Why is it
that in many Radio Frequency system and components, most of the time the
impedance is 50 ohms?( Sometimes this value is even the default
value for the PCB board ). Why not 60 ohms or 70 ohms? How is the value is
determined? What?s the meaning behind it? This article will help you to unlock
the mystery.

We know
that RF transmission needs an antenna and coaxial cable, we always hope RF
signals to travel longer distance. In order to transmit a longer distance, we
often want to transmit a signal with a larger power to cover a larger
communication range. But in fact, coaxial cable is lossless itself, just like
the wires we normally use. If the transmission power is too larger, the wires
will heat up or even fuse. So there was an expectation that we would try to
find a coaxial cable that could delivery high power with low loss.

Around 1922, Bell
Laboratory did many experiments and finally found that coaxial cable with the
characteristic impedance of 30 ohms and 77 ohms were suitable for this kind of
high-power transmissions with low loss. Among them, 30 ohms coaxial cable can
transmit the maximum power and 77 ohms coaxial cable transmission signal loss
is the minimum.

The
arithmetic mean value of 30 ohms and 77 ohms is 53.5 ohms, and the geometric
mean of 30 ohms and 70 ohms is 48 ohms. What we often call the 50 ohm system
impedance is usually an engineer compromise between 53.5 ohms and 48 ohms,
considering maximum power transmission and minimum loss as much as possible.
And though practice, the system impedance of 50 ohms is also matching with port
impedance half-wavelength dipole aerial and
quarter-wave monopole antenna, the resulting refection loss is minimal.

In our common systems,
such as TV and broadcast receiving system, the system impedance is basically 75
ohms. Because of 75 ohm Radio Frequency transmission system, signal
transmission loss is minimal. Signal transmission loss is the important
consideration in the TV and broadcast receiving system. But as to the station
with transmitters, 50 ohms is common. Because the maximum power transmission is
the main factor we consider and the loss is also important. This is why our
intercom system, we normally see the parameters is 50 ohms.

If the impedance is
match to 50 ohms, It can be mathematically rigorous. Any component, circuit, or
wires has losses in practice, and design any system component has certain Radio
Frequency bandwidth. So match to 50 ohms, engineering just to ensure that all the
brand internal frequency points fall near 50 ohms. On the Smith circle diagram,
it?s as close to the center of circle as possible. Ensure that the Radio
Frequency transmission signal in the band has no reflection loss ASAP. So as to
obtain the maximum energy transmission.

Why
so many engineers like to use 50 ohms to do the impedance transmission of PCB (
Sometimes this value is even the default value for the PCB board ), Why not 60 ohms or 70 ohms?

For a wire with a
certain width, three major factors can affect the impedance of the PCB wire.
First of all, EMI (electromagnetic interference) in the near field of PCB
routing is proportional to the height of reference plane. The lower the height,
the smaller the radiation. Secondly, the crosstalk will vary significantly with
the line height. If you reduce the height by half, the crosstalk will be
reduced to nearly a quarter. Finally, the lower the height, the smaller the
impedance, which is not easily affected by capacitive load. All three factors
allow the designer to keep the line as close to the reference as possible. What
prevents you from dropping the line height to zero is that most chips cannot
drive a transmission line with an impedance of less than 50 ohms. The
exceptions to this rule are the Rambus, which can drive 27 ohms, and National?s
BTL series, which can drive 17 ohms.

For example, the 8080
processor?s very old NMOS structure, which operates at 100KHz, has no EMI,
crosstalk, and capacitive load problems, and it cannot drive 50 ohms. For this
processor, high impedance means low power consumption, and you want to use
thin, high lines with high impedance as much as possible. The purely
perspective should also be considered. For example, in terms of density, the
distance between layers of multilayer plates is very small, and the line-width
process required for 70 ohm impedance is
difficult to achieve. In this case, you should use 50 ohms, which has a wider line width and is
easier to make.

How is the impedance of coaxial cable? In the field of Radio Frequency, the consideration are different from those in PCBS. But in the Radio Frequency industry, coaxial cable has the same impedance range. According to IEC publications(1976), 75 ohm is a common impedance standard for coaxial cables, because you can match some common antenna configurations. It also define a 50 ohm cable based on solid-state polyethylene, because the skin effect of 50 ohm impedance is minimized for external shielding with a fixed diameter and a fixed dielectric constant of 2.2(the dielectric constant of solid-state polyethylene).

