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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 :
Capacitive Reactance If we draw a similar graph for capacitive reactance ,against frequency we get:
Syllabus Sections: Tuned Circuits and resonance
At resonance X_{C} = X_{L} 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.
So now that you have seen how the equation has been derived from first principles what could it be used for ? Well if you wanted to know what value of a inductor is required to bring a system to resonance whether that is an antenna or trap the formula would be used if you know the all but one of the factors in the equation so you would need in this example to know the Frequency and the Capacitor's value. Knowing how to use a scientific calculator would also help and manipulation of formula.!!!  2H2 15
Recall the equivalent circuit of a crystal and that it
exhibits series and parallel resonance. 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.
A piece of uncut Quartz crystal
2H2
15 Recall that crystals are manufactured
for either series or parallel operation and will only be
stable and correct on the marked frequency when used in the
intended manner. 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.1 Understand that at resonance X_{C} = X_{L} 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:
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 circuitA series tuned circuit has its lowest impedance at resonance hence the term acceptor circuit. Parallel tuned circuitA 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.
2H4 15 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
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)
 2H4 15 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.
2H4 15 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
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.
 2H4 15 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 F_{1} & F_{2} compared to the resonant frequency F_{R }= 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 ?
= = =
2H5 15 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
2H5 15 Apply the formula for R_{D} given component values The formula is : Thus at resonance in a parallel tuned circuit of L C & R, the dynamic resistance R_{D} 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 R_{D} in this equation
2H5 15 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 Qfactor 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)




