A **switched-capacitor filter** is an electronic filter that uses capacitors and switches (usually implemented with MOSFETs) to emulate resistors and thereby implement analog filtering functions like low-pass, high-pass, band-pass, and more. Unlike traditional resistor-capacitor (RC) filters, switched-capacitor (SC) filters can be easily adjusted by changing the clock frequency that controls the switches, making them highly versatile in applications like signal processing.
### Key Concepts:
1. **Capacitors and Resistors**:
- In traditional RC filters, resistors and capacitors determine the time constant and hence the frequency response of the filter.
- In a switched-capacitor filter, the resistor is replaced by a "switched-capacitor" network that mimics the behavior of a resistor.
2. **Operation Principle:**
- **Switched Capacitor as an Equivalent Resistor**:
A capacitor is periodically switched between two nodes at a given clock frequency, transferring charge between them. Over time, this charge transfer simulates the behavior of a resistor.
- The effective resistance \( R_{eff} \) created by the switching process is determined by:
\[
R_{eff} = \frac{1}{f_c \cdot C}
\]
Where:
- \( f_c \) is the switching clock frequency.
- \( C \) is the capacitance value.
- The result is that the behavior of the filter can be controlled by changing the switching frequency, rather than by adjusting physical resistors.
3. **Switching Mechanism**:
- A switched-capacitor filter has switches (MOSFETs or other types) that periodically connect and disconnect a capacitor from different points in the circuit.
- The capacitor is alternately connected to two different nodes: one node (typically a voltage reference) to charge the capacitor, and another to discharge it, creating a periodic charge transfer.
- This periodic charging and discharging approximates the current flow that you would normally get through a resistor in an RC filter.
4. **Sampled Charge Transfer**:
- When the capacitor is connected between two nodes (say between an input and ground) through a switch, it will charge to the voltage across those nodes.
- After being disconnected from the first node, it is switched to the second node, where the stored charge is transferred.
- This creates an effective current flow, just as a resistor would, although the current is transferred in discrete packets rather than continuously.
5. **Frequency Control**:
- One major advantage of SC filters is that the equivalent resistance is proportional to the clock frequency. By changing the clock frequency, the effective resistance can be tuned, which adjusts the filter's cutoff frequency.
6. **Advantages**:
- **Accuracy**: Since the effective resistance is based on the clock frequency and capacitance, it can be very precisely controlled, unlike resistors, which have manufacturing tolerances.
- **Tuning**: By adjusting the clock frequency, the filter characteristics can be tuned without needing to physically change components.
- **Integration**: SC filters are commonly used in integrated circuits (ICs), as resistors occupy a lot of chip area, while capacitors and switches can be made much smaller.
### Example: Low-Pass Switched-Capacitor Filter
To understand the operation of a switched-capacitor low-pass filter, let's consider a simple example:
1. **Basic Structure**:
- A capacitor is alternately switched between two points: the input and ground, and the output and ground.
- A second capacitor (sometimes called the integrator capacitor) is placed at the output, smoothing the voltage.
2. **Charge Transfer**:
- During the first phase of the clock, the input capacitor charges to the input voltage.
- During the second phase, the input capacitor discharges into the output.
- This transfer of charge acts as the filtering action, and the output capacitor smooths the charge to give a continuous analog signal.
3. **Cutoff Frequency**:
- The cutoff frequency \( f_{cutoff} \) of the filter is determined by the clock frequency \( f_c \) and the capacitance \( C \):
\[
f_{cutoff} = \frac{f_c}{2 \pi R_{eff} C}
\]
Where \( R_{eff} \) is the equivalent resistance created by the switching mechanism.
### Application Example:
A practical example of switched-capacitor filters is in **Sigma-Delta ADCs (Analog-to-Digital Converters)**, where they are used as anti-aliasing filters. The precise control of the cutoff frequency via clocking is crucial for ensuring accurate signal sampling without distortion.
### Summary of Operation:
1. **Capacitor Charging**: A capacitor is switched between nodes, causing it to charge and discharge.
2. **Charge Transfer**: The periodic charge transfer mimics the flow of current in a resistor.
3. **Frequency Tuning**: The filter’s behavior is governed by the clock frequency, allowing easy tuning.
4. **Digital Clock Control**: Unlike traditional filters, SC filters are digitally controlled by a clock signal, offering better control over the filter characteristics.
Switched-capacitor filters are highly versatile, efficient, and widely used in integrated circuit designs where precise analog filtering is needed.