Higher Order Filters

Study Unit 6: Filter Design - (Part 3) Higher Order Filters

Lecture Outcomes

  • Topics for today’s discussion:
    • Filter response types
    • Higher order filter design
    • Cascade design
    • Generalised impedance converters
    • Direct synthesis

Common Filter Approximations

  • Filter approximations and their characteristics:
    • Butterworth
    • Chebyshev
    • Elliptical (Cauer)
    • Bessel (Thomson)

Higher Order Filter Design: Cascaded Design Approach

  • Approach: Cascaded design is used, where the overall filter function H(s) is a product of individual filter sections.
    • H(s) = H1(s) \times H2(s) \times … \times H_{n/2}(s)
  • Tables with normalised filter coefficients are used to simplify the design process.
  • Low-Pass Filter Response:
    • To obtain the actual component values, multiply the normalised coefficients from the table with the cutoff frequency.
    • Let f0 be the normalized cutoff frequency from the table, and fc be the desired cutoff frequency. Multiply the normalized coefficients with cutoff frequency: f0 = f0(table) \times f_c
  • High-Pass Filter Response:
    • To obtain the actual component values for a high-pass filter, divide the cutoff frequency by the normalised coefficients.
    • Let f0 be the normalized cutoff frequency from the table, and fc be the desired cutoff frequency. The normalized coefficients will be divided: f0 = \frac{fc}{f_0(table)}

Generalised Impedance Converter (GIC)

  • GIC Configuration:
    • A GIC typically involves two operational amplifiers (OA1 and OA2) and five impedances (Z1, Z2, Z3, Z4, Z5) arranged in a specific configuration.
    • The effective impedance (ZA) seen by the circuit is determined by the formula:
      • ZA = \frac{Z1 Z3 Z5}{Z2 Z4}

Generalised Impedance Converter (GIC): Popular Synthesis Implementations

  • Synthesised Inductor:
    • By selecting appropriate components for the impedances (Z1 to Z5), a GIC can simulate an inductor.
    • If ZA = L, then L = \frac{R1 R3 R5 C2}{R4}
  • Synthesised FDNR (Frequency-Dependent Negative Resistor):
    • By selecting appropriate components for the impedances (Z1 to Z5), a GIC can simulate an FDNR.
      • ZA = \frac{Z1 Z3 Z5}{Z2 Z4}
    • ZA = \frac{(\frac{1}{j\omega C1}) R3 (\frac{1}{j\omega C5})}{R2 R4} = - \frac{1}{\omega^2 D}
    • D = \frac{R2 R4 C1 C5}{R_3}

Generalised Impedance Converter (GIC): Filter Design with GIC Inductor

  • Circuit configuration using a GIC to synthesize an inductor:
    • When Z_4 is a capacitor the transfer function is:
    • \frac{v{out}}{v{in}} = \frac{R}{R + j\omega L} = \frac{R}{R + \frac{1}{j\omega C}}

Generalised Impedance Converter (GIC): Filter Design with GIC FDNR

  • Circuit configuration using a GIC to synthesize an FDNR:
    • The impedance transformation is represented as:
    • L \Rightarrow \frac{D}{j \omega}
    • \frac{v{out}}{v{in}} = \frac{R}{R + j\omega L}

Higher Order Filter Design: Direct Synthesis Approach

  • Approach:
    • Use a passive RLC ladder prototype and determine the configuration, which dictates the filter approximation.
  • Tables with normalised filter coefficients are used.
  • Low-Pass Filter Response:
    • Application of the \frac{1}{j\omega} transform leads to three design equations for the new components.
      • C{new} = \frac{1}{kz R} C_{old}
      • R{j,new} = (\frac{kz}{\omegac}) L{j,old}
      • D{j,new} = (\frac{1}{kz \omegac}) C{j,old}
  • High-Pass Filter Response:
    • Direct application of grounded GIC:
      • R{new} = \frac{kz}{R_{old}}
      • C{j,new} = (\frac{1}{kz \omegac L{j,old}})
      • L{j,new} = (\frac{kz}{\omegac C{j,old}})