similar magnitude, because the fields generated by currents flowing in opposing direc- tions in nearby wires will mainly cancel each other out. Common-mode current field strength grows linearly with the signal frequency (20 dB per decade), while differential- mode fields grow with the square of the frequency (40 dB per decade), as long as the wavelength is longer than the cable segment of interest. Differential-mode currents can be predicted quite well from transmission-line models, but common-mode currents in a cable usually have to be measured with a current probe clamped around the cable, as the parasitic capacitances against ground causing them are difficult to estimate [126]. A transmission-line technique aimed at reducing both differential and common-mode emis- sions are balanced twisted-pair lines. Twisting the wires reduces the effective size of the loop area l · d, which is one factor in the field strength of differential-mode emissions. Driving the lines with voltages whose sum equals at all times the ground potential helps to minimize the common-mode current, as from both wires currents of equal magnitude but opposite direction will flow through the capacitance to ground.
A.4
Transmission lines and antennas
Figure A.1 shows a model circuit for an infinitely short cable segment of length dz. Here, functions R(z), L(z), G(z) and C(z) represent the total resistance, inductance, insula- tion conductivity and capacity in the cable from its start (z = 0) up to this segment. For moderate cable lengths, currents, and voltages, the resistance dR/dz and insulation conductivity dG/dz per cable length are negligibly small. For signals with a wavelength significantly longer than the cable, the relationship between voltage and current is primar- ily determined by the impedance of the load connected to the receiving end. For signals of higher frequency, however, the voltage changes at the transmitting end happen very quickly compared to the round-trip time, and the load connected to the far end loses its immediate influence on the current flowing into the cable. The current flowing into the transmission line depends, apart from the applied voltage, for signals of high frequency, primarily on the inductance dL/dz of the two conductors, and the capacitance dC/dz, which between them determine the impedance
Z = s dR + jωdL dG + jωdC ≈ r dL dC (A.25)
of the transmission line. With the approximation dR = dG = 0, the transmission line impedance is independent of the signal frequency ω/2π and the cable length l.
The propagation of a voltage waveform u(z, t) on a transmission line is described by ∂2u ∂z2 = dL dC dz2 ∂2u ∂t2 + dR dL + dG dC ∂u ∂t + dR dG dL dC u ≈ dL dCdz2 ∂2u ∂t2, (A.26)
the “telegrapher’s equation” [125, p. 54].
Solutions to the dR = dG = 0 simplification of this differential equation can be repre- sented as the sum u(z, t) = up(z, t) + ur(z, t) of two waveforms that travel along the line
with speeds
dR
dL
dC
dz
u(z,t)
i(z,t)
1/dG
Figure A.1: Model circuit for a short segment of a transmission line.
and −c, respectively. The corresponding current on the line is i(z, t) = ip(z, t) − ir(z, t) =
up(z, t)/Z − ur(z, t)/Z.
If we apply to a cable a waveform (such as a short pulse) up(0, t) at one end (z = 0), it
will travel as a voltage
up(z, t) = up(0, t − z/c) (A.28)
and current
ip(z, t) =
up(z, t)
Z (A.29)
along the line until it reaches the far end. Let’s assume the cable end (z = l) is terminated with a load of impedance Z2. If Z2 differs from the impedance Z of the cable, then the
current and voltage i(l, t) = u(l, t)/Z2 that we can measure at the far end will also differ
from the ip(l, t) = up(l, t)/Z propagated along the cable. The only way to accommodate
the difference between these two voltages in the solution space of the telegrapher’s equa- tion is the addition of a second waveform ir(z, t) = ur(z, t)/Z traveling in the opposite
direction, such that the sum of both is consistent with the voltage over the terminating load: Z2 = up(l, t) + ur(l, t) ip(l, t) − ir(l, t) = Z · up(l, t) + ur(l, t) up(l, t) − ur(l, t) ⇒ ur(l, t) up(l, t) = Z2− Z Z2+ Z (A.30)
In other words, unless the impedance of what is connected to the end of the line matches the impedance of the line (Z2 = Z), the signal will be reflected back, for example with
unchanged voltage if the end is open (Z2 = ∞ ⇒ ur(0, t) = up(0, t − 2l/c)) or inverted if
the end is a short circuit (Z2 = 0 ⇒ ur(0, t) = −up(0, t − 2l/c)). In order to avoid signal
distortions caused by such reflections at the end of transmission lines, every interface has to be designed with the same impedance as the interconnecting cable, which is by convention chosen to be 50 Ω for RF laboratory equipment such as antennas, amplifiers, receivers, and oscilloscopes. In coaxial cables and connectors, this is achieved by choosing the inner mantle diameter D and the outer core diameter d such that Z = (138 Ω/√εr)·log10D/d =
50 Ω. For polyethylene, the most common insulator material in coaxial cables, we have εr= 2.3, which results in a signal propagation speed of c = c0/√εr= 0.66 c0 [126, 74].
