Unit 5 - Notes

ELE205

Unit 5: Passive and Active Filters

1. Introduction to Filters

A filter is a frequency-selective network that passes a specified band of frequencies with little or no attenuation (Passband) and attenuates signals of frequencies outside this band (Stopband).

Classification of Filters

Filters are classified based on several criteria:

Based on Component Type:
- Passive Filters: Constructed using R, L, and C. They require no external power supply but cause signal attenuation (insertion loss).
- Active Filters: Constructed using R, C, and active devices (Op-Amps, Transistors). They require a power supply and can provide gain.
Based on Frequency Characteristics:
- Low Pass Filter (LPF): Passes frequencies from $0$ to cut-off frequency $f_c$ .
- High Pass Filter (HPF): Passes frequencies above $f_c$ to infinity.
- Band Pass Filter (BPF): Passes a specific range ( $f_1$ to $f_2$ ).
- Band Stop/Elimination Filter (BSF/BEF): Attenuates a specific range ( $f_1$ to $f_2$ ).
Based on Relation between Series and Shunt Arms:
- Constant-K (Prototype) Filters: The product of series and shunt impedance is independent of frequency ( $Z_1 Z_2 = K^2$ ).
- m-Derived Filters: Derived from constant-k to provide sharper cut-off characteristics.

2. Symmetrical T and $\pi$ (Pi) Sections

Symmetrical networks are two-port networks where the input and output impedances are identical ( $Z_{in} = Z_{out}$ ) when terminated correctly.

T-Section Network

A T-section consists of two series arms and one shunt arm arranged in the shape of a 'T'.

Total Series Impedance: $Z_1$
Total Shunt Impedance: $Z_2$
Configuration: Two resistors/impedances of value $Z_1/2$ are in series, and one impedance $Z_2$ is in shunt between them.

$\pi$ (Pi)-Section Network

A $\pi$ -section consists of one series arm and two shunt arms arranged in the shape of a ' $\pi$ '.

Total Series Impedance: $Z_1$
Total Shunt Impedance: $Z_2$
Configuration: One impedance $Z_1$ is in series, and two impedances of value $2Z_2$ are in shunt at the input and output ports.

3. Characteristic Impedance ( $Z_0$ ) and Propagation Constant ( $\gamma$ )

For a symmetrical network composed of pure reactances (L and C, assuming negligible resistance), the behavior is defined by $Z_0$ and $\gamma$ .

Characteristic Impedance ( $Z_0$ )

The impedance measured at one pair of terminals of an infinite chain of symmetrical networks, or the input impedance when the output is terminated in $Z_0$ .

For T-Network ( $Z_{0T}$ ):

Z_{0T} = \sqrt{\frac{Z_1^2}{4} + Z_1 Z_2} = \sqrt{Z_1 Z_2 \left(1 + \frac{Z_1}{4Z_2}\right)}

For $\pi$ -Network ( $Z_{0\pi}$ ):

Z_{0\pi} = \sqrt{\frac{Z_1 Z_2}{1 + \frac{Z_1}{4Z_2}}} = \frac{Z_1 Z_2}{Z_{0T}}

Propagation Constant ( $\gamma$ )

The propagation constant defines how the amplitude and phase of a signal change as it passes through the network.

\gamma = \alpha + j\beta

$\alpha$ (Attenuation Constant): Measured in Nepers. Determines signal loss.
$\beta$ (Phase Shift): Measured in Radians. Determines phase delay.

Relation to Impedances:
For a symmetrical T or $\pi$ network:

\cosh \gamma = 1 + \frac{Z_1}{2Z_2}

\sinh \frac{\gamma}{2} = \sqrt{\frac{Z_1}{4Z_2}}

Analysis of Pure Reactive Networks

In a filter composed purely of L and C, $Z_1$ and $Z_2$ are imaginary ( $jX$ ). Their ratio $Z_1/4Z_2$ is a real number.

1. Passband Condition ( $\alpha = 0$ ):
For the signal to pass without attenuation, $\gamma$ must be purely imaginary ( $\gamma = j\beta$ ). This occurs when:

-1 < \frac{Z_1}{4Z_2} < 0

In this region, the characteristic impedance

Z_0

is purely resistive (Real), allowing power transfer.

2. Stopband Condition ( $\alpha \neq 0$ ):
Attenuation occurs when:

\frac{Z_1}{4Z_2} < -1 \quad \text{or} \quad \frac{Z_1}{4Z_2} > 0

In this region,

Z_0

is purely reactive (Imaginary), meaning the network reflects power rather than absorbing/transmitting it.

4. Design of Constant-K Filters

A Constant-K filter is a basic passive filter where the product of series impedance ( $Z_1$ ) and shunt impedance ( $Z_2$ ) is a constant real number, denoted by $k^2$ or $R_0^2$ , independent of frequency.

Z_1 Z_2 = R_0^2 = \frac{L}{C} = k^2

Here,

R_0

is the design impedance (nominal load resistance).

