From the previous page you should now have an understanding of :

Resonance

  1. What is Resonance and where can it be found ?

  2. What is Capacitive Reactance ?

  3. What is Inductive Reactance ?

Now to keep writing Capacitive Reactance and Inductive Reactance is rather long winded so we can adopt two symbols :-

Capacitive Reactance in an AC circuit is represented by

Inductive Reactance in an AC circuits represented by

In the formulae that follow they use certain symbols :-

= frequency in Hz = Inductance in Henries = Capacitance in Farads = 3.142


Inductive Reactance

If we draw a graph of inductive reactance , against Frequency, we get :-

Note: Inductive reactance is considered to be positive.


Capacitive Reactance

If we draw a similar graph for capacitive reactance ,against frequency we get:-

Note: capacitive reactance is considered to be negative.


Syllabus Sections:-

Tuned Circuits

3i.1 Understand that at resonance XC = XL and the formula for resonant frequency.

The formula is :-

You need to be familiar with the equation but where does it come from ??

We know that at resonance in a tuned circuit the reactance of the capacitor is equal to the reactance of the inductor and the two equation associated with reactance are:-


The is necessary to maintain standard notation, for each Hz there are 2 radians (= 360o Degrees)

Now = so

= so now divide both sides by
thus and now multiply each side by
thus this time take the square root of both sides
thus the equation at the top of this section.

We may add an example here soon.

Apply the formula to find values of f, L or C from given data.

At resonance XC = XL that means that the capacitive and inductive impedance are equal we can use the following equation

Manipulating an equation can be quite daunting for some so this part will be further explained even more fully in the maths section.
From To find C this is what you have to do multiply both side by
giving then divide both sides by
giving now we need to "square" both sides to remove the "square root on the left hand side
giving which also equals

From to find L divide both sides by C
giving which we can also look at as

>




3i.2 Identify resonance curves for series and parallel tuned circuits

For this part of the syllabus all you need to remember is the shape of the resonance curves for series and parallel tuned circuits.

Series tuned circuit

A series tuned circuit has its lowest impedance at resonance hence the term acceptor circuit.

Parallel tuned circuit

A parallel tuned circuit has its highest impedance at resonance hence the term rejecter circuit. This high impedance at resonance is used for instance for "traps" in aerials to electrically shorten a dipole aerial making it operational in two bands - see section on antenna and feeders for more information.


3i.3 Understand the concept of the magnification factor Q as applied to the voltages and currents in a resonant circuit.

The concept of Q is not easy to understand, but do not be put off by the name "Q" it is not a 007 special agent.

Let's liken Q to something that you can understand. If you deposited 100 in a bank and a week later returned to with draw the money and received 200 you would say that the bank was rather good and that your money had been magnified by 2

Now if went you went back to the bank you received 1000 you would first think that the bank was fantastic and then may be that your 100 had been magnified by 10 or had a magnification factor of 10

This magnification factor is also known as Q so we could say that the Q factor of the bank was 10.

So let's go back into the world of amateur radio and consider Q and resonance.

Series Resonance

We have spoken about the reactance of the coil (inductor) which is like resistance but the coil still has "normal" resistance and for these calculations we call this and identify it in the circuit diagram as a resistor.

In this example let == 10k = 10R and with = 10mV RMS at the resonant frequency.

as and cancel each other out, so from Ohm's law =

We are now going to use a maths notation 10-3 if you do not understand this then click here.

From the figures and the equation given above we have :-

=

Thus = = 1 mA

Voltage across or = x or

Thus Voltage across or =

Thus Voltage across or = 10 volts with only an input voltage of 10mV thus there is an apparent "GAIN" in output volts.

Gain

This "gain" between the input and the output is known as magnification factor which in this case = 1000 (10mV rising to 10V = 1000 times) and thus the

magnification factor at resonance = in this case 10k / 10 = 1000

This magnification factor is given the name

Thus = or =

(Quality Factor)

Parallel Resonance

By varying the frequency of the input current, the point where = approaches and a circulating current increases to a maximum in the parallel loop limited only by . The voltage across the circuit = x .

