Prof. Dr. Qiuting HuangIntegrated Systems Laboratory

Electronic Circuits

9. Switched Capacitor Filters

Motivation

ETH 2Integrated Systems Laboratory

Transmission of voice signals requires an active RC low-pass filter with very low 𝑓𝑓cutoff = 3.4 kHz.

𝑓𝑓cutoff = 12𝜋𝜋𝜋𝜋𝜋𝜋

→ 𝑅𝑅 = 1𝜋𝜋2𝜋𝜋𝑓𝑓cutoff

⏟≈𝜋𝜋=10 pF

4.7 MΩ

Such a resistor may occupy a large area when realized on an integrated circuit.

How can we build this filter without resistor?

Transfer charge Δ𝑄𝑄 from potential 𝑉𝑉1 to potential 𝑉𝑉2 at a fixed rate 𝑓𝑓c = 1

𝑇𝑇c Phase ① (Φ1 closed, Φ2 open): 𝑄𝑄1 = 𝐶𝐶𝑉𝑉1 Phase ② (Φ1 open, Φ2 closed): 𝑄𝑄2 = 𝐶𝐶𝑉𝑉2 Transferred charge per time 𝑇𝑇𝑐𝑐:

Δ𝑄𝑄 = 𝐶𝐶 𝑉𝑉1 − 𝑉𝑉2

Average current 𝐼𝐼2,avg = Δ𝑄𝑄𝑇𝑇c

= 𝜋𝜋 𝑉𝑉1−𝑉𝑉2𝑇𝑇c

Equivalent resistor: 𝑅𝑅eq = 𝑇𝑇c𝜋𝜋

= 1𝑓𝑓c𝜋𝜋

ETH 3Integrated Systems Laboratory

Switched Capacitor Operating Principle

𝐶𝐶2𝑉𝑉out 𝑇𝑇c = −𝐶𝐶1𝑉𝑉in 𝑇𝑇c−𝐶𝐶1𝑉𝑉in 𝑇𝑇c − 𝐶𝐶2𝑉𝑉out 𝑇𝑇c = 0𝐶𝐶2𝑉𝑉out 2𝑇𝑇c = −𝐶𝐶1𝑉𝑉in 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 2𝑇𝑇c𝐶𝐶2𝑉𝑉out 3𝑇𝑇c = −𝐶𝐶1𝑉𝑉in 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 2𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 3𝑇𝑇c𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝐶𝐶2𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c

Phase ①: 𝐶𝐶1 is charged to 𝑉𝑉in.Same amount of charge is moved to 𝐶𝐶2.

Phase ②: 𝐶𝐶1 is discharged.

Inverting Integrator Using Switched Capacitors

ETH 4Integrated Systems Laboratory

𝐶𝐶2𝑉𝑉out−𝐶𝐶2𝑉𝑉out

𝐶𝐶1𝑉𝑉in −𝐶𝐶1𝑉𝑉in Initial condition:𝑉𝑉out = 0

𝐶𝐶2𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 𝑇𝑇𝑐𝑐 = −𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 𝑇𝑇𝑐𝑐−𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 𝑇𝑇𝑐𝑐 − 𝐶𝐶2𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 𝑇𝑇𝑐𝑐 = 0𝐶𝐶2𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 2𝑇𝑇𝑐𝑐 = −𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 𝑇𝑇𝑐𝑐 − 𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 2𝑇𝑇𝑐𝑐𝐶𝐶2𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 3𝑇𝑇𝑐𝑐 = −𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 𝑇𝑇𝑐𝑐 − 𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 2𝑇𝑇𝑐𝑐 − 𝐶𝐶1𝑉𝑉𝑖𝑖𝑖𝑖 3𝑇𝑇𝑐𝑐𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝐶𝐶2𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c

Phase ①: 𝐶𝐶1 is charged to 𝑉𝑉in.Same amount of charge is moved to 𝐶𝐶2.

Phase ②: 𝐶𝐶1 is discharged.

Inverting Integrator Using Switched Capacitors

ETH 5Integrated Systems Laboratory

Output signal 𝑉𝑉out looks like a continuous-time signal for

sufficiently small 𝑇𝑇c.

