Broadband Active Loop Antennas
Often, the vintage radio enthusiast is faced with the problem of providing an adequate external antenna for AM radio receivers without a build-in ferrite rod antenna. Typically, theses receivers are designed to be connected to a relatively large outdoor random wire or vertical wire antenna and also need to be provided with good grounding. Indoor random wire antennas and a counterpoise ground have been successfully used with these receivers from the 1920s to the 1950s. However they are often no longer an option because of the overwhelming man made electromagnetic noise levels in modern day buildings.
One solution is to resort to indoor magnetic loop antennas that are much less prone to picking up local electrical noise which is the prevalent component of electromagnetic noise in buildings. Magnetic loop antennas fall into two categories: Tuned loops and broadband loops. Both are typically connected to an amplifier providing an output voltage Uout at a low output impedance that is fed into the receiver using a dummy antenna that mimics the behavior of a large outdoor random wire antenna. In it's simplest incarnation, a dummy antenna is just a capacitor of 50pf to 200pF in series with a resistor of 100Ω to 300Ω.
Tuned loops are typically multi-turn loops having an inductance of around 300μH for the AM broadcast band that are complemented with a variable capacitor to a tuned circuit. The disadvantage of this design is of course the necessity to tune the antenna as well as the receiver to the desired frequency. This is where broadband loop designs intended to cover an entire broadcast band range come in.
To understand the basic design principles of broadband loop antennas, we need to look at an obvious problem: The voltage pickup of the loop from an incident electromagnetic wave is inherently frequency dependent. In fact, the voltage pickup (amplitude) U0 of the loop is given by 
where n is the number of turns of the loop, B is the amplitude of the magnetic flux density component of the incident electromagnetic wave perpendicular to the area A of the loop and ω is the frequency (in rad/s) of the incident electromagnetic wave.
If we connected a high input impedance RF amplifier to the loop, the input voltage of this RF stage would be the voltage pickup U0 of the loop, which is the open circuit voltage of the loop viewed as a generator. Hence, the following circuitry would have to be designed to provide a 1/ω frequency dependency to achieve a flat overall frequency response curve desirable with broadband active antennas. Although this is theoretically feasible a more common and also much simpler approach is the following:
Instead of using the open circuit voltage U0 of the loop antenna (viewed as a generator) as input for a high input impedance RF amplifier, we shall use it's short circuit current Isc as input for a low input impedance RF amplifier. When connecting a low input impedance RF amplifier to the terminals of the loop, the current Isc is given by
where the impedance Z is composed of the loop's inductive reactance jωL, it's series loss resistance Rs and the input resistance Rin of the RF amplifier. The radiation resistance of the loop is neglectable for all practical purposes . The absolute value of Isc is therefore given by
Provided that the series loss resistance of the loop and the RF amplifier's input impedance are small compared to the inductive reactance of the loop the above equation can by approximated by
Since the loop's series loss resistance Rs is given by the absolute value of it's inductive reactance ωL and Q-factor by  Rs=ωL/Q, Rs << ωL will hold true for all reasonable loops with a Q factor of 10 and above and our only worry remaining is whether Rin << ωL holds true.
Under these conditions Isc is a more favorable input quantity since it does not exhibit an explicit frequency dependency. Using a transimpedance amplifier  that takes the input current Isc and outputs a voltage Uout = rIsc, where r is the transimpedance of the amplifier in Ohms, we get an output voltage of
from this active broadband loop setup. Obviously, increasing the output voltage Uout of our active loop antenna can be achieved by increasing it's area A and/or increasing the transimpedance r of the amplifier. But what about the number of turns of the loop? The voltage pickup U0 of the loop increases linearly with the number of turns n. However, the inductance L of the loop typically increases with n2. Hence it immediately follows from equation (1) that a single turn loop will give the highest output voltage Uout. In this case of n=1, equation (1) then reduces to
A rule of thumb when looking at an unknown loop antenna model is therefore: Single turn loops are either broadband active loops or tuned shortwave loops (where a small inductance L is needed to reach the desired frequency range) while multi-turn loops are almost always tuned loops.
It becomes apparent that minimizing the inductance L of the loop will increase the output voltage Uout. Looking at approximations for the inductance L of circular and square single turn loops  we see that L is typically proportional to the square root of the loop's area A. In fact, this vindicates our previous assumption from equation (1) that increasing the loop's area increases Uout. Also, we see that increasing the diameter of the wire used for making the loop will decrease L and therefore increase Uout. This is why some practical active broadband loop designs employ copper pipes normally used for plumbing instead of wire.
Let us at this point come back to the original goal of the active magnetic loop antenna. We need an indoor antenna less prone to electrical noise pickup as a substitute for an outdoor random wire or vertical wire antenna. The question that arises at this point is: How does the active magnetic loop antenna perform in terms of output voltage compared to what it is supposed to be a substitute for? For random wire antennas this is a question difficult to answer. However, for an ideal vertical wire antenna over a perfectly conducting ground we can give a precise answer:
For the AM broadcast band with wavelengths approximately from 200m to 600m all practical vertical wire receiving antennas are electrically short, i.e. their physical height is much smaller than a quarter wavelength of the incident electromagnetic wave. In this case, the voltage pick-up U0 from an incident electromagnetic wave is 
where E is the amplitude of the component of the electrical field strength parallel to the wire and h is the physical length (height) of the wire. In antenna theory, the voltage pick-up in relation to the electrical field strength of the incident electromagnetic wave is usually written as
where heff is called the effective height of the antenna. Obviously in case of an electrically short vertical wire antenna over a perfect ground we have heff=h/2. Hence, in order to compare our active loop antenna to the vertical wire antenna we need to find an expression for the effective height of the active loop antenna.
Since we are in the far field of the transmitter we can assume the incident electromagnetic wave to be a plane wave and hence electrical field strength E and magnetic flux density B are related by
with c being the speed of light. Plugging this relation into equation (2) then immediately yields
and therefore we obtain the effective height heff of the active loop antenna to be
For any comparison to the vertical wire antenna, we need to plug some numerical values for a typical single turn loop into the above equation: Let's take a single turn loop with an area of 1m2 and an inductance of 5μH, leading to an effective height of approximately
It all now hinges on the transimpedance r of the amplifier. Using a simple common base bipolar junction transistor broadband amplifier stage followed by a common collector (emitter follower) stage we might get a transimpedance of around 1.5kΩ yielding a rather disappointing effective height of 1m. However, using a transimpedance amplifier employing an RF op-amp or a multi-stage transistor
broadband amplifier we can easily obtain a transimpedance of 15kΩ, giving the active broadband loop antenna an impressive effective height of heff=10m, equivalent to a vertical wire antenna of h=20m.
Let us conclude our considerations of broadband active loop antennas by looking at the problems of symmetry and ground reference. The loop antenna will be least susceptible to local electrical noise when it is symmetric and floating (i.e. has no ground reference) since almost all sources of electrical noise in buildings are ground referenced.
Hence, we require the transimpedance amplifier to have a symmetric input with no ground reference. On the other side, however, an asymmetric output that can be ground referenced is favorable. This is simply because some vintage radio receivers connected to the output via a dummy antenna might not have a symmetric input with no ground reference. Furthermore, modern day equipment such as an oscilloscope or spectrum analyzer is typically ground referenced by the electrical safety grounding.
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