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Constant K Low Pass Filter Constant k Low pass Filter Presentation by: S.KARTHIE Assistant Professor/ECE SSN College of Engineering Objective At the end of this section students will be able to understand, • What is a constant k low pass filter section • Characteristic impedance, attenuation and phase constant of low pass filter. • Design equations of low pass filter. Low Pass Ladder Networks • A low-pass network arranged as a ladder or repetitive network. Such a network may be considered as a number of T or ∏ sections in cascade. Low Pass Ladder Networks • a T section may be taken from the ladder by removing ABED, producing the low-pass filter section shown A B L/2 L/2 L/2 L/2 D E Low Pass Ladder Networks • Similarly a ∏ -network is obtained from the ladder network as shown L L L C C C C 2 2 2 2 Constant- K Low Pass Filter • A ladder network is shown in Figure below, the elements being expressed in terms of impedances Z1 and Z 2. Z1 Z1 Z1 Z1 Z1 Z1 Z2 Z2 Z Z 2 2 Z2 Constant- K Low Pass Filter • The network shown below is equivalent one shown in the previous slide , where ( Z1/2) in series with ( Z1/2) equals Z1 and 2 Z2 in parallel with 2 Z2 equals Z2. AB F G Z1/2 Z1/2 Z1/2 Z1/2 Z1 Z1 Z1 2Z Z2 Z2 2Z 2 2Z 2 2Z 2 2Z 2 2 D E J H Constant- K Low Pass Filter • Removing sections ABED and FGJH from figure gives the T & ∏ sections which are terminated in its characteristic impedance Z OT &Z0∏ respectively. Z1 Z1 2 2 Z1 2 2 ZOT 2 oπ 2Z 2Z oπ Z ZOT Z ππ Z ππ Constant- K Low Pass Filter • We know that the relationship between characteristic impedance of T and ∏ network is Z 0T Z 0πππ === Z1Z 2 • For low pass section Z 1= jωL & Z 2=1/jωC 1 L Z 0T Z 0πππ === jωωωL ××× === jωωωC C 2 L Let R k === C Constant- K Low Pass Filter 2 • Hence, Z 0T Z 0πππ === R k • Therefore, from the above equations 2 Z 0T Z 0πππ === Z1Z 2 === R k === 'Cons tan t 'k • A ladder network composed of reactances, the series reactances being of opposite sign to the shunt reactances are called ‘constant-k’ filter sections . Constant- K Low Pass Filter • Positive (i.e. inductive) reactance is directly proportional to frequency, and negative (i.e. capacitive) reactance is inversely proportional to frequency. • Thus the product of the series and shunt reactances is independent of frequency. • The constancy of this product has given this type of filter its name . Constant- K Low Pass Filter •Z0T and Z 0∏ will either be both real or both imaginary together . Also, when Z 0T changes from real to imaginary at the cut-off frequency, so will Z0∏. • The two sections will thus have identical cutoff frequencies and thus identical passbands. • Constant-k sections of any kind of filter are known as prototypes . Constant- K Low Pass Filter Cutoff Frequency (f C): Passband 1 X Z1 • With Z1 === jωωωL & Z 2 === jωωωC Stopband • To determine cutoff frequency, the Z1 = -4Z 2 condition is f Z2 Z1 === 0 & Z1 === −−− 4Z 2 Constant- K Low Pass Filter 1 jωωωCL === −−−4 • With Z 1= - 4Z 2, jωωωCC 4 2 1 ωωωCL === ⇒⇒⇒ ωωωC === ωωωCC LC 4 4πππ 2f 2 === C LC The Cutoff frequency of the constant k Low pass filter is 1 fC === πππ LC Constant- K Low Pass Filter • We know that characteristic impedance of T and ∏ networks is Z1 Z 0T === Z1Z 2 1 +++ 4Z 2 Z1Z 2 Z 0πππ === Z1 1+++ 4Z 2 Constant- K Low Pass Filter 1 • With Z1 === jωωωL & Z 2 === jωωωC • The characteristic impedance of a T-Networks becomes ωωω2LC Z OT === R k 1 −−− ... ((()(A))) 4 4 === ωωω2 • but LC C , hence the above equation becomes ωωω2 ZOT === R k 1 −−− 2 ωωωC Constant- K Low Pass Filter • The Characteristic impedance of a T-section constant k low pas s filter is f 2 ZOT === R k 1−−− 2 fC Constant- K Low Pass Filter Characteristic impedance curve ZOT Passband Stopband Rk Imaginary Real f fC Constant- K Low Pass Filter • Similarly the characteristic impedance of a ∏ networks is R k ZOπππ === f 2 1 −−− 2 fC Constant- K Low Pass Filter Characteristic impedance curve ZO∏ Passband Stopband Real Imaginary Rk fC f Constant- K Low Pass Filter • When the frequency is very low, ω is small and the term (ω2LC/4) in equation (A) is negligible. hence, L ZOT === Rk === C • The characteristic impedance then becomes equal to L which is purely resistive . C • This value of the characteristic impedance is known as the design impedance or the nominal impedance of the section. Constant- K Low Pass Filter Attenuation & Phase shift constant: • Since Z 1 &Z 2 are of opposite type, we know that γγγ Z1 sinh === 2 4Z 2 ααα βββ jωωωL sinh +++ j === 2 2 1 4 jωωωC ααα βββ ωωω2LC ωωω2 ωωω sinh +++ j === j === j === j 2 2 2 4 ωωωC ωωωC ααα βββ f sinh +++ j ===j 2 2 fC Constant- K Low Pass Filter • In passband ααα === 0 jβββ f ∴∴∴ sinh === j 2 fC βββ f sin === 2 fC • The phase shift constant is f βββ === 2sin −−−1 fC Constant- K Low Pass Filter • In attenuation band, ααα Z1 cosh === 2 4Z 2 • With similar substitution we get, f ααα === 2cosh −−−1 fC Constant- K Low Pass Filter Phase shift constant curve βββ π fC f Constant- K Low Pass Filter Attenuation Curve α fC f Passband Stopband Constant- K Low Pass Filter Design Equations • The expression for Inductance and Capacitance is obtained using cutoff frequency . R k 1 L === C === πππfC πππR k fC 1 fC === πππ LC Constant- K Band Stop Filter Design Equations • The inductance and capacitance value of the constant k band pass filter is R k (((f 2 −−− f1 ))) 1 L1 === C1 === πππ 1ff 2 4πππR k ((()(f 2 −−− f1 ))) R k (((f 2 −−− f1 ))) L 2 === C2 === 4πππ((()(f 2 −−− f1 ))) πππR k 1ff 2 Summary • A Ladder networks consisting of opposite type impedances and which satisfy the relationship 2 Z 0T Z 0πππ === Z1Z 2 === R k === 'Cons tan t 'k is called constant k filter sections. • The characteristic impedance of a filter is purely resistive in the passband which equal to L ZOT === R k === C this value of characteristic impedance is called nominal or design impedance of the network. Summary • The cutoff frequency of a filter is obtained using relation Z1 === −−−4Z2 • The cutoff frequencies of a T and ∏ networks is same. Hence these networks are called prototypes . Thank You.
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