λ dipole: D = 1.64
1/4 λ monopole: D = 3.28

However, in order to realize the gain for the monopoles, they must be above a large (perfect)
ground plane. This is usually not the case with low power devices, so it is safer to use 1.5 for the
directivity of all antennas.

Antenna gain G is defined as D * er where er is the radiation efficiency.

e r can be calculated when the radiation resistance Rr of the antenna is known. This is where
the small size of antennas for short range devices causes a problem.
2

R r = 20π 2  
L
 
λ
For a dipole of length L:

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A monopole will be ½this

 NAµ eff 
2

For a loop antenna: R r = 31200         
 λ       
2

N = number of turns            µ e f f = effective permeability of a core if used A = area of loop

The terms N and µ e f f are included in case you want to calculate the radiation resistance for a low
frequency coil with a ferrite core. In the high frequency case, N=1 and µ e f f = 1.
The only restriction on the loop for this equation to be accurate is that the circumference be less
than 0.3 λ . This will always be the case at high frequencies because a loop larger than this will
have too much inductance to allow it to be tuned.

The conductivity of copper,    σ = 5.7 * 107 Siemens/M

2 πf µ
The surface resistance is:         Rs =
σ

R s × length
The Ohmic resistance is:       R ohmic =
π× dw

dw = loop wire diameter or equivalent diameter of copper trace

length = length of loop

For a rectangular loop of length length 1 on one side and length 2 on the other.

R ohmic = R s
(length 1 + length 2 )
w+t
w = the copper trace width in meters
t = the copper trace thickness in meters
-6
For 1oz copper, t = 34 * 10 M

Usually neglected but very significant is the equivalent series resistance of any tuning capacitors.
In most circuits, the capacitor Q is much higher than the coil Q, because the resistance of COG or
NPO capacitors is usually around 0.45 Ohms. However, R and R      r      ohmic calculated above are
usually much less than .45 Ohms, so the resistance of the capacitor is significant. High Q
capacitors with a series resistance of 0.1Ohms are available, but this is also usually higher than
Rr or Rohmic. Since the total resistance is determined mostly by the ceramic tuning capacitor, it is

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not necessary to make the loop traces wide to reduce Rohmic .

let Rloss = Rohmic + Rcap

where Rcap = the total resistance of the tuning capacitors

Rr
R r + R loss

It may be useful to calculate the Q of the loop and the tuned impedance, Rp:

2πf × L            XL
Q=                  =
(R loss + R r ) (R loss + R r )
R p = 2πf × Q

RP is the tuned parallel resistance of the loop. Both the Q and RP will be quite high and RP has
to be transformed down to a lower impedance using capacitors or taps on the loop to match it to
the transmitter IC.
At the highest power setting for the Melexis transmitter ICs, the differential output stage current is
Therefore, in order go avoid saturating the output stage, the load resistance must ≤ 200 Ohms. If
the loop impedance of a 315 MHz transmitter is around 10K, the impedance ratio is 50. The tap
ratio is the square root of the impedance ratio or 7.1. If a 4pF capacitor is needed to tune the
loop inductance calculated above, the capacitor across the output collectors as shown in the
Melexis evaluation boards should be about 7.1* 4pF = 27pF (nearest standard value). The tuning
capacitor for the loop can then be calculated: Since the series combination of the two capacitors
is 4pF, the tuning capacitor would be 4.7pF.

The output collectors can also be connected to the loop by feeding Vcc to the loop at the center
and connecting the collectors to a tap on either side of the Vcc feed. The distance can be
estimated by using the impedance ratio and then making the distance between the collectors
equal to the total loop length divided by the impedance ratio. Thus if the loop is 140mm long, and
the impedance ratio is 4.4, the distance between the collectors would be 32mm. This may not be
exact but would be a good starting point. The prototype board could be made with traces from
the collectors parallel to the loop and then shorting wires could be soldered between the loop and
the traces from the collectors. The final layout would have the connections placed at the same
location as the shorting jumpers.

