8.4.1. SERIES CONNECTION OF R AND C, OR R AND L

If a series connection of a resistor and a capacitor, or of a resistor and an inductor is connected across an ac-voltage they stand for two different impedances. Leaving aside that the capacitor and the inductor have always a small ohmic resistance, we can simplify the situation by looking at them at first as solely capacitive or inductive reactances Xc or X1.

Recollecting our knowledge about phase relations at L and C, we find that the voltages appearing in the circuits shown in fig. 69 must have special relations. As we know:

fig. 69

In a series connection of resistances the curent in both components is equal.

Intending to draw a phasor diagram we start therefore with the phasor of the current Itot. We know in both circuits the voltage at the resistor Vr must be exactly in phase with that current.

While the voltage at the capacitor must be lagging for 90 degrees in relation to the current and the voltage at the inductor must be leading for 90 degrees. As we know too: phasors are added geometrically.

fig. 70

Therefore the overall voltage Vtot will be found by shifting the start of Vr up to the end of Vc or V1 and by drawing a line from the noughtpoint up to the end of Vc or V1 we get the overall voltage Vtot necessary to let the current Itot flow through the circuit.

The voltages found at those components are depending on Ohm's Law, therefore Vr = I x R, Vc = I x Xc, and V1 = I x X1

These formulas demonstrate too: the relation between the voltages is equal to the relation between the reactances. In order to get an imagination of the behaviour of one of those circuits we can therefore draw instead of the voltage-triangle a triangle made up from the resistance, the reactance and showing the overall impedance.

fig. 71

This triangle shows very clearly: the impedance of the circuit can be calculated by using the old formula of PHYTHAGORAS. This combination introduced here can be used for so called PASSES.

fig. 72

If we observe how the OUTPUT VOLTAGE is changing while the frequency of the INPUT VOLTAGE is increased over a certain range we observe that the output voltage is:

- either changing from low to high values (HIGH-PASS)
- or from high values to low values (LOW-PASS)

The combination of R and C or R and L offers four different possibilities depending on where the components are positioned.

fig. 73

fig. 75a

fig. 75b

As we can see from the graphs showing the outputvoltage is not suddenly cut off totally at a special frequency, but the outputvoltage is fading out over a wide range of frequency.

We can calculate the output-voltages at various frequencies at each PASS-combination by the following formulas:

“HIGH” - Passes

“LOW” - Passes

Nevertheless for technicians it is necessary to compare different passes in relation to their ability to pass or to cut off the input signal.

Therefore there was defined a certain “LIMITTING FREQUENCY” which is considered as the frequency from which on the output-voltage is defined as “cut-off”. This limitting frequency is reached if the output-voltage is equal or lower than 70.7% of the input-signal. This limiting frequency can be calculated by the following formulas:

limit frequency with RC - combinations	limit frequency with RL - combinations

CHECK YOURSELF.

1. What does the term PASS mean 7

2. What is the difference between a HIGH - and a LOWPASS.

3. An RL Highpass should have a limitting frequency of 120 Hz. You have a coil with 150 mH. What is the resistance necessary for this pass.

4. What is the limitting frequency of a Low pass which is consisting of a resistance R=120 Ohms and a capacitor of 2 mikroFarad?

5. The limitting frequency of an amplifier should be 28 Hz. The coupling-capacitor. The coupling-capacitor has a capacity of 4.7 nF. Which resistance must have the resistor?

8.4.2. COMBINATION OF L AND C, RESONANT (TUNED) CIRCUITS

Combinations of inductors and capacitors have always a special characteristic referring to their RESPONSE to different frequencies. If we want to understand their behaviour we have two main possibilities:

FREE OSCILLATING CIRCUIT

Let us suppose a capacitor is charged with a certain voltage. As soon as this capacitor is connected across an inductor there starts to flow a current. The amount of current is slowly increasing because of the self-induced voltage across the coil.

figure

figure

figure

On the one hand this current is discharging the capacitor which lets drop the voltage of the capacitor. On the other hand the increasing current is building up a magnetic field around the coil. The current will reach its maximum just when the capacitor is discharged totally. At that very instant the voltage at the capacitor is Cero while the current is at its maximum, and therefore the magnetic field has its maximum too.

figure

There is no charge left at the capacitor, therefore the capacitor cannot deliver any current anymore. Combinations of inductors and capacitors have always a special characteristic referring to their RESPONSE to different frequencies. If we want to explain their behaviour we have two main possibilities:

FREE OSCILLATING CIRCUIT

Let us suppose a capacitor is charged with a certain voltage.

