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 \omegaL and \omegaH 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 \omegaL and \omegaH range.

Ideal Notch Response

• Frequencies between \omegaL and \omegaH 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 \omegaL and \omegaH.

Generic Filter Response and its Characteristics

• The ratio of the output and input is represented as: H(s) = \frac{Xo}{Xi} = \frac{Vo}{Vi}
• Polynomial format:
  • H(s) = \frac{N(s)}{D(s)} = \frac{ams^m + a{m-1}s^{m-1} + \cdots + a1s^1 + a0}{bns^n + b{n-1}s^{n-1} + \cdots + b1s^1 + b0}
• Rewritten to show roots (zeros and poles):
  • H(s) = \frac{N(s)}{D(s)} = H0 \frac{(s - z1)(s - z2)\cdots(s - zm)}{(s - p1)(s - p2)\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{\omega \to 0} ZC = \infty (i.e., open circuit)
  • High frequencies: \lim{\omega \to \infty} ZC = 0 (i.e., short circuit)

The Differentiator

Circuit Analysis

• Step 1: Current equations (KCL)
  • i1 = \frac{v{in} - 0}{Z_C}
  • i2 = \frac{0 - v{out}}{R}
  • \frac{v{in}}{ZC} = -\frac{v_{out}}{R}
• Step 2: Finding the transfer function
  • H(s) = \frac{v{out}}{v{in}} = -\frac{R}{Z_C} = -RCs
• Meaning:
  • H(j\omega) = -j\omega RC = -j(\omega/\omega0) = (\omega/\omega0) \angle -90^\circ
  • \omega_0 = \frac{1}{RC}

The (Miller) Integrator

• Transfer function:
  • H(s) = \frac{v{out}}{v{in}} = -\frac{Z_C}{R} = -\frac{1}{RCs}
• Meaning:
  • H(j\omega) = -\frac{1}{j\omega RC} = -\frac{1}{j(\omega/\omega0)} = \frac{j}{(\omega/\omega0)} = \frac{1}{(\omega/\omega_0)} \angle +90^\circ
  • \omega_0 = \frac{1}{RC}

The (Deboo) Integrator

• Transfer function:
  • H(s) = \frac{v{out}}{v{in}} = \frac{1}{RCs}

Low-Pass Filter with Gain

• Transfer function:
  • H(j\omega) = -\frac{R2}{R1} \frac{1}{j\omega R2 C + 1} = H0 \frac{1}{1 + j(\omega/\omega_0)}
• Design equations:
  • H0 = -\frac{R2}{R_1}
  • \omega0 = \frac{1}{R2 C}

High-Pass Filter with Gain

• Transfer function:
  • H(j\omega) = -\frac{R2}{R1} \frac{j\omega R1 C}{j\omega R1 C + 1} = H0 \frac{j(\omega/\omega0)}{1 + j(\omega/\omega_0)}
• Design equations:
  • H0 = -\frac{R2}{R_1}
  • \omega0 = \frac{1}{R1 C}

Band-Pass Filter with Gain

• Transfer function:
  • H(j\omega) = -\frac{R2}{R1} \frac{j\omega R1 C1}{j\omega R1 C1 + 1} \frac{1}{j\omega R2 C2 + 1} = H0 \frac{j(\omega/\omegaL)}{1 + j(\omega/\omegaL)} \frac{1}{1 + j(\omega/\omegaH)}
• Design equations:
  • H0 = -\frac{R2}{R_1}
  • \omegaL = \frac{1}{R1 C_1}
  • \omegaH = \frac{1}{R2 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- = \angle v+
  • Consider the current through the top branch:
    • \angle i{R1} \propto \angle \frac{vi - v-}{R1} \propto \angle \frac{0^\circ - (+90^\circ)}{R_1} \propto \angle -90^\circ