ADC & DAC
DAC
I/O和解释
$$ \begin{gather} \begin{aligned} V_o &= \frac{ D }{ 2^n } V_r \end{aligned} \end{gather} $$
重要类型 - R-2R Ladder DAC
ADC
I/O和原理
| Category | Concept |
|---|---|
| $V_{ref}$ | Reference Voltage |
| Resolution(LSB) | $\dfrac{ V_r }{ 2^n }$ |
| SC | Start of Conversion |
| EOC | End of Conversion |
| $D_i$ | 数字输入的二进制第$i$位 |
$$ \begin{gather} \begin{aligned} D &= \frac{ V_i }{ V_r } 2^n \end{aligned} \end{gather} $$
Successive Approximation ADC
总结
$$ \begin{gather} \begin{aligned} 数字输入 \times 分辨率 &= 模拟输出 模拟输入 % 分辨率 &= 数字输出 \end{aligned} \end{gather} $$
Opamp Revision
Imperfection in Opamp
- Ideal Opamp + Current Sources = Model of a Real Opamp
Bias Current
定义
- 图中的的放大器是理想的
$$ \begin{gather} \begin{aligned} I_B(+) &\approx I_B(-) \ I_B &= \frac{ I_B(+) + I_B(-) }{ 2 } \end{aligned} \end{gather} $$
以Inverting Configuration 为例介绍非理想时的输出电压
$$ \begin{gather} \begin{aligned} V_o &= - \frac{ V_{in} R_2 }{ R_1 } + I_B R_2 \\ &= -\frac{ V_{in} R_2 }{ R_1 } - I_B R_1 \end{aligned} \end{gather} $$
| Category | Concept |
|---|---|
| Refer to Output(RTO) | $I_BR_2$ |
| Refer to Input(RTI) | $-I_BR_1$ |
- 等效电路
Compensation
$$ \begin{gather} \begin{aligned} V_o &= -V_{in} + R(I_B(-) - I_B(+)) \\ &= -V_{in} + RI_{os} \end{aligned} \end{gather} $$
Input Offset Current
Definition
$$ \begin{gather} \begin{aligned} I_{os} &= |I_B(+) - I_B(-)| \end{aligned} \end{gather} $$
用Bias 和 Offset Current 来表示电流
$$ \begin{gather} \begin{aligned} I_B(+) &= \frac{ I_{os} }{ 2 } + I_B \\ I_B(-) &= - \frac{ I_{os} }{ 2 } + I_B \end{aligned} \end{gather} $$
Ex. Integrator
$$ \begin{gather} \begin{aligned} V_{out} &= \end{aligned} \end{gather} $$
Figure of Merit
Ex.1 - Integrate and Hold
电路
$S_1$接$V_{in}$时 Integrator
$$ \begin{gather} \begin{aligned} V_{out} &= - \int_{}^{ } \frac{ V_{in} }{ RC }dt \end{aligned} \end{gather} $$
$S_1$接地时 Memory
$$ \begin{gather} \begin{aligned} V_{out} &= Constant \\ | \frac{ dV_{out} }{ dt }| &= \frac{ 1 }{ C } \frac{ dQ }{ dt } \\ &= \frac{ 1 }{ C } (I_B + \frac{ V_{os} }{ R }) \\ &= \frac{ 1 }{ \tau } (V_{os} + I_BR) \end{aligned} \end{gather} $$
- 为了使得$| \dfrac{ dV_{out} }{ dt}|$尽可能小; $V_{out}$不变时，$V_{in}$ 尽可能大
$$ \begin{gather} \begin{aligned} R \ge 2 k \Omega \end{aligned} \end{gather} $$
Figure of Merit
$$ \begin{gather} \begin{aligned} F &= V_{os} + I_{os} R \end{aligned} \end{gather} $$
Ex.2 - Differential Amplifier
$$ \begin{gather} \begin{aligned} V_{out} &= 2 V_{os} + 20k \cdot I_{os} \\ F &= 2 V_{os} + 20k \cdot I_{os} \end{aligned} \end{gather} $$
Grounding & Common-Mode Rejection
Power Dissipation Capacitance for CMOS Logic Current
$$ \begin{gather} \begin{aligned} C_{PD} &= \frac{ Q_{switch} }{ V_{CC} } \\ &= \frac{ I_{max} \tau_{switch} }{ V_{CC} } \\ &= \frac{ \tau_{switch} }{ R_{switch} } \end{aligned} \end{gather} $$