As soon as this capacitor is connected across an inductor there starts to flow a current. The amount of current is slowly increasing, because of the selfinduced voltage across the coil.

On the one hand this current is discharging the capacitor which lets drop the voltage of the capacitor. On the other hand the increasing current is building up a magnetic field around the coil.

The current will reach its maximum just when the capacitor is discharged totallyted across an inductor.

At that very instant the voltage at the capacitor is Cero while the current is at its maximum, and therefore the magnetic field has its maximum too.

There is no charge left at the capacitor, therefore the capacitor cannot deliver any current anymore. The current will have to vanish, but it will not stop to flow immediately. As soon as the current will be Cero the magnetic field must have vanished too. But before this can be the case, the magnetic field has to collapse first. The collapsing field will induce a voltage across the coil which will have a direction opposite to the voltage connected to it when the current started to flow.

This selfinduced voltage will cause a current to flow. This current will have the same direction as before, and it will charge the capacitor again but now in opposite direction.

As soon as the magnetic field has vanished totally the capacitor will be charged again to a voltage of the same amount as it was in the beginning, but in opposite direction.

Now the same process will start again, and cause a second halfwave of a sinusoidal ac-current and voltage. Summarizing: If we inject some electric energy to a parallel connection of a capacitor and an inductor there will appear an ac-voltage across the circuit with a frequency depending on the inductance and on the capacity.

But in reality these oscillations will fade out very soon, because the current flowing in this circuit to and for has to pass some obstacles. So for example the resistance of the wires forming the coil, or the resistance of the interconnecting wires. There will vanish also some of the charges stored in the capacitor by moving through the insulating dieelectricum.

All in all, after a short time we will find no more oscillations.

We can explain this effect also from another point of view:

ENERGY CONSIDERATIONS

If we look at the process explained in the last chapter from the point of view of energy, we will find that this LC combination is behaving very similar like a pendulum.

fig. 77

fig. 78

A pendulum starts with a lifted mass which means there is POTENTIAL MECHANICAL ENERGY

When released, the mass gains more and more velocity during its movement downward to the lowest point. In terms of energy: the potential energy is turned into CINETIC ENERGY.

This cinetic energy will cause the mass moving on upwards after passing the lowest point and - by moving upward again - turning the cinetic energy back into POTENTIAL MECHANICAL ENERGY.

TUNED CIRCUIT

Starts with seperated charges on the plates of the capacitor which means ELECTRIC ENERGY.

Once connected to the inductor, the capacitor starts to discharge and push current through the inductor. The current will cause a magnetic field in the inductor and - as soon as the capacitor is totally discharged - the former electric energy is turned into MAGNETIC ENERGY.

So the capacitor is free of charges now, it cannot supply any current anymore, and therefore the magnetic field starts to collapse now.

The collapsing magnetic field induces a voltage and causes the current to go on flowing as before.

This will charge the capacitor no in opposite direction as before. This effect goes on till the capacitor is charged again and the magnetic field has been turned into ELECTRIC ENERGY again.

CHECK YOURSELF:

1. Describe the construction of a tuned circuit.
2. Describe what happens in such a circuit after some energy into it.
3. Explain the similarities between pendulum and resonant circuit.
4. What is the reason for the fast vanishing of oscillations in such a circuit?

(introduction...)

