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.
  • $H(s) = H<em>1(s) \times H</em>2(s) \times … \times H_{n/2}(s)$
• 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 $f<em>0$ be the normalized cutoff frequency from the table, and $f</em>c$ be the desired cutoff frequency. Multiply the normalized coefficients with cutoff frequency: $f<em>0 = f</em>0(table) \times f_c$
• High-Pass Filter Response:
  • To obtain the actual component values for a high-pass filter, divide the cutoff frequency by the normalised coefficients.
  • Let $f<em>0$ be the normalized cutoff frequency from the table, and $f</em>c$ be the desired cutoff frequency. The normalized coefficients will be divided: $f<em>0 = \frac{f</em>c}{f_0(table)}$

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:
    • $Z<em>A = \frac{Z</em>1 Z<em>3 Z</em>5}{Z<em>2 Z</em>4}$

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 $Z<em>A = L$, then $L = \frac{R</em>1 R<em>3 R</em>5 C<em>2}{R</em>4}$
• Synthesised FDNR (Frequency-Dependent Negative Resistor):
  • By selecting appropriate components for the impedances (Z1 to Z5), a GIC can simulate an FDNR.
    • $Z<em>A = \frac{Z</em>1 Z<em>3 Z</em>5}{Z<em>2 Z</em>4}$
  • $Z<em>A = \frac{(\frac{1}{j\omega C</em>1}) R<em>3 (\frac{1}{j\omega C</em>5})}{R<em>2 R</em>4} = - \frac{1}{\omega^2 D}$
  • $D = \frac{R<em>2 R</em>4 C<em>1 C</em>5}{R_3}$

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:
  • $\frac{v<em>{out}}{v</em>{in}} = \frac{R}{R + j\omega L} = \frac{R}{R + \frac{1}{j\omega C}}$

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

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

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.
    • $C<em>{new} = \frac{1}{k</em>z R} C_{old}$
    • $R<em>{j,new} = (\frac{k</em>z}{\omega<em>c}) L</em>{j,old}$
    • $D<em>{j,new} = (\frac{1}{k</em>z \omega<em>c}) C</em>{j,old}$
• High-Pass Filter Response:
  • Direct application of grounded GIC:
    • $R<em>{new} = \frac{k</em>z}{R_{old}}$
    • $C<em>{j,new} = (\frac{1}{k</em>z \omega<em>c L</em>{j,old}})$
    • $L<em>{j,new} = (\frac{k</em>z}{\omega<em>c C</em>{j,old}})$