Designing Ground
CM Rejection
定义
$$ \begin{gather} \begin{aligned} CMRR &= | \frac{ A_{dm} }{ A_{cm} }| \end{aligned} \end{gather} $$
Influence on Error
RTI
$$ \begin{gather} \begin{aligned} V_o &= A_{cm} V_{cm} + A_{dm} V_{diff} \\ &= A_{dm} \Big( V_{diff} + V_{cm} \frac{ A_{cm} }{ A_{dm} }\Big) \\ &= A_{dm} (V_{diff} + \frac{ V_{cm} }{ CMRR }) \\ RTI &= \frac{ V_{cm} }{ CMRR } \end{aligned} \end{gather} $$
等效电路和分析
$$ \begin{gather} \begin{aligned} V_o &= A(V(+) - V(-)) \\ &= A(V(+) - (V_o \pm \frac{ V_{CM} }{ CMRR })) \\ \frac{ V_o }{ V(+) } &= \frac{ A \Big[ 1 + \dfrac{ 1 }{ CMRR } \Big] }{ 1 + A } \\ & \approx 1 + \frac{ 1 }{ CMRR } \end{aligned} \end{gather} $$
- $CMRR$越大，Error越小
CMRR of Differential Amplifier with Imbalanced Resistors
Gains
$$ \begin{gather} \begin{aligned} V_o &= \frac{ V_{CM} - \dfrac{ V_{diff} }{ 2 } - \dfrac{ R_2 }{ R_1 + R_2 } (V_{CM} + \dfrac{ V_{diff} }{ 2 })}{ R_1 } R_2(1 - \varepsilon) + \frac{ R_2 }{ R_1 +R_2 } (V_{CM} - \frac{ V_{diff} }{ 2 }) \\ &= A_{dm} V_{dm}+ A_{cm} V_{cm} \\ A_{diff} &= \frac{ R_2 }{ R_1 } \Big( 1 - \frac{ R_1 + 2R_2 }{R_1 + R_2 } \times \frac{ \varepsilon }{ 2 }\Big) \\ A_{dm} &= \frac{ R_2 }{ R_1 + R_2 } \varepsilon \end{aligned} \end{gather} $$
CMRR when $\varepsilon$ is Small
$$ \begin{gather} \begin{aligned} CMR(dB) &= 20 l \Big[ \frac{ 1 + \dfrac{ R_2 }{ R_1 } }{ \varepsilon} \Big] \end{aligned} \end{gather} $$
Instrumentaiton Amplifier
电路
-
红色部分是Buffer Stage
-
第二部分是Unity Gain Amplifier
-
电流$I$方向是3-1-2-4, 因此$V_3 - V_1 = V_4 - V_2$
电流/电压 分析
$$ \begin{gather} \begin{aligned} I &= \frac{ V_3 - V_4 }{ 2R_a + R_b } \\ &= \frac{ V_1 - V_2 }{ R_b } \\ V_o &= V_3 - V_4 \\ &= (V_1 - V_2) \frac{ 2R_a + R_b }{ R_b } \end{aligned} \end{gather} $$
Gain & CMRR Considering $\varepsilon$
$$ \begin{gather} \begin{aligned} A_{diff} &= \frac{ 2R_a + R_b }{ R_b } \\ A_{cm} &= \frac{ \varepsilon }{ 2 } \\ CMRR &= \frac{ 4R_a +2R_b }{ \varepsilon R_b } \end{aligned} \end{gather} $$
IA Application (ECG)
测量方法
| Category | Concept |
|---|---|
| LL-RA | |
| LL-LA | |
| LA-RA |
Difference Amplifier Problem
Circuit
$$ \begin{gather} \begin{aligned} \Delta &\ll R_1 \\ Sense &= O/P \\ &= \Big( In(+) - In(-) \Big) + Ref \\ G &= 1 \\ Ref &= 0V \end{aligned} \end{gather} $$
DM Gain
- $In(+) = V_{diff} = -In(-)$
$$ \begin{gather} \begin{aligned} A_{dm} &\approx 1 \end{aligned} \end{gather} $$
$$ \begin{gather} \begin{aligned} V_p &= V_n \\ (Ref - In(+)) \frac{ R_1 }{ R_1 + R_1 + \Delta } &= (Sense - In(-)) \frac{ R_1 }{ R1 + R_1 - \Delta } \\ V_{diff} &= Sense - Ref \end{aligned} \end{gather} $$
CM Gain
- $In(+) = In(-) = V_{CM}$
