Study Unit 6: Filter Design - (Part 1) First Order Filters

Study Unit 6: Filter Design - (Part 1) First Order Filters

Lecture Outcomes

• Topics for today’s discussion:
  • General filter responses
  • First order filters
  • Differentiators and integrators
  • Low-pass, high-pass, and band-pass filters
  • Phase shifters

Common Filter Responses

Ideal Low-Pass Response

• Frequencies below $\omega$ are passed.
• Frequencies above $\omega$ are rejected.
• Notation:
  • Green: Filter response
  • Blue: Output components
  • Red: Rejected components
• Illustrations:
  • Input signal depicts various frequencies.
  • Filtered signal shows only frequencies below $\omega$.

Ideal High-Pass Response

• Frequencies above $\omega$ are passed.
• Frequencies below $\omega$ are rejected.
• Notation:
  • Green: Filter response
  • Blue: Output components
  • Red: Rejected components
• Illustrations:
  • Input signal contains various frequencies.
  • Filtered signal shows only frequencies above $\omega$.

Ideal Band-Pass Response

• Frequencies between $\omega<em>L$ and $\omega</em>H$ are passed.
• Frequencies below $\omega_L$ are rejected.
• Frequencies above $\omega_H$ are rejected.
• Notation:
  • Green: Filter response
  • Blue: Output components
  • Red: Rejected components
• Illustrations:
  • Input signal contains a range of frequencies.
  • Filtered signal only contains frequencies within the $\omega<em>L$ and $\omega</em>H$ range.

Ideal Notch Response

• Frequencies between $\omega<em>L$ and $\omega</em>H$ are rejected.
• Frequencies below $\omega_L$ are passed.
• Frequencies above $\omega_H$ are passed.
• Notation:
  • Green: Filter response
  • Blue: Output components
  • Red: Rejected components
• Illustrations:
  • Input signal has a spectrum of frequencies.
  • Filtered signal is missing the frequencies between $\omega<em>L$ and $\omega</em>H$.

Generic Filter Response and its Characteristics

• The ratio of the output and input is represented as: $H(s) = \frac{X<em>o}{X</em>i} = \frac{V<em>o}{V</em>i}$
• Polynomial format:
  • $H(s) = \frac{N(s)}{D(s)} = \frac{a<em>ms^m + a</em>{m-1}s^{m-1} + \cdots + a<em>1s^1 + a</em>0}{b<em>ns^n + b</em>{n-1}s^{n-1} + \cdots + b<em>1s^1 + b</em>0}$
• Rewritten to show roots (zeros and poles):
  • $H(s) = \frac{N(s)}{D(s)} = H<em>0 \frac{(s - z</em>1)(s - z<em>2)\cdots(s - z</em>m)}{(s - p<em>1)(s - p</em>2)\cdots(s - p_n)}$
• Stability:
  • Revise these concepts from previous modules (like ELI 220).

Philosophy of the Module

• Comments about the approach:
  • Balancing math with intuition and circuit understanding.
  • Balancing analysis with design.
• Extremely useful tool for intuitive reasoning: Asymptotic frequency insight:
  • Low frequencies (DC): $\lim<em>{\omega \to 0} Z</em>C = \infty$ (i.e., open circuit)
  • High frequencies: $\lim<em>{\omega \to \infty} Z</em>C = 0$ (i.e., short circuit)

The Differentiator

Circuit Analysis

• Step 1: Current equations (KCL)
  • $i<em>1 = \frac{v</em>{in} - 0}{Z_C}$
  • $i<em>2 = \frac{0 - v</em>{out}}{R}$
  • $\frac{v<em>{in}}{Z</em>C} = -\frac{v_{out}}{R}$
• Step 2: Finding the transfer function
  • $H(s) = \frac{v<em>{out}}{v</em>{in}} = -\frac{R}{Z_C} = -RCs$
• Meaning:
  • $H(j\omega) = -j\omega RC = -j(\omega/\omega<em>0) = (\omega/\omega</em>0) \angle -90^\circ$
  • $\omega_0 = \frac{1}{RC}$

The (Miller) Integrator

• Transfer function:
  • $H(s) = \frac{v<em>{out}}{v</em>{in}} = -\frac{Z_C}{R} = -\frac{1}{RCs}$
• Meaning:
  • $H(j\omega) = -\frac{1}{j\omega RC} = -\frac{1}{j(\omega/\omega<em>0)} = \frac{j}{(\omega/\omega</em>0)} = \frac{1}{(\omega/\omega_0)} \angle +90^\circ$
  • $\omega_0 = \frac{1}{RC}$

The (Deboo) Integrator

• Transfer function:
  • $H(s) = \frac{v<em>{out}}{v</em>{in}} = \frac{1}{RCs}$

Low-Pass Filter with Gain

• Transfer function:
  • $H(j\omega) = -\frac{R<em>2}{R</em>1} \frac{1}{j\omega R<em>2 C + 1} = H</em>0 \frac{1}{1 + j(\omega/\omega_0)}$
• Design equations:
  • $H<em>0 = -\frac{R</em>2}{R_1}$
  • $\omega<em>0 = \frac{1}{R</em>2 C}$

High-Pass Filter with Gain

• Transfer function:
  • $H(j\omega) = -\frac{R<em>2}{R</em>1} \frac{j\omega R<em>1 C}{j\omega R</em>1 C + 1} = H<em>0 \frac{j(\omega/\omega</em>0)}{1 + j(\omega/\omega_0)}$
• Design equations:
  • $H<em>0 = -\frac{R</em>2}{R_1}$
  • $\omega<em>0 = \frac{1}{R</em>1 C}$

Band-Pass Filter with Gain

• Transfer function:
  • $H(j\omega) = -\frac{R<em>2}{R</em>1} \frac{j\omega R<em>1 C</em>1}{j\omega R<em>1 C</em>1 + 1} \frac{1}{j\omega R<em>2 C</em>2 + 1} = H<em>0 \frac{j(\omega/\omega</em>L)}{1 + j(\omega/\omega<em>L)} \frac{1}{1 + j(\omega/\omega</em>H)}$
• Design equations:
  • $H<em>0 = -\frac{R</em>2}{R_1}$
  • $\omega<em>L = \frac{1}{R</em>1 C_1}$
  • $\omega<em>H = \frac{1}{R</em>2 C_2}$

Phase Shifter

• Transfer function:
  • $H(j\omega) = \frac{-j\omega RC + 1}{j\omega RC + 1}$
• Insight into the circuit:
  • Consider phase at opamp inputs:
    • $\angle v_+ = \angle +90^\circ$
    • $\angle v<em>- = \angle v</em>+$
  • Consider the current through the top branch:
    • $\angle i<em>{R1} \propto \angle \frac{v</em>i - v<em>-}{R</em>1} \propto \angle \frac{0^\circ - (+90^\circ)}{R_1} \propto \angle -90^\circ$