9. switched capacitor filters - eth zeleccirc/docs/restricted/lecture09.pdf · 9. switched...
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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.