At the same time the overall resistance to the input current rises to a maximum and whilst the current through is at a minimum.

At resonance this parallel tuned circuit is called a rejecter circuit. Take another look at the graph above.


Understand and apply the formula for Q factor given circuit components values

The formula is :-= and

From the above = and we know that thus

Also from above = and we know that thus

In all the formulae the following is true:-

= frequency in Hz = inductance in henries = capacitance in Farads = 3.142

so for a questions in the exam it is now just a matter of applying the formula to the variable given making sure that the correct multiplication factors are use as it is unlikely that you will be given the values of the variables as above but as milli henries and kHz or MHz and with the capacitors microfarads, so be prepared to do some manipulation.


Recall the definitions of the half power point and the shape factor of resonance curves.

The half power point is where the level of the response has fallen to (0.707 or 70.7%) of the maximum response or the -3dB level.

The shape factor is the ratio of the bandwidth at -60dB and -6dB

Assuming that we loosely couple a signal generator into the inductor of a parallel LC circuit and measure the voltage across with an RF voltmeter

Now let us sweep the input frequency and draw graph of the resulting voltage changes.

is the resonant frequency giving a peak voltage, .

The points either side are generally accepted as limits of usable frequency - This range to is the bandwidth [e.g.. 12kHz for speech] for a parallel LC circuit, which could be the load in a power valve RF amplifier.

Recall and apply the equation for Q given the resonant frequency and the half power points on the resonance curve.

The circuit above shows a parallel tuned circuit with an RF voltmeter, such as is used in a common form of Q meter. Q meters give a measure of "Goodness" of a component or circuit. From the above resonance chart we have

The equation for the resonant frequency and half power points is :-

=

which is also

the Bandwidth F1 & F2 compared to the resonant frequency FR = SELECTIVITY

Q of a response curve

If we had a response curve where the centre frequency FR was 10MHz and a band width of 200kHz (F1 - F2) what will be the "Q" of the circuit ?

= = =


3i.4 Understand the meaning of dynamic resistance.

This expression dynamic resistance is used in parallel tuned circuits of inductor, resistor and capacitor. When such a circuit is at its resonant frequency the tuned circuit can be represented entirely by by resistance. This resistance is called dynamic impedance or dynamic resistance. This is an apparent resistance but exist with alternating current of the resonant frequency.

The parallel resistance of a tuned circuit at resonance is x . This is known as the DYNAMIC resistance

An example of this is shown, a parallel tuned circuit in the "ANODE" of a RF amplifier.

Let = 1000 Ohms

and = 100

To tune to resonance we adjust until the DYNAMIC RESISTANCE is HIGHEST, i.e. for a max dip on the ammeter.

The coupling resistance to the next stage (Dynamic Resistance) is x = 100,000 Ohms

The series RF resistance of the coil

= = 10Ohms


Understand and apply the formula for RD given component values

The formula is :-

Thus at resonance in a parallel tuned circuit of L C & R, the dynamic resistance RD can be calculate from the formula where L (the inductance) is in Henries, C (the capacitance) is in Farads and R (the resistance) is in ohms.

The general expression for Dynamic Resistance is :-

= which does not include any frequency component.

It is obvious from this that the larger compared to , the higher will be the Dynamic Resistance and .

So we can also express the Q factor using RD in this equation


Understand the effect of damping resistors in a tuned circuit.

If we set a perfect tuned circuit into oscillation and then removed the source of the initial oscillation, it will carry on oscillating indefinitely. If then, we introduce some resistance into the circuit a little power is dissipated each cycle and the oscillation will die off exponentially. This loss of oscillation occurs whether the resistance is in parallel or in series with the tuned circuit. The resistance we have introduced is called DAMPING by damping resistor and reduces the Q-factor of a resonant circuit.

The picture above is of an actual oscilloscope trace where a tuned circuit is set in oscillation by a square wave which is superimposed on the trace just to show you where the square wave triggers the oscillation. Each sharp edge of the square wave causes the tuned circuit to oscillate and the damping effect of the imperfect tuned circuit is well shown. Note the shape of the curve of the reducing amplitude, the percentage of amplitude lost is the same for each per cycle hence the exponential curve.