You can prove that 50
ohm is the best from the basic physics. The skin effect loss L (in decibels) of
the cable is proportional to the total skin effect resistance R (unit length)
divided by the characteristic impedance Z0. The total skin effect resistance R
is the sum of the shielding layer and the intermediate conductor resistance.
The skin effect resistance of the shielding layer is inversely proportional to
its diameter d2 at high frequencies. The skin effect resistance of the inner
conductor of coaxial cable is inversely proportional to its diameter d1 at high
frequencies. The total series resistance R, therefore, is proportional to (1/d2
+1/d1). Combining these factors, given the dielectric constant ER of d2 and the
corresponding isolation material, you can use the following formula to reduce
skin loss.

In any basic books of
electromagnetic fields and microwaves, you both can find Z0 is the function of
d2, d1 and ER(Note: relative dielectric constant of insulation layer).

Put formula 2 into
formula 1, multiply numerator and denominator by d2, we can get:

Formula 3 separate the
constant term (60)/ *(1/d2), effective item ((1+d2/d1 )/ln(d2/d1 )) to
determine the minimum point. The minimal value point of formula 3 is only
controlled by d2/d1, independent of ER and fixed value d2. Taking d2/d1 as the
parameter, make a graph for L, showing that when d2/d1=3.5911(note: solve a transcendental equation), get the
minimum value. Assuming that the dielectric constant of solid polyethylene is
2.25, d2/d1=3.5911, the characteristic impedance is 51.1 ohms. Long times ago,
radio engineer used this value, approximately 50 ohms, as the optimal value for
coaxial cable for case of use. This prove that L is the smallest around 0 ohm.
But it doesn?t effect you use other impedance. For example, if you make a 75
ohm cable with the same shielding layer diameter (note: d2) and insulator(note:
ER). The skin loss will increase by 12%. For different insulators, the optimal
impedance generated by the optimal d2/d1 ratio will be slightly different
(note: for example, the air insulation corresponds to about 77 ohms, the
engineer value of 75 ohms is convenient to use).

Other additions: the
above derivation also explains why 75 ohm TV cable cut surface is lotus root
hollow core structure and 50 ohm communication cable is solid core. Another
important tip, as long as the economic situation permits, as far as possible to
choose a large outside diameter cable (note: d2), in addition to improving the
strength, the main reason is that, the larger the outside diameter, the larger
the inside diameter (the optimal diameter ratio d2/d1), of course, the
conductor of RF loss is smaller.

Why is 50 ohms the
impedance standard of RF transmission line? One of the most popular versions of
the story comes from Harmon Banning

**Impedance matching in RF circuit design**

Impedance matching is
the basic requirement of RF design and testing. Reflection of signals caused by
impedance mismatches can cause serious problems.

Matching seems like
trivial common sense when you are dealing with theoretical circuits that
consist of ideal power supplies, transmission lines, and loads.

Assume that the load
impedance ZL is fixed. We need to do is include a source impedance(ZS) equal to
ZL, and then design the transmission line so that its characteristic
impedance(ZO) is also equal to ZL.

But, let?s consider for
a moment the difficulty of implementing this solution in a complex RF(radio
frequency) circuit consisting of many passive components and integrated
circuits. If the engineer has to modify each component and specify the size of
each microstrip based on the impedance selected as the basis for all other
impedances.

In addition, this
assumes that the project has entered the PCB phase. What if we want to use
discrete modules to test and characterize the system using off-the-shelf

cables as
interconnections? In this case, compensating for mismatched impedance is more
impractical.

The solution is simple:
Choose a standard impedance that can used in many RF(radio frequency) systems
and ensure that the corresponding design components and cables, etc., haven
chosen this impedance: the industry has chosen this standard impedance in ohms
and the number is 50.

50? (ohm)

The first thing to
understand is, for 50? impedance, there is no thing special in nature. Although you may
think, if you spend enough time working with RF engineers, you will feel that
it isn?t a fundamental constant. It is not even a fundamental constant of
electrical engineer, for example, remember that simply changing the physical
size of coaxial cable changes its characteristic impedance.

In spite of this, 50
ohm is very important. Because most RF(radio frequency) systems are designed
around this impedance. It?s hard to determine exactly why 50?
become the standard of RF(radio frequency) impedance, but can reasonably
assume that found 50? under the condition of the coaxial cable of the early is a good
compromise.

