A switched capacitor filter works by using switched capacitors to effectively emulate resistors, allowing them to be combined with conventional capacitors to create integrated filters.
Switched capacitor (SC) filters are analog filters that use capacitors and electronic switches instead of traditional resistors. This technique allows filter circuits, which would typically require discrete resistors and capacitors or more complex integrated active components, to be built entirely with capacitors, switches, and sometimes op-amps (for active filters) on a single integrated circuit chip.
The Core Principle: Emulating a Resistor
The fundamental idea behind a switched capacitor filter is that a capacitor switched between two points at a high frequency can simulate the behavior of a resistor connecting those points.
Here's how this "equivalent resistance" is created:
- A Small Capacitor: A small capacitor ($C_S$) is used.
- Two Switches: Two switches are controlled by a non-overlapping clock signal oscillating at a frequency ($f_{CLK}$).
- Charge Transfer:
- When the first switch is closed, $C_S$ is connected across a voltage difference ($V_1 - V_2$) and charges up.
- When the second switch is closed, $C_S$ is disconnected from $V_1$ and connected to $V_2$, discharging its stored charge into the circuit node at $V_2$.
- Charge Transfer Rate: With each clock cycle, a specific amount of charge ($\Delta Q$) is transferred from $V_1$ to $V_2$. This charge is given by $\Delta Q = C_S \times (V_1 - V_2)$.
- Average Current: Since this charge transfer happens $f{CLK}$ times per second, the average current ($I{avg}$) flowing from $V_1$ to $V2$ is the total charge transferred per second: $I{avg} = \Delta Q \times f_{CLK} = C_S \times (V_1 - V2) \times f{CLK}$.
- Equivalent Resistance: By Ohm's Law ($R = V/I$), the equivalent resistance ($R_{eq}$) between $V_1$ and $V_2$ is $(V_1 - V2) / I{avg}$. Substituting the current equation:
$R_{eq} = \frac{V_1 - V_2}{C_S \times (V_1 - V2) \times f{CLK}} = \frac{1}{CS \times f{CLK}}$.
So, a switched capacitor circuit behaves like a resistor whose value is determined by the capacitor's size and the clock frequency.
Why Replace Resistors?
While traditional resistors work well, integrating precise, large-value resistors onto a silicon chip is difficult and consumes significant chip area. Capacitors, on the other hand, can be fabricated with much greater precision relative to each other, and their values can be scaled more effectively.
According to the provided reference, "The presence of switched capacitors to replace the resistor(s) within filters, both passive and active (using op amps), enables filters as fully integrated circuits." This is a key advantage.
Furthermore, the equivalent resistance of a switched capacitor circuit can be easily tuned by changing the clock frequency ($f_{CLK}$). This allows for filters with tunable cutoff frequencies, a feature not easily achievable with fixed integrated resistors.
Forming Filters
Just like conventional RC filters use resistors and capacitors, SC filters use switched capacitor circuits (acting as resistors) alongside conventional capacitors.
- Passive SC Filters: Switched capacitors are combined directly with conventional capacitors. "SC filters can be combined with conventional capacitors to produce low-pass filters." For example, replacing the resistor in a simple RC low-pass filter with a switched capacitor circuit results in an SC low-pass filter.
- Active SC Filters: Switched capacitors replace resistors in circuits incorporating operational amplifiers (op-amps), such as active filter topologies (e.g., Sallen-Key, state-variable filters). This allows complex, high-order filters to be implemented on-chip.
The cutoff frequency of an SC filter is primarily determined by the ratio of capacitor values and the clock frequency, rather than absolute resistance and capacitance values, making the filter characteristics more stable against process variations inherent in manufacturing.
Advantages of Switched Capacitor Filters
Switched capacitor filters offer several significant advantages over their continuous-time counterparts when implemented as integrated circuits:
- Integrability: As highlighted by the reference, they enable filters to be built as "fully integrated circuits".
- Accuracy: The filter characteristics depend on capacitor ratios and clock frequency, which can be controlled with high precision on-chip. The reference states that "SC filters are accurate".
- Tunability: The filter's corner frequency can often be adjusted simply by changing the clock frequency.
- Efficiency: The switching process can be energy-efficient, particularly at lower voltages. The reference notes that "SC filters are... efficient."
- Density: Switched capacitor implementations can be more compact than traditional integrated resistor-capacitor networks for achieving certain filter performance levels.
In summary, a switched capacitor filter replaces resistors with small capacitors that are rapidly switched, effectively creating an equivalent resistance. This technique allows for the integration of accurate, efficient, and often tunable filters onto single silicon chips by combining these "switched resistors" with conventional capacitors.