Low Pass Filter (Constant-K)

Structure: Inductors in series ( $Z_1 = j\omega L$ ), Capacitor in shunt ( $Z_2 = 1/j\omega C$ ).
Cut-off Frequency ( $f_c$ ): The frequency where the passband ends.
$f_c = \frac{1}{\pi \sqrt{LC}}$
Design Formulas:
Given the Load Resistance $R_0$ and Cut-off frequency $f_c$ :
$L = \frac{R_0}{\pi f_c}$
$C = \frac{1}{\pi R_0 f_c}$

High Pass Filter (Constant-K)

Structure: Capacitor in series ( $Z_1 = 1/j\omega C$ ), Inductor in shunt ( $Z_2 = j\omega L$ ).
Cut-off Frequency ( $f_c$ ):
$f_c = \frac{1}{4\pi \sqrt{LC}}$
Design Formulas:
Given $R_0$ and $f_c$ :
$C = \frac{1}{4\pi R_0 f_c}$
$L = \frac{R_0}{4\pi f_c}$

Limitations of Constant-K Filters

Slow Attenuation: The slope of attenuation in the stopband is gradual, not sharp.
Impedance Mismatch: $Z_0$ varies with frequency in the passband. It equals $R_0$ only at specific frequencies, leading to reflections at other frequencies.

5. m-Derived Filters

To overcome the slow attenuation of Constant-K filters, m-derived filters are used. They modify the $Z_1$ or $Z_2$ elements by a factor $m$ (where $0 < m < 1$ ).

Features

Sharp Cut-off: Provides infinite attenuation at a specific resonance frequency $f_\infty$ strictly close to $f_c$ .
Impedance Matching: Characteristic impedance remains consistent with the Constant-K prototype (allowing them to be cascaded).

Types

Series m-derived:
- The series arm is $m Z_1$ .
- The shunt arm consists of a series resonant circuit (part of original $Z_2$ and $Z_1$ ).
Shunt m-derived:
- The shunt arm is $Z_2 / m$ .
- The series arm is a parallel resonant circuit.

Relation between $m$ , $f_c$ , and $f_\infty$

For a Low Pass Filter:

m = \sqrt{1 - \left(\frac{f_c}{f_\infty}\right)^2}

where

f_\infty

is the frequency of infinite attenuation.

6. Composite Filters and Applications

A single filter section (Constant-K or m-derived) rarely satisfies all design requirements (sharp cut-off AND constant impedance). A Composite Filter is a cascade of different filter sections designed to have the same characteristic impedance $R_0$ .

Structure of a Composite Filter

It typically consists of three distinct stages connected in series:

Internal Sections (Sharp Cut-off): m-derived sections (with low $m$ , typically $m \approx 0.3$ ) to provide sharp attenuation immediately after $f_c$ .
Passband Flatness: Constant-K sections to provide high attenuation far from $f_c$ .
Terminating Half-Sections: Bisected $\pi$ -sections with $m=0.6$ . These are used at the input and output because an m-derived section with $m=0.6$ maintains a nearly constant characteristic impedance over the entire passband, ensuring excellent matching with the load.

Applications

Telecommunications (channel selection).
Audio processing (crossover networks).
Radio receivers (Intermediate Frequency filtering).

7. Active Filters

Active filters utilize active components like Operational Amplifiers (Op-Amps) combined with Resistors and Capacitors (RC networks). Inductors are generally avoided.

Why avoid Inductors?

Inductors are bulky, heavy, and expensive.
They have internal resistance causing power loss.
They are susceptible to electromagnetic interference.
Difficult to fabricate in Integrated Circuits (ICs).

8. Comparison: Active vs. Passive Filters

Feature	Passive Filters	Active Filters
Components	R, L, C	Op-Amp, R, C (No Inductors)
Gain	Always $\le 1$ (Attenuation)	Can provide Gain ( $> 1$ )
Power Source	Not required	Required
Loading Effect	Affected by load impedance	Negligible (High $Z_{in}$ , Low $Z_{out}$ )
Frequency Range	Very high (RF/Microwave)	Audio to low MHz (limited by Op-Amp bandwidth)
Cost & Size	Bulky (due to L)	Small, lightweight, economical
Design Flexibility	Difficult to tune	Easy to tune

9. Advantages of Active Filters

Gain Availability: The Op-Amp can amplify the input signal, eliminating insertion loss.
Isolation (Buffering): High input impedance and low output impedance prevent the filter from loading the source or being affected by the load.
Size and Weight: Absence of inductors makes them suitable for miniaturization and integration on chips.
Tunability: Frequency characteristics can be easily adjusted by varying a resistor or capacitor.

10. Designing First-Order Active Filters

First-order filters have a roll-off rate of 20 dB/decade (6 dB/octave). They contain a single reactive element (Capacitor) in the RC network.

A. First-Order Active Low Pass Filter

Uses a non-inverting Op-Amp configuration. An RC circuit is placed at the non-inverting terminal.

Circuit Configuration:

Input signal $V_{in}$ connects to a Resistor $R$ .
The other end of $R$ connects to the Op-Amp Non-Inverting input (+) and a Capacitor $C$ .
The Capacitor $C$ connects to Ground.
Feedback resistors $R_f$ and $R_1$ set the passband gain.

Design Equations:

Cut-off Frequency ( $f_c$ ):
$f_c = \frac{1}{2\pi RC}$
$\text{Or, } \omega_c = \frac{1}{RC}$
Passband Gain ( $A_F$ ):
$A_F = 1 + \frac{R_f}{R_1}$
Transfer Function:
$\frac{V_{out}}{V_{in}} = \frac{A_F}{1 + j(\frac{f}{f_c})}$

B. First-Order Active High Pass Filter

The positions of R and C are swapped compared to the LPF at the input.