Antennas are interface devices that convert between the voltage and current signal propa- gated along a metallic transmission line and the electromagnetic field propagated through
150 A.4. TRANSMISSION LINES AND ANTENNAS free space. Free space has an impedance
Z0 = E/H =
r µ0
ε0 ≈ 377 Ω,
(A.31) that is, the peak electric field strength of a freely propagating electromagnetic wave mea- sured in V/m is 377 times the peak magnetic field strength measured in A/m. At least, this is the case if we are sufficiently far away from the transmitting antenna, in its “far field”. In the “near field” of a transmitting antenna, the size of which is typically less than a wavelength, the magnitude relationship between magnetic and electric fields can differ significantly from E/H = Z0.
One of the simplest narrow-band antennas for electric fields is the λ/2 dipole or free- space-resonant dipole. It consists of a metal wire, whose length is half the wavelength λ = c0/f of the intended transmission or reception frequency f . In a receiving dipole,
the electric field in the surrounding space exercises a force on its electrons. This in turn leads to a voltage and current distribution over the dipole that can be propagated into a transmission line connected to the center end points of the dipole, whose two half segments are otherwise insulated from each other.
The dipole is also an LC resonator that filters a narrow frequency band out of the spectral composition of the surrounding electric field. For optimal frequency adjustment, the dipole has to be shortened by a factor V ≈ 0.90–0.95 depending on the diameter of the wire [74, p. 57].
A receiving antenna in a field with an energy flux density of
S = E
2
377 Ω, (A.32)
where E is the root mean square of the electric field magnitude, extracts from the field the signal power
P = SAe, (A.33)
where Ae is the effective aperture of the antenna. For a λ/2 dipole, this value is
Ae=
λ2
4π · G, (A.34)
where G = 1.64 is the directional gain of a dipole compared to a (hypothetic) isotropic antenna whose gain is identical in all directions.
The effective length le of an antenna is a quantity that characterizes the voltage
U = E · le (A.35)
created at an open output connector of an antenna by an electric field strength E. Its value
le=
λ
π (A.36)
for a λ/2 dipole is somewhat shorter than the actual length of the dipole due to the non-uniform distribution of current on the dipole. In order to extract most energy from a receiving antenna, the connected amplifier must have the same impedance as the antenna, in which case it will see as its input voltage
which is half the antenna’s open-circuit voltage. [74]
An effective antenna has to transform the voltage/current relationship (impedance) be- tween the transmission line and free space, otherwise much of the signal energy would be reflected back, either into the cable in the case of a transmission antenna, or into the surrounding space in the case of a receiving antenna. The advantage of a λ/2 dipole is that its impedance at its resonance frequency is about 60 Ω, which makes it suitable for connection to a 50 Ω cable without an impedance matching transformer. Even though the near field and the current distribution can differ significantly when the same passive antenna is used for transmission or reception, the impedance and directional gain are identical in either case.