The transmitted signal can now be calculated using the output power of the transmitter IC and
the gain of the transmitting antenna. Remembering from above that an antenna’s gain is the

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Rr
Gt = D
(R r + R loss )
The power density at any point at a distance R from the transmitter is:
G t Pt   E2
S=           =
4πR 2 377
Where Pt is now the power output of the transmitter IC. This assumes the transmitter is matched
to the antenna such that
Vpeak
R ant =
I peak

Vpeak I peak
Pt =
2
Therefore, the peak electric field strength at any distance D from the transmitter is:

5.48
E=          G t Pt
D

Receiving the signal

If Gr is the gain of the receiving antenna, the power received by the receiver is:
λ2
Pr =      Gr × S
4π

Where S is the power density at the receiver antenna.
This is an interesting equation, because it says that the received power decreases as the
frequency increases since the wavelength gets shorter. Why then do we usually think high
frequencies are better for longer range? This is because there are other factors which come into
play:
1. As the frequency increases, the antenna becomes smaller, so if available antenna space is
fixed, you can use bigger antennas with higher gain at higher frequencies.
2. As the wavelength decreases, propagation in buildings gets better because the shorter
wavelengths are not attenuated as much by openings such as door frames, and re-radiation by
short metal objects is more likely.
3. Regulations usually allow for higher transmitted power at higher frequencies.
Combining the receiving and transmitting equations, we get:

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2
 λ                  23.87 
2
300
Pr = Pt G t G r                      f D
 = Pt G t G r                   where       ≈ 23.87
 4πD                MHz                       4π
Where f is in MHz.

However, this equation assumes free space conditions and must be modified when the
transmitter and receiver are in buildings:

n
 λ                  23.87 
n

Pr = Pt G t G r                      f D
 = Pt G t G r        
 4πD                MHz 
2
where n = 3 to 4 for buildings depending on the type of construction.

Converting this to range:

24        Pt G t G r
D≈           n
f MHz          Pr

If you prefer working with dBm instead of watts and microvolts,

Pr ( dBm) = Pt ( dBm) + G t ( dB) + G r ( dB) − (10n ) log D meters − (10n ) log f MHz + 10n × log 23.87

An example:

Consider a 315MHz loop transmitting antenna 20mm X 50mm made out of 1oz copper with a
width of .025 inches.
-3   2
The equivalent loop area is about 1 x 10 M
-6
The equivalent wire diameter of the trace is 168 x 10 M
The loop inductance is 53.61nH, and the loop reactance is then 106 Ohms
-3
The radiation resistance is 39.44 x 10 Ohms
-3
The loss resistance due to the surface resistance of the traces is 9.93 x 10 Ohms
-3
The total resistance due to the tuning capacitor is approximately 400 x 10 Ohms

2
A.J. Nadler (from RFMD), RF Silicon Transceiver Family Targets ISM Band Low-power
Wireless Data Communication, Applied Microwave & Wireless, March 1998.

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39.44
Therefore, the radiation efficiency is:                         = .08776 = −21.1dB
(39.44 + 9.93 + 400 )
If the transmitter IC has a 5mA output current and a 1V peak swing, the maximum matched
output power is 2.5mW = +3.9Bm
-6
Therefore the equivalent isotropic transmitted power is 5mW x .096 x 1.5 = 369 x 10 W
At a distance of 3 meters, the field strength will be 34.6mV/M. The FCC limit at this frequency is
6.04mV/M, so the transmitter power will need to be reduced approximately 15dB to meet the
-6
limits.    The 6.03mV/M corresponds to an effective radiated power of            10.95 x 10 W
Alternatively, a smaller loop antenna can be used so the transmitter size can be reduced, but this
will require more transmitted power and therefore more current being drawn from the battery.

The maximum range will be determined by the allowable transmitter power and the receiver
antenna gain.     Assume the receiver has an antenna with a gain of 1 (isotropic) to be
-15
conservative. If the receiver sensitivity is –97dBm (3.0uV) or 180 x 10 W (50 Ohms), we get
from the range equation using the FCC limits converted to ERP:

315MHz range = 6.73 to 594 Meters
418MHz range = 6.63 to 766 Meters
915MHz range = 6.67 to 1693 Meters

The minimum range was calculated using n = 4 and the maximum using n = 2. The most
interesting result is that the ranges can vary so much and they are all almost equal when the
building conditions are the worst.

www.melexis.com
Contact Melexis Direct:
Europe and Japan
E-mail: sales_europe@melexis.com
Phone: 32-13-670-780
USA and rest of the world
Email: sales_usa@melexis.com
Phone: (603)-223-2362

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