Tuned LC circuits.

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Frequency

A tuned circuit made of a coil L (unit: Henry) and a capacitor C (unit: Farad) has the following resonance frequency:

f.res = 1/(2.pi.root (L.C))

Example: a coil of 0.2 mH (0.0002 Henry) is connected to a 500 pF tuner capacitor (0.000,000,000,500 Farad).
The lowest frequency we can tune to is 503 kHz.

If we turn the value of the tuner capacitor to a value of 48.8 pF, the resonance frequency will be 1611 kHz.
Here in Europe 1611 kHz is the highest frequency on medium wave.
So, with this LC circuit we can tune over the entire MW band.
In practice both coil and detector diode will have a certain capacitance, so we must set the tuner capacitor to a lower value for tuning at 1611 kHz.
If the coil has too much capacity it is possible that we can't reach the highest frequency.
In this case we can add some space between the turns of the coil, this will reduce capacity.

Bandwidth

At resonance frequency the impedance of a parallel LC circuit will reach it's highest value, the voltage over the circuit will then also reach it's highest value.
Above and below the resonance frequency the voltage will decrease.
There are two frequency's where the voltage is 0.707 times the voltage at f.res.
One frequency is just below f.res, this is frequency fl.
And one frequency ia above f.res, this is frequency fh.

The voltage reduction to a factor 0.707 is a reduction of 3 dB
The current is there also reduced to 0.707, and so the power in the circuit is halve the power of f.res. (0.707 x 0.707 = 0.5).

The bandwidth of the tuned circuit is:   BW =fh - fl

Circuit Q

The Q of the circuit is:: Q=f.res / BW

The higher the Q, the smaller the bandwidth, which is better for separating adjacent stations.
And also, the higher the Q, the higher the voltage which the received station gives over the circuit, so a more sensitive receiver.

Curve of frequency respons for two tuned circuits. At the upper curve, the bandwidth is 2.4 kHz. The Q is 1000 / 2.4 = 416 In the lower curve the bandwidth is 5 kHz. And here the Q is 1000 / 5 = 200.

The Q of a circuit can vary from less then 100 for coils made with massive wire, up to 400 or more for coils made with litzwire.

The Q of a LC circuit will decrease when we connect a antenna or detector to it, this is because the antenna or detector gives extra parallel resistance to the circuit.
By doing this the selectivity of the circuit will reduce.

Parallelresonance

In a parallel LC circuit, the impedance will be high in resonance.
If the coil and capacitor has no losses, the impedance would even be infinite at resonance.
In practice this is not possible, there will always be losses, for instance in the resistance of the coilwire.
So, the impedance is not infinite, but has a certain value, it looks like there is a resistor connected parallel to the LC circuit, this we call the parallel resistance of the circuit "Rp".

Rp=2.pi.f.L.Q

Parallelcircuit with taps on the coil

As discribed above the parallel LC circuit has a certain parallel resistance.
For maximum sensitivity in a receiver, the load impedance (speaker or audio transformer) must have about the same value.
If we only have a speaker or transformer with a lower impedance, then we can connect the detector diode at a tap on the coil, instead of the top of the coil.
The circuit will then not be loaded too much, and the Q and selectivity will not drop.

The table below gives the value of impedance and voltage on the tap's of the coil.
Also the current is given which can be given at the tap, the current will increase at a lower tap.

The used diode must also have about the same resistance as the impedance at the tap.
Because of the lower voltage at the tap's the diode efficiency will decrease however, and this will reduce the sensitivity of the receiver.
The best option is the diode at the top of the coil, and the use of a load impedance with a high enough value.

Series resonance

In a series connected LC circuit, the impedance will be low at resonance.
If there are no losses in coil and capacitor, the impedance will be zero Ohm at resonance.
But here are also always losses, in serie resonance we keep a certain resistance Rs.
The higher the Q, the lower the series resistance Rs.

Rs=(2.pi.f.L) / Q

The Q factor of a unloaded LC circuit is determined by the following factors: series resistance in the circuit, parallel resistance across the circuit and magnetic losses.

The  series resistance in the LC circuit, 
the lower the series resistance the higher the Q.

The total series resistance in the circuit (Rs) is the sum of:

The wire resistance of the coil, thicker wire or litzwire with much strands helps to reduce wire resistance.

The resistance of the capacitor plates, silvered plates gives the lowest resistance.
Plates with oxide gives more resistance than clean plates.

Contact resistance between rotor and frame of the variable capacitor, preferable the variable capacitor has a spring connected to rotor and frame, this provides a low resistance.
When the contact is made with a slider contact at the rotor, this must be clean and free of oxide.

The parallel resistance across the LC circuit,
the higher the parallel resistance, the higher the Q.

Parallel resistance across the circuit (Rp) is caused by dielectric losses.

There are:
Dielectric losses in the coil former
Dielectric losses in the insulation of the coilwire
Dielectric losses in the insulators of the tuning capacitor
Dielectric losses in materials placed near the coil or tuning capacitor, look  here for more information about this subject.
If the capacitor plates are not clean: dielectric losses in the dirt and oxide on the capacitor plates.

Also there can be a leaking resistance in insulators, e.g. by moisture.

All these losses together makes a parallel resistance (Rp) across the LC circuit.

Magnetic losses
This occurs when a magnetic material (iron) is placed near the coil.
Non magnetic materials (plastic, wood, aluminium etc.) don't give magnetic losses.

Why is the Q factor not constant for all frequencies?

The series resistance Rs gives a reduction of Q.
If we leave other losses out of consideration, the Q will have a value of:

Q= 2.pi.f.L / Rs

If the value of Rs is constant, the value of Q will increase with increasing frequency (f).
A series resistance in the circuit will especially give a reduction of Q at lower frequencies.

On the other hand:
if we only look at the losses caused by the parallel resistance Rp, the Q will have a value of:

Q=Rp / (2.pi.f.L)

Here we see, that at a constant value of Rp, the Q will decrease with increasing frequency (f).
A parallel resistance across the circuit will especially give a reduction of Q at higher frequencies.

The value of Q will both depend on series and parallel resistance, it is possible that a LC circuit gives a increasing Q at increasing frequency (because of series resistance) and than Q will reduce (because of the parallel resistance).
In this case we have a peak in Q somewhere in the frequency band.

Often the losses caused by the parallel resistance are the highest, and we only see a reduction of Q at higher frequencies in the medium wave band.
Than there is also a peak in Q, but this occurs at a frequency lower than we use.

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