𝐶𝐶2𝑉𝑉out−𝐶𝐶2𝑉𝑉out

𝐶𝐶1𝑉𝑉in −𝐶𝐶1𝑉𝑉in Initial condition:𝑉𝑉out = 0

𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝐶𝐶2𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c

Inverting Integrator Using Switched Capacitors

ETH 6Integrated Systems Laboratory

𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c −𝐶𝐶1𝐶𝐶2𝑉𝑉in 𝑛𝑛𝑇𝑇c = −

𝐶𝐶1𝐶𝐶2�𝑘𝑘=0

𝑖𝑖−1

𝑉𝑉in[(𝑛𝑛 − 𝑘𝑘)𝑇𝑇c]

small 𝑇𝑇c 𝑉𝑉out 𝑛𝑛𝑇𝑇c = −𝐶𝐶1𝑇𝑇c𝐶𝐶2

lim𝑇𝑇c→0

�𝑘𝑘=0

𝑖𝑖−1

𝑉𝑉in (𝑛𝑛 − 𝑘𝑘)𝑇𝑇c ⋅ 𝑇𝑇c = −𝐶𝐶1𝑇𝑇c𝐶𝐶2

�0

𝑖𝑖𝑇𝑇c𝑉𝑉in 𝑡𝑡 d𝑡𝑡

Note: differentiation would require an input-output relation: 𝑉𝑉in 𝑛𝑛𝑇𝑇c = const ⋅ ∑𝑘𝑘 𝑉𝑉out[ 𝑛𝑛 − 𝑘𝑘 𝑇𝑇c]

Transform for time-discrete signals is needed in order to solve difference equation and calculate transfer function

Initial condition:𝑉𝑉out = 0

Z-TransformDefinition 𝑍𝑍 𝑥𝑥 𝑛𝑛𝑇𝑇c = 𝑋𝑋 𝑧𝑧 = �

𝑘𝑘=−∞

∞

𝑥𝑥 𝑘𝑘𝑇𝑇c 𝑧𝑧−𝑘𝑘

Time delay Z 𝑥𝑥 𝑛𝑛 − 𝑘𝑘 𝑇𝑇c = 𝑧𝑧−𝑘𝑘𝑋𝑋(𝑧𝑧)

Integration 𝑇𝑇c1 − 𝑧𝑧−1

Differentiation1 − 𝑧𝑧−1

𝑇𝑇c

Mapping to Laplace domain 𝑠𝑠 = 𝑧𝑧−1𝑇𝑇c

or 𝑠𝑠 = 1−𝑧𝑧−1

𝑇𝑇c(forward or backward Euler transform)

Mapping to 𝑗𝑗𝑗𝑗-axis 𝑧𝑧 = 𝑒𝑒𝑗𝑗𝑗𝑗𝑇𝑇c = 𝑒𝑒𝑗𝑗2𝜋𝜋𝑓𝑓𝑓𝑓c

ETH 7Integrated Systems Laboratory

Solve difference equation of SC inverting integrator Difference equation: 𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝐶𝐶2𝑉𝑉out (𝑛𝑛 − 1)𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c Apply Z-Transform: 𝐶𝐶2𝑉𝑉out 𝑧𝑧 = 𝑧𝑧−1𝐶𝐶2𝑉𝑉out 𝑧𝑧 − 𝐶𝐶1𝑉𝑉in 𝑧𝑧 Transfer function: 𝑇𝑇 𝑧𝑧 = 𝑉𝑉out(𝑧𝑧)

𝑉𝑉in(𝑧𝑧)= − 𝜋𝜋1

𝜋𝜋2−𝜋𝜋2𝑧𝑧−1= −𝜋𝜋1

𝜋𝜋2

11−𝑧𝑧−1

First order low-pass filter with unity gain and 𝑓𝑓cutoff = 3.4 kHz.

𝜏𝜏 = 𝑅𝑅𝐶𝐶 = 12𝜋𝜋𝑓𝑓cutoff

≈ 47𝜇𝜇s

𝐶𝐶 = 10 pF → 𝑅𝑅 = 4.7 MΩ

SC realization (𝑅𝑅 → 𝑇𝑇c𝜋𝜋𝑅𝑅

): 𝜏𝜏 = 𝜋𝜋𝜋𝜋𝑅𝑅𝑇𝑇c

Ratio of capacitors can be realized more accurately than absolute values of 𝑅𝑅 and 𝐶𝐶.

𝑓𝑓c = 100 kHz ≫ 3.4 kHz → 𝐶𝐶𝜋𝜋 = 𝜋𝜋𝜏𝜏𝑓𝑓c

≈ 2.1 pF

ETH 8Integrated Systems Laboratory

Example: SC Low-pass Filter

Exactly the same circuit can be operated as non-inverting integrator only by changing the switching schedule.

Phase ①: 𝐶𝐶1 is charged to 𝑉𝑉in.

