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 === ωωω 2 • but LC C , hence the above equation becomes
ωωω2 Z OT === 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 Z Oπππ === 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