$$ \begin{gather} \begin{aligned} (Ref- V_{CM}) \frac{ R_1 }{ 2R_1 + \Delta} &= (Sense -V_{CM}) \frac{ R_1 }{ 2R_1 - \Delta } \\ (Ref -V_{CM}) (2R_1 - \Delta) &= (Sense - V_{CM}) (2R_1 + \Delta)) \\ A_{CM} &= \frac{ Sense - Ref }{ V_{CM} } \\ & \approx \frac{ \Delta }{ R_1 } \end{aligned} \end{gather} $$
CMRR
$$ \begin{gather} \begin{aligned} CMMR &= \frac{ R_1 }{ \Delta } \end{aligned} \end{gather} $$
Noise & Opamp
SNR & NEB
- Signal to Noise Ratio (SNR) and Noise Equivalent Bandwidth (NEB)
$$ \begin{gather} \begin{aligned} SNR &= 20 \times lg( \frac{ V_s(rms) }{ V_n(rms) }) \end{aligned} \end{gather} $$
$$ \begin{gather} \begin{aligned} NEB &= n \times \frac{ GBW }{ Noise \ Gain } \end{aligned} \end{gather} $$
- n is Brick-Wall Factor
Opamp Noise Model
Circuit
- $i_n, e_n$ are Spectral Density
White Noise Passing LPF
RMS Input Noise
Formulas
- $R_1 || R_2 = R_3$
$$ \begin{gather} \begin{aligned} E_{R1-R2} &= E_{RP} = \sqrt{ 4kT \times 1.57 f_H \times (R_1 || R_2)} \\ E_{R3} &= E_{RP} \\ E_{nn} &= i_n (R_1 || R_2) \sqrt{ 1.57 f_H } \\ E_{np} &= E_{nn} \\ E_{n} &= e_n \sqrt{ 1.57 f_H } \\ E_{ni} &= \sqrt{ E_{R1 -R2}^2 + E_{R3}^2 + E_{nn}^2 + E_{np}^2 + E_n^2} \\ &= \sqrt{ 1.57 f_H} \sqrt{ 8kTR_P + 2R_P^2 i_n^2 + e_n^2 } \end{aligned} \end{gather} $$
- $E_{ni}$ refers to the non-inverting input
$$ \begin{gather} \begin{aligned} E_{noise} &= E_{ni} (1 + \frac{ R_2 }{ R_1 }) \end{aligned} \end{gather} $$
Relationship in Graph
Appendix
CheckList
| Category | Concept |
|---|---|
| 什么情况下使用Bias 或 Offset | Impedance (Very) Unbalanced: Bias; Impedance Matched: Offset |
| Figure of Merit | A function of all significant contributions of the components to the total system error, which changes monotonically with the overall error |
| Shielding | Covering it with Metal |
| twisted pair | 减少Differential Noise |
| Supply Decoupling | 对于大电流电路端，需要单独使用宽轨线进行布线, 防止磁场干扰、过热、噪声问题 |
| Ground Bounce | 多个芯片共享参考地，产生回路电流干扰和寄生电容、电感 |
| Supply Decoupling | 在供电引脚之间防止无极性电容(去耦电容), 隔离噪声干扰 |
| Cold End | 连接Load和Ground的电路 |
| Hot End | 连接Power和Input Signal的电路 |
| Thermocouple | 两种不同材料的导线连在一起，当有$\Delta T$时，就会产生电动势 |
| White Noise | Noise that has flat Spectral Density |
| 3 Sources for Noise | 两个Voltage Inputs, 每个输入端的Current Noise |
| 2 Reasons for Noise | Recombination & Generation of electrons in semiconductors; Thermal Agitation of electrons, $E_r = \sqrt{4 R k T \Delta f}$, $\Delta f$ is bandwidth |
| Noise Equivalent Bandwidth | 在指定带宽内，能产生和实际噪声功率相等的均匀白噪声所对应的等效宽带 |
| Gain-Bandwdith | The point of Unitary-Gain, 能够提供一定增益的最大频率带宽 |
公式
- Dynamic Range (DR)
$$ \begin{gather} \begin{aligned} DR &= 20 \times lg( \frac{ V_{max} }{ V_{min} }) \\ &= (6.02N + 1.76) dB \end{aligned} \end{gather} $$
- Energy Spectral Density
$$ \begin{gather} \begin{aligned} V &= \sqrt{ E_n \Delta f } \end{aligned} \end{gather} $$