Phase ②: Charge is transferred to 𝐶𝐶2.

Charge on 𝐶𝐶2 is inverse compared to inverting integrator.

−𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = −𝐶𝐶2𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c − 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c 𝐶𝐶1𝑉𝑉in 𝑧𝑧 = 𝐶𝐶2 1 − 𝑧𝑧−1 𝑉𝑉out 𝑧𝑧

ETH 9Integrated Systems Laboratory

Non-Inverting SC Integrator

𝐶𝐶2𝑉𝑉out 𝑛𝑛𝑇𝑇c = 𝐶𝐶2𝑉𝑉out 𝑛𝑛 − 1 𝑇𝑇c + 𝐶𝐶1𝑉𝑉in 𝑛𝑛𝑇𝑇c

𝑉𝑉out 𝑧𝑧𝑉𝑉in(𝑧𝑧)

=𝐶𝐶1𝐶𝐶2

11 − 𝑧𝑧−1

Switched Capacitor Tow-Thomas Biquad

ETH 10Integrated Systems Laboratory

All resistors are replaced by switched capacitors.

Non-inverting integrator can be realized with only one stage.

Design equations:𝐶𝐶𝜋𝜋4𝐶𝐶𝜋𝜋3

= −𝑘𝑘

𝐶𝐶𝜋𝜋2𝐶𝐶𝜋𝜋3𝐶𝐶1𝐶𝐶2

= 𝑗𝑗0𝑇𝑇c 2

𝐶𝐶𝜋𝜋1𝐶𝐶1

=𝑗𝑗0𝑇𝑇c𝑄𝑄

Design equations:𝐶𝐶𝜋𝜋4𝐶𝐶𝜋𝜋3

= −𝑘𝑘

𝐶𝐶𝜋𝜋2𝐶𝐶2

=𝐶𝐶𝜋𝜋3𝐶𝐶1

= 𝑗𝑗0𝑇𝑇c

𝐶𝐶𝜋𝜋1𝐶𝐶1

=𝑗𝑗0𝑇𝑇c𝑄𝑄

Design equations:𝐶𝐶𝜋𝜋4𝐶𝐶𝜋𝜋3

= −𝑘𝑘

𝐶𝐶𝜋𝜋2𝐶𝐶2

=𝐶𝐶𝜋𝜋3𝐶𝐶1

= 𝑗𝑗0𝑇𝑇c𝐶𝐶𝜋𝜋3𝐶𝐶𝜋𝜋1

= 𝑄𝑄

Switched Capacitor Tow-Thomas Biquad (2)

ETH 11Integrated Systems Laboratory

𝑇𝑇 𝑧𝑧 = 𝑉𝑉out 𝑧𝑧𝑉𝑉in(𝑧𝑧)

= −𝜋𝜋𝑅𝑅4𝜋𝜋𝑅𝑅3

𝐶𝐶𝑅𝑅2𝐶𝐶𝑅𝑅3𝐶𝐶1𝐶𝐶2

𝑧𝑧−2+𝑧𝑧−1 −2−𝐶𝐶𝑅𝑅1𝐶𝐶1

+1+𝐶𝐶𝑅𝑅1𝐶𝐶1

+𝐶𝐶𝑅𝑅2𝐶𝐶𝑅𝑅3𝐶𝐶1𝐶𝐶2

General form of continuous-time low-pass filter is transformed to discrete-time filter by backward Euler transform 𝑠𝑠 = 1−𝑧𝑧−1

𝑇𝑇c:

𝑇𝑇 𝑠𝑠 =𝑘𝑘𝑗𝑗0

2

𝑠𝑠2 + 𝑗𝑗0𝑄𝑄 𝑠𝑠 + 𝑗𝑗02

→ 𝑇𝑇 𝑧𝑧 ≈𝑘𝑘 𝑗𝑗0 𝑇𝑇𝑐𝑐

2

𝑧𝑧−2 + 𝑧𝑧−1 −2 −𝑗𝑗0𝑇𝑇𝑐𝑐𝑄𝑄 + 1 +

𝑗𝑗0𝑇𝑇𝑐𝑐𝑄𝑄 + 𝑗𝑗0 𝑇𝑇𝑐𝑐

2

SC Ladder Filter

ETH 12Integrated Systems Laboratory

Ladder filter can be realized without inductors and without resistors.

All 𝑅𝑅𝑖𝑖 are replaced by corresponding 𝐶𝐶𝜋𝜋𝑛𝑛.