Understanding the transfer function for a high pass filter is fundamental for anyone designing analog circuits, processing audio signals, or analyzing control systems. This mathematical representation describes how the filter attenuates low-frequency components while allowing high-frequency signals to pass through with minimal loss. By examining the relationship between the output and input in the frequency domain (s-domain), engineers can predict circuit behavior, calculate cutoff frequencies, and cascade multiple stages to achieve steeper roll-off characteristics Easy to understand, harder to ignore..
The Core Concept: What Is a Transfer Function?
Before diving into the specifics of high-pass topologies, You really need to define the transfer function itself. In linear time-invariant (LTI) systems, the transfer function, denoted as $H(s)$, is the ratio of the Laplace transform of the output signal $Y(s)$ to the Laplace transform of the input signal $X(s)$, assuming zero initial conditions:
$H(s) = \frac{Y(s)}{X(s)}$
Here, $s$ is the complex frequency variable ($s = \sigma + j\omega$). For steady-state sinusoidal analysis, we substitute $s = j\omega$, where $\omega$ is the angular frequency in radians per second. The magnitude $|H(j\omega)|$ tells us the gain (or attenuation) at a specific frequency, while the phase $\angle H(j\omega)$ reveals the time delay or phase shift introduced by the circuit.
For a high pass filter (HPF), the ideal magnitude response is zero for frequencies below the cutoff frequency ($\omega_c$) and unity (or a specific gain $A_v$) for frequencies above it. Real-world approximations, however, exhibit a transition band and a roll-off rate typically measured in decibels per decade (dB/decade) or decibels per octave (dB/octave) Took long enough..
First-Order Passive RC High Pass Filter
The simplest implementation uses a single resistor and a capacitor. In this configuration, the capacitor is placed in series with the signal path, and the resistor is connected to ground (shunt). The output voltage is measured across the resistor.
Derivation of the Transfer Function
Using the voltage divider rule in the s-domain, the impedance of the capacitor is $Z_C = \frac{1}{sC}$ and the impedance of the resistor is $Z_R = R$ It's one of those things that adds up..
$H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{Z_R}{Z_R + Z_C} = \frac{R}{R + \frac{1}{sC}}$
Simplifying the algebra by multiplying the numerator and denominator by $sC$:
$H(s) = \frac{sRC}{sRC + 1}$
It is standard practice to factor out the time constant $\tau = RC$ and define the cutoff frequency (corner frequency) as $\omega_c = \frac{1}{RC} = \frac{1}{\tau}$.
$H(s) = \frac{s / \omega_c}{s / \omega_c + 1} = \frac{s}{s + \omega_c}$
This is the standard form of the first-order high pass filter transfer function.
Frequency Response Analysis
Substituting $s = j\omega$:
$H(j\omega) = \frac{j\omega}{j\omega + \omega_c}$
Magnitude Response: $|H(j\omega)| = \frac{\omega}{\sqrt{\omega^2 + \omega_c^2}} = \frac{1}{\sqrt{1 + (\omega_c / \omega)^2}}$
- At DC ($\omega \to 0$): $|H| \to 0$. The capacitor acts as an open circuit, blocking DC completely.
- At Cutoff ($\omega = \omega_c$): $|H| = \frac{1}{\sqrt{2}} \approx 0.707$ (-3 dB). The phase shift is $+45^\circ$.
- At High Frequencies ($\omega \gg \omega_c$): $|H| \to 1$ (0 dB). The capacitor acts as a short circuit, passing the signal directly to the output.
Phase Response: $\phi(\omega) = \angle H(j\omega) = 90^\circ - \arctan\left(\frac{\omega}{\omega_c}\right)$
- Phase leads by $90^\circ$ at DC.
- Phase leads by $45^\circ$ at $\omega_c$.
- Phase approaches $0^\circ$ at infinite frequency.
Roll-off Rate: The asymptotic slope in the stopband is +20 dB/decade (or +6 dB/octave). This means for every tenfold decrease in frequency below $\omega_c$, the signal amplitude drops by a factor of 10.
First-Order Active High Pass Filter (Op-Amp Based)
Passive filters suffer from loading effects; the output impedance changes based on the connected load. Here's the thing — active filters using operational amplifiers (op-amps) provide buffering and voltage gain. The most common topology is the non-inverting active high pass filter Took long enough..
Circuit Topology and Transfer Function
The circuit places the RC high-pass network at the non-inverting input (+) of the op-amp. The feedback network (resistors $R_f$ and $R_1$) sets the passband gain That's the part that actually makes a difference. Simple as that..
$H(s) = \left(1 + \frac{R_f}{R_1}\right) \frac{sRC}{sRC + 1} = A_{v(pass)} \frac{s / \omega_c}{s / \omega_c + 1}$
Where $A_{v(pass)} = 1 + \frac{R_f}{R_1}$ is the passband voltage gain.
Key Advantages:
- High Input Impedance: The op-amp input draws negligible current, preventing loading of the source.
- Low Output Impedance: The op-amp output can drive heavy loads without changing the filter characteristics.
- Adjustable Gain: The passband gain is independent of the filter cutoff frequency components ($R$ and $C$), allowing independent tuning of gain and frequency response.
Second-Order High Pass Filters: Sallen-Key Topology
To achieve a steeper roll-off (-40 dB/decade in the stopband), a second-order filter is required. The Sallen-Key topology (also known as a Voltage-Controlled Voltage Source or VCVS) is the industry standard for active second-order filters due to its low component count and minimal sensitivity to component tolerances.
Standard Transfer Function Form
The general normalized second-order high pass transfer function is:
$H(s) = H_0 \frac{s^2}{s^2 + \frac{\omega_c}{Q}s + \omega_c^2}$
Where:
- $H_0$ is the passband gain. Now, * $\omega_c$ is the corner frequency (rad/s). * $Q$ (Quality Factor) determines the "peaking" near the corner frequency and the sharpness of the transition.
Sallen-Key Component Relationships
For the standard unity-gain Sallen-Key HPF (where $R_1 = R_2 = R$, $C_1 = C_2 = C$), the design equations simplify significantly:
- Cutoff Frequency: $\omega_c = \frac{1}{RC}$ (or $f_c = \frac{1}{2\pi RC}$)
- Quality Factor: $Q = \frac{1}{3 - K}$, where $K$ is the op-amp gain ($K = 1 + \frac{R_f}{R_g}$).
Critical Design Insight: As the gain $K$ approaches 3, the $Q$ factor approaches infinity, leading to oscillation (the circuit becomes an oscillator). For a maximally flat Butterworth response, $Q =