Of course, the
important question isn?t the source of this particular value, but the benefit
of having this normalized impedance. A perfectly matched design is much simpler
because manufactures of IC, fixed attenuators, antennas , and so on can consider this impedance to build their
component. Furthermore, PCB layout becomes simpler because so many engineers
have the common goal of designing
microstrips and striplines with characteristic impedance of 50.

Application notes based
on Analog Devices (MT-094.pdf), You can press the following way to create 50? microstrip: 1 OZ copper, 20 mil wide go
line, line and gap between the ground plane of 10 mil (assuming the dielectric
material is RF-4).

Before we continue, we
should to understand, not each high frequency system or components are designed
for 50?. Can choose other values, in fact, 75? impedance is still very common; The characteristic of coaxial cable
is proportional to the natural logarithm of the ratio of its outer diameter
(D2) to its inner diameter(D1).

This mean that the
larger spacing between the inner and outer conductors corresponds to the higher
impedance. The larger spacing between the two conductors also result in lower
capacitance. Therefore, 75? coaxial cable capacitance is lower than 50? coaxial cable capacitance, this makes 75? cable is more suitable for high
frequency digital signal. This signal requires a low capacitance to avoid excessive attenuation of high-frequency
content associated with a rapid transition between logic low and logic high.

Reflection coefficient:

Considering the
importance of impedance matching in the RF design. We are not surprised to find
that there is a specific parameter used to indicate the matching quality,
called the reflection coefficient, the symbol for ?(Greek
capital letters gamma). It?s the ratio of the complex amplitude of the
reflected wave to the complex amplitude of the incident wave. However, the
relationship between the incident wave and the reflected wave is determined by
the source impedance (ZS) and the load impedance(ZL), so the reflection
coefficient can be defined as:

If the ?source? is the
transmission line in this case, we can change ZS to Z0 and get the reflection
coefficient as follows:

In a typical system,
the size of the reflection coefficient is some number between 0 and 1. Let?s
look at the three simplest cases mathematically to help us understand how the
reflection coefficient corresponds to the actual circuit behavior:

a. If the match is perfect ( ZL=Z0 ), the
numerator is zero, so the reflection coefficient is zero. This makes sense
because a perfect match does not cause reflection.

b. If the load impedance is infinite ( i. e. open
circuit, ZL= infinity), the reflection coefficient becomes infinity divided by
infinity, i.e., 1, and the reflection coefficient of 1 corresponds to total
reflection, i.e. all wave energy is reflected. This also make sense, because a
transmission line connected to an open circuit corresponds to a complete
discontinuity(see previous lecture). -- the load cannot absorb any energy, and
therefore must be completely reflected.

C. If the load impedance is zero (short circuit,
ZL=0), the size of the reflection coefficient becomes Z0 divided by Z0. So we
have a |?| = 1, it?s justified. Because a short circuit
also corresponds to a complete discontinuity in impedance that cannot absorb
any incident wave energy.

voltage standing
wave ratio (VSWR)

Another parameter
used to describe impedance matching is the voltage standing wave ratio (VSWR),
which is defined as follows:

From the point of
view of the obtained standing wave(VSWR), VSWR is close to impedance matching.
It conveys the ratio of the highest standing wave amplitude to the lowest
standing wave amplitude. There are many VSWR videos that help you visualize the
relationship between impedance mismatch and VSWR amplitude characteristics. The
following figure shows the VSWR characteristics of three different reflection
coefficients.

Waveform in three VSWR cases: greater impedance mismatch leads to greater
differences between the highest and lowest amplitude positions along the
standing wave.

VSWR is usually
expressed as the ratio: A perfect match would be 1:1, meaning that the peak
amplitude of the signal is always the same ( There is no standing wave). A
ratio of 2:1 indicates that the reflection has resulted in a standing wave
whose maximum amplitude is twice its minimum amplitude.

Conclusion:

1. The use of
standardized impedance makes RF design more practical and efficient.

2. Most of the
impedance of the RF system is 50?. Some systems use 75?. The
latter value is more suitable for high-speed digital signals.

3. The quality
of the impedance matching can be achieved by reflection coefficient (?) said
in math. Match exactly corresponds to ?=0, and
completely discontinuous (including all the energy is reflected) corresponding
to the ?=1.

4. Another
method of quantifying the quality of impedance matching is the voltage standing
wave ratio(VSWR).