Constant-k Filters

Constant-k filters are the oldest and simplest class of passive LC filter prototypes derived from image parameter theory. They are the foundation of classical filter design, and understanding their structure, image impedance behaviour, and cutoff frequency equations is important for network analysis courses and GATE. The constant-k type serves as the basis from which more sophisticated filter types (like m-derived sections) are developed.

Image Parameter Theory

The design of constant-k filters is based on image parameter theory. This theory analyses a two-port network by defining the image impedance at each port as the impedance seen looking into that port when the other port is terminated in its own image impedance. In other words, the image impedance is the self-consistent termination that makes the network appear to be matched at both ports simultaneously. For a symmetric T or pi network, both image impedances are equal.

The image transfer function describes how a signal is attenuated or phase-shifted as it propagates through one section of the filter when terminated in image impedances. In the passband, the image impedance is real and the attenuation is zero (ideal propagation). In the stopband, the image impedance becomes imaginary, meaning no power is transferred.

The Constant-k Condition

A constant-k filter section is defined by the condition Z1 * Z2 = k^2, where Z1 is the total series arm impedance, Z2 is the shunt arm impedance, and k is a real constant independent of frequency. This constant k has the dimensions of ohms and is equal to the characteristic resistance of the filter, which is also the image impedance at zero frequency for a low-pass section. For a low-pass constant-k filter with series inductance L and shunt capacitance C, Z1 = jwL and Z2 = 1/(jwC), so Z1 * Z2 = L/C = k^2. Therefore k = sqrt(L/C), and the filter is designed so that L/C equals the square of the desired characteristic impedance.

Cutoff Frequency and Image Impedance

The cutoff frequency of a constant-k low-pass filter is the frequency at which the image impedance transitions from real (passband) to imaginary (stopband). For the T-section LPF, the image impedance is Zi_T = k * sqrt(1 - (f/fc)^2), where fc = 1/(pi * sqrt(LC)). Below fc, Zi_T is real and positive, meaning signal passes without attenuation. Above fc, the square root argument becomes negative, Zi_T is imaginary, and no power propagates through the section.

The pi-section image impedance is Zi_pi = k / sqrt(1 - (f/fc)^2). At f = 0, Zi_pi = k. As f approaches fc, Zi_pi approaches infinity. Note the duality: Zi_T and Zi_pi are related by Zi_T * Zi_pi = k^2, confirming the constant-k relationship. A chain of identical T or pi sections cascaded together forms a filter with sharper rolloff than a single section.

T-section vs Pi-section

Both T and pi sections implement the same constant-k filter characteristic when terminated in their respective image impedances. The T-section divides the series element into two halves (L/2 in each series arm) with the shunt element (C) in the centre. The pi-section divides the shunt element into two halves (C/2 at each end) with the full series element (L) in the middle. When identical sections are cascaded, T-sections connect at the midpoints of the series arms, while pi-sections connect at the midpoints of the shunt arms. The choice between T and pi depends on the source and load termination and on how the sections are cascaded.

High Pass Constant-k Filter

A high-pass constant-k filter is obtained from the LPF prototype by applying the LP-to-HP frequency transformation: replace L with 1/(w^2 * C') and C with 1/(w^2 * L'), effectively swapping the roles of L and C. In the HPF constant-k T-section, the series arms contain capacitors (C/2 each) and the shunt arm contains an inductor (L). The cutoff frequency formula remains fc = 1/(pi * sqrt(LC)) with the same k = sqrt(L/C). The image impedance expressions are mirror images of the LPF case.

Solved Example

Given:
Design a constant-k low-pass T-section filter with:
Characteristic impedance k = R0 = 600 ohm
Cutoff frequency fc = 10 kHz

Why this formula applies:
For constant-k LPF: k = sqrt(L/C) and fc = 1/(pi*sqrt(LC))
Two equations with two unknowns L and C.

Formula:
k = sqrt(L/C)  =>  L = k^2 * C
fc = 1 / (pi * sqrt(LC))

Substitution:
From fc: sqrt(LC) = 1 / (pi * fc) = 1 / (pi * 10000) = 31.83e-6
LC = (31.83e-6)^2 = 1.013e-9

And L = k^2 * C = 360000 * C
So: 360000 * C^2 = 1.013e-9
C^2 = 2.814e-15
C = 53.05 nF

Calculation:
L = 360000 * 53.05e-9 = 19.1 mH

Verification:
k = sqrt(L/C) = sqrt(0.0191 / 53.05e-9) = sqrt(360000) = 600 ohm  (correct)
fc = 1 / (pi*sqrt(0.0191*53.05e-9)) = 1/(pi*31.83e-6) = 10,000 Hz  (correct)

Final Answer:
For T-section: each series arm L/2 = 9.55 mH, shunt capacitor C = 53.05 nF.
Characteristic impedance k = 600 ohm confirmed.

Exam Tip: For a constant-k filter, always remember two key equations: k = sqrt(L/C) and fc = 1/(pi*sqrt(LC)). These two equations with two unknowns (L and C) completely determine the filter given k and fc. GATE often asks to find L or C given the other parameters.

• The constant-k condition Z1*Z2 = k^2 holds at all frequencies. k equals the characteristic resistance and is real.
• For LPF: Z1 = jwL (series), Z2 = 1/jwC (shunt), so k = sqrt(L/C).
• Cutoff frequency fc = 1/(pi*sqrt(LC)) for both LPF and HPF constant-k filters.
• T-section splits series element into two halves (L/2 each); pi-section splits shunt element into two halves (C/2 each).
• Image impedance Zi_T = k*sqrt(1-(f/fc)^2): real and positive in passband, imaginary in stopband.
• Constant-k filters have poor stopband attenuation rate near cutoff; m-derived sections improve this.

Quick Revision

• Constant-k condition: Z1*Z2 = k^2. For LPF, k = sqrt(L/C).
• Cutoff frequency: fc = 1/(pi*sqrt(LC)). Design equations: k = sqrt(L/C) and fc = 1/(pi*sqrt(LC)).
• T-section: L/2 series, C shunt, L/2 series. Pi-section: C/2 shunt, L series, C/2 shunt.
• Image impedance Zi_T*Zi_pi = k^2. At f=0: Zi_T = Zi_pi = k.
• Passband: f < fc, image impedance real, zero attenuation. Stopband: f > fc, image impedance imaginary.
• Trap: fc for constant-k is 1/(pi*sqrt(LC)), NOT 1/(2*pi*sqrt(LC)). The pi factor (not 2*pi) comes from image parameter theory.
• HPF constant-k: series arms are capacitors C/2, shunt arm is inductor L. Same fc formula applies.