SERIES TUNED CIRCUIT AT AC

If we connect an ac-voltage source across an LC combination as shown in fig. 79 This can be looked at as a series connection of two impedances connected to an ac-voltage source.

fig. 79

To simplify the problem we can ommit first the ohmic parts of the two components and concentrate on the reactances only. We know, that their reactances are depending on the frequency of the voltage they are connected to. Refering to a series connection of resistors and the rule, that the overall resistance of two resistors connected in series is the some of the two original resistances, we can easily derive that the overall impedance of that series connection will be the sum of the two original reactances.

fig. 80

If we do the addition of the two reactances by graphical means, as shown in fig. 80 we find, that the overall impedance will have high values at deep and at high frequencies, and it will have a minimum at a certain frequency. This certain frequency is called the RESONANT FREQUENCY or the TUNED FREQUENCY and it will be exactly that frequency at which the reactance of the inductor and the reactance of the capacitor will be equal. Summing up our findings we can also say: the current in this circuit will be maximum at the resonant frequency.

PARALLEL-TUNED CIRCUIT AT AC

If we want to derive, what happens in a parallel combination of an inductor and a capacitor, connected to an ac-voltage (as shown in fig. 81), we can again use the graphical method.

fig. 81

But this time we have to add the ADMITTANCES of the reactances of the two components to find the overall admittance in that circuit.

The admittances equal the reciprocal of the reactance. Adding the values of the admittances we get as the overall admittance Ytot. But we should not forget, that this represents the overall admittance, and in order to be able to compare it with our findings in the chapter before we have to turn this graph into a graph representing the overall impedance which is again the reciprocal value of the admittance. The result is shown in fig. 81. And we find that here the impedance has a peak value exactly at the so called resonant frequency.

SUMMARIZING

SERIES TUNED CIRCUITS HAVE A MINIMUM IMPEDANCE AT RESONANT FREQUENCY!

PARALLEL TUNED CIRCUITS HAVE A MAXIMUM IMPEDANCE AT RESONANT FREQUENCY!

fig. 82

THE RESONANT FREQUENCY

Up to now we do not know, how to calculate the resonant frequency, for any combination of an inductor and a capacitor. As we stated above, the resonant frequency appears if the reactance of the inductor equals the reactance of the capacitor.

Therefore it is easy to derive the formula for the resonant circuit.

8.4.4.1. QUALITY OF TUNED CIRCUITS

For radio technology tuned circuits are mainly means to filter out a special frequency from a spectrum of frequencies. Therefore radio technicians would wish it would be possible to construct tuned circuits with a graph as shown in fig. 83. A tuned circuit with such a characteristics gle frequency. But a tuned circuit in reality will never have such a characteristics. Its characteristics will always be much smoother.

fig. 83

But different combinations of different inductors and capacitors, will show different characteristics. And it is now necessary to differentiate them.

To find out the quality of a tuned circuit mostly we use a connection like shown in fig. 84. If we vary the frequency of the ac-source we will measure changing voltages at the voltmeter. This can be understood very easy if we look at the circuit as a series connection of a resistor and a tuned circuit of parallel type.

fig. 84

Recollecting the overall impedance of the parallel type we can easily predict that the current at resonant frequency will be minimum and therefore the voltage V at the Voltmeter will be maximum. If we plot these values, we find also for the resonant frequency certain resistance in the tuned circuit. This resistance is called the RESONANT RESISTANCE.

fig. 85

fig. 86

If we imagine we would connect additional resistances in parallel and we would repeat the same experiment, we can predict too, that the graphs get as flatter as lower the parallel resistance gets. A flatter graph shows that the circuit is less able to filter. The QUALITY of tuned circuits can be calculated by the following formulas:

parallel Tuned circuit: series Tuned circuit:

8.4.4.2. BANDWIDTH

Sometimes it is important to have a tuned circuit which lets through not only a single frequency, but a whole bond of frequencies. In this case it could be important to have a flatter graph. To be able to define the bandwidth of tuned circuit, there is again taken the value of 0.707 as limit. This means it the output voltage has dropped to 70.7 % of the output at fres the limit is reached.

CHECK YOURSELF

1. What is the difference of a series and a parallel tuned circuit?

2. What is the meaning of the terms: resonant frequency. Quality, bandwidth?

3. You have found in a radio a parallel connection of a capacitor of 10nF and an inductor with 100mH. What is the resonant frequency?

4. You want to built a tuned circuit for MW. You have a variable capacitor of 500pF. What inductance must the coil have? (500pF is the maximum value of the variable capacitor).