1. 4.2.3 Resonant filters. Following this session you should:- Recognise and sketch the characteristics for a simple band-pass filter; Be able to draw the circuit for a band pass filter based on a parallel LC circuit; Be able to select and use the formula X L =2  fL; Recall that resonance occurs in a parallel LC network when X L =X C, and hence be able to calculate the resonant frequency; Be able to select and use the formula f O  where f O is the resonant frequency  Appreciate that in practical inductors their resistance r L, has the effect of lowering the value of f O ;  Be able to select and use the formula for dynamic impedance (resistance), R D to calculate the output voltage of an unloaded filter at resonance, where R D= L r L C

2. Resonant filters cont d Know that the Q-factor is a measure of the selectivity of the band pass filter; Be able to calculate the Q-factor, either from the frequency response graph or by using component values; Be able to select and use the formulae; Q= 2  f O L and Q= f O for an unloaded circuit; r L bandwidth

3. What is resonance? A very simple mechanical analogy of resonance can be explained by the following example. Imagine a small mass is attached to the end of helical spring and pulled down slightly and released. The mass and spring will oscillate up and down with gradually decreasing amplitude. If while the mass is oscillating we give it a small downward tap each time it starts to move downwards the oscillations can be made to grow into large oscillations.

4. Examples of mechanical resonance are very common and can have devastating effects.

5. Tacoma bridge, USA

6. The Millenium Bridge, London

8. We have already encountered capacitive reactance X C and represents the magnitude of the effective opposition in ohms of a capacitor to alternating current.

9. Inductive reactance is the magnitude of the effective opposition in ohms of an inductor (coil) to alternating current. and is given by X L (  ) =2  fL where f is the frequency of the AC signal in Hz, and L the inductance of the coil in henrys.

10. When an alternating voltage is applied to a circuit containing a capacitor and a coil, the response of the circuit is a maximum when the frequency of the applied voltage equals the natural frequency or resonant frequency of the circuit. Electrical resonance occurs in the same way that a mechanical structure exhibits mechanical resonance if the forcing frequency equals the natural frequency of vibration of the structure. Eg; The Tacoma Suspension bridge. The form of the electrical ‘response’ depends on whether the capacitor and coil are connected in series or in parallel. For the purpose of the course we shall be examining resonance of a parallel lC circuit.

11. In the above circuit L is a coil, it is simply a length of wire coiled around a piece of magnetic material. Ideally it has minimal dc resistance and its inductive reactance at an ac frequency f is X L = 2  fL(ohms) Where F is the frequency of the AC supply and L is the inductance of the coil in henrys. Parallel Resonance dc resistance of the coil in ohms

12. In the circuit shown resonance occurs when the capacitive reactance X C equals the inductive reactance X L ie 2  fL = 1 2  fC f= 1 2  LC This frequency is known as the resonant frequency of the circuit and is given the symbol f O

13. At resonance the impedance( combined effect of any opposition to the AC current due to any capacitance, inductance and resistance in the circuit) is a maximum. At resonance the current is a minimum and the voltage developed across the circuit is a maximum.

14. It was stated earlier that an ideal inductor should have inductance but no dc resistance. In practice this is impossible to achieve and therefore a coil will have some dc resistance given as r L The impedance of a parallel resonant circuit is given as R D = L (ohms) Cr L This quantity is termed the dynamic impedance (resistance) of the circuit at resonance and from the equation it can be seen that the lower the dc resistance of the coil, the higher is the dynamic impedance of the parallel circuit. A high dynamic impedance is required in most practical circuits.

15. It is important to remember that this simplification can only be applied to the circuit at resonance, and when it is unloaded. In deriving the formula for R D several assumptions have been made, one of which is that r L is small, and so the formula works well for values of r L <25Ω.

16. “Q” factor or Selectivity of a resonant circuit. The frequency response of a parallel LC network, shows a maximum voltage developed across the circuit at the resonant frequency f o. This occurs because Z is a maximum and I is a minimum at resonance( for the same I at other frequencies it would be smaller due to Z being smaller). This phenomenon is made use of in radio receivers in which C or L is varied until f o equals the frequency of the wanted signal.

17. The frequency response of a parallel LC circuit shows a peak at the resonant frequency. The sharpness of the peak is described by Q, the quality factor.

18. The quality factor for an unloaded circuit is defined by:- Where f o is the resonant frequency and the bandwidth is measured between the two –3dB points (where the output is at least 0.707 of the maximum value-ie think back to the bandwidth of an Op-Amp) If a tuned circuit in a radio receiver has a very narrow peak, it is able to tune individually to stations that differ only slightly in frequency. It has a high Q and we say that the receiver is highly selective

19. From the Frequency response curve for the resonant circuits A and B it can be seen that the circuit with the higher Q factor and therefore the most selective circuit is circuit A. ie using the equation for Q for the same resonant frequency f 0 circuit B has the greater bandwidth but the lower Q factor.

20. Finally the Q-factor or selectivity of the network can also be evaluated from component values using the following equation:-

21. Ideally the transition of a signal from pass to reject should be clear cut. However the response in practice will usually be somewhat rounded in a more gradual change from pass to stop. The examples overleaf demonstrate this fact quite clearly.

23. 6.The following circuit shows a band pass filter connected to a signal generator with V IN set to 10 V. The inductor has a resistance r L of 5 Ω. V IN is kept at 10 V and the frequency increased to give the maximum value of V OUT.

24. (a) Calculate the frequency at which V OUT is a maximum.[2] …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. (b)By calculating the Dynamic Resistance R D of the filter, determine the maximum value of the voltage V OUT with V IN set to 10 V.[4] …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………… At Resonance X C =X L, 145288Hz = 1666.6 

25. (c) Determine the bandwidth of this filter.[3] …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. ………………………………………………………………………………………….

26. (d) Sketch the frequency response of the filter using the axes below. Label all important values. …………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. frequency /Hz V OUT