NOTE: There is no such thing as a perfect tuned circuit so all circuits have some inherent damping which eventually stops their oscillation and such is the case with the tuned circuit pictured on the oscilloscope trace as it is the inherent damping of the inductor that caused the damping of the oscillations.

There are equations to calculate the equivalent series resistance caused by a parallel one but these are not required at this level.

Let's look at the real world situation an RF amplifier.

If we use a near perfect tuned circuit in an RF amplifier, once is excited it s oscillations would die out very slowly and any modulation at the amplifier input would be lost in the tuned circuit, so we "DAMP" it to the point where the oscillation dies out much faster than any modulation waveform. This damping, by the way, reduces the Q.

Another definition of Q is the ratio of Energy stored / energy lost (per cycle)

Q = Energy stored / energy lost (per cycle)


3i.5 Recall the equivalent circuit of a crystal and that it exhibits series and parallel resonance.
A piece of uncut Quartz crystal The equivalent circuit of a crystal, which is a thin slice of quartz held in a holder, is shown immediately to the left. Points A and B are representing the crystal holder.

A crystal resonator (or crystal for short) consists of a piece of "quartz" set between two conducting plates. If a voltage is applied to the plates the resultant electric field causes stresses in the crystal to occur such that it vibrates at a certain frequency that is in built due to the size and cut of the quartz.

The equivalent circuit of a crystal with discrete components is a resistor inductor and capacitor in series and another capacitor in parallel as shown in the diagram with the point "A" and "B" each side of the parallel capacitor.

The crystal has two modes of resonance :-

Accept a frequency - or low impedance as a series tuned circuit used to pass a frequency in a receiver

Block at another frequency very close frequency or hi impedance or parallel tuned circuit used in an oscillator in a transmitter.

When obtaining crystals the frequency is quoted for 32pf capacity parallel resonance.

If that same crystal is used in the series resonance mode the frequency will be slightly different.


3i.6 Recall that voltages and circulating currents in tuned circuits can be very high and understand the implications for component rating.

Currents flowing in tuned circuits can be greater than the input current due to what is called the "magnification factor" which is also known as "Q" and discussed above.

Similarly voltage too can be higher than the input voltage.

THERE ARE MANY VALUES OF INDUCTANCE AND CAPACITANCE THAT FORM A "RESONANT" CIRCUIT AND HIGH CIRCULATING CURRENT, FOR A GIVEN FREQUENCY.

IT IS THE L/C RATIO WHICH DETERMINES THE BANDWIDTH AND SELECTIVITY OF A TUNED CIRCUIT AND ITS DYNAMIC RESISTANCE.

1.THE HIGHER THE L TO C RATIO THE HIGHER THE DYNAMIC RESISTANCE THE HIGHER THE Q, ( Rd OPPOSES THE CIRCULATING CURRENT),THE NARROWER THE BANDWIDTH, THE BIGGER THE VOLTAGE.

2.THE LOWER THE L TO C RATIO THE LOWER THE Q THE WIDER THE BANDWIDTH THE LOWER THE VOLTAGE. BOTH OF THE ABOVE HAVE CIRCULATING CURRENT BUT BECAUSE THE Q AND THE DYNAMIC RESISTANCE IS LOW THE VOLTAGE ACROSS THE TUNED CIRCUIT IN 2 IS ALSO LOW

Electronic components have what is called a "rating" and it is especially important that the rating is not exceeded else catastrophic failure of the component could occur and a domino effect occur which causes more damage than to the individually component.

CONSIDER THE TRAP DIPOLE ANTENNA AND A HIGH POWER BEING TRANSMITTED. FAILURE OF THE COMPONENT CAN OCCUR IN THIS EXAMPLE FOR TWO REASONS. 1. AT RESONANCE THE VOLTAGE RATING OF THE CAPACITOR CAN BE EXCEEDED. 2. OFF RESONANCE THE CURRENT CAPACITY OF THE COIL AND CAPACITOR CAN BE EXCEEDED ( CHECK THE REACTANCE)


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