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theory:sensor_technology:stc_common_mode_rejection_ratio_cmrr [2018/10/17 03:40]
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-====== Common Mode Rejection Ratio (CMRR) ====== 
- 
-The Common Mode Rejection Ratio is is the ratio between the output of an amplifier due to a common signal with repect to a differential signal. Consider a certain amplifier with two inputs $V_{A}$ and $V_{B}$. We define ​ 
-  * the gain of the amplifier as $A$ - meaning that differential voltages are amplified $A$ times 
-\begin{equation} 
-V_{O} = A (V_{A}-V_{B}) 
-\end{equation} 
-  * an amplification of common signals of $B$ - meaning that the average of $V_{A}$ and $V_{B}$ is amplified $B$ times  
-\begin{equation} 
-V_{O} = B (V_{A}+V_{B})/​2 
-\end{equation} 
- 
-The Common Mode Rejection Ratio is defined as 
-\begin{equation} 
-CMRR = \frac{A}{\left |B  \right |} 
-\label{eq:​CMRR} 
-\end{equation} 
-  
-How does this work with the simple OpAmp amplifier of <imgref CMRR>? 
- 
-<​imgcaption CMRR|Simple OpAmp based amplifier>​{{.measurement_theory:​cmrr.png?​300}}</​imgcaption>​ 
- 
-We can start with superposition of the output $V_{O}$ due to voltage $V_{B}$ and $V_{A}$. To simplify the problem, we start with the output $V_{O}$ due to voltage $V_{A}$ and $V^{+}$: 
-\begin{equation} 
-\begin{aligned} 
-V_{O}&​=V^{+} \cdot \frac{R_{4}+R_{3}}{R_{3}}-V_{B} \cdot \frac{R_{4}}{R_{3}} \\ 
-&= V^{+} \cdot \frac{m \cdot R+R}{R}-V_{B} \cdot \frac{m \cdot R}{R} \\ 
-&= V^{+} \cdot \frac{m +1}{R}-V_{B} \cdot m  
-\end{aligned} 
-\label{eq:​Vout2} 
-\end{equation} 
- 
-where $V^{+}$ can be calculated as 
- 
-\begin{equation} 
-\begin{aligned} 
-V^{+} &= \frac{R_{2}}{R_{1}+R_{2}} \cdot V_{A} \\ 
-&= \frac{m \cdot R}{m \cdot R + R} \cdot V_{A} \\ 
-&​=\frac{m}{m+1} \cdot V_{A}. 
-\end{aligned} 
-\label{eq:​Vplus} 
-\end{equation} 
-Which yields 
- 
-\begin{equation} 
-\begin{aligned} 
-V_{O}&= \frac{m}{m+1} \cdot V_{A} \cdot \frac{m +1}{R}-V_{B} \cdot m \\ 
-&= m \cdot \left ( V_{A} - V_{B}  \right )  ​ 
-\end{aligned} 
-\label{eq:​Vouttotal} 
-\end{equation} 
- 
-This means that in the ideal situation, the gain $A$ is equal to $m$, and there is no effect of a common signal. However, we made an assumption that the factor $m$ in $R_{4}$ is equal to the factor $m$ in $R_{2}$. What would happen if we distinguish between $m_{4}$ and $m_{2}$? 
- 
-Start again with 
-\begin{equation} 
-\begin{aligned} 
-V_{O}&​=V^{+} \cdot \frac{R_{4}+R_{3}}{R_{3}}-V_{B} \cdot \frac{R_{4}}{R_{3}} \\ 
-&= V^{+} \cdot \frac{m_{4} \cdot R+R}{R}-V_{B} \cdot \frac{m_{4} \cdot R}{R} \\ 
-&= V^{+} \cdot \frac{m_{4} +1}{R}-V_{B} \cdot m_{4} 
-\end{aligned} 
-\label{eq:​Vout2_imbalance} 
-\end{equation} 
- 
-where $V^{+}$ can be calculated as 
- 
-\begin{equation} 
-\begin{aligned} 
-V^{+} &= \frac{R_{2}}{R_{1}+R_{2}} \cdot V_{A} \\ 
-&= \frac{m_{2} \cdot R}{m_{2} \cdot R + R} \cdot V_{A}  \\ 
-&​=\frac{m_{2}}{m_{2}+1} \cdot V_{A}. 
-\end{aligned} 
-\label{eq:​Vplus_imbalance} 
-\end{equation} 
-Now we find 
-\begin{equation} 
-V_{O}= \frac{m_{2}}{m_{2}+1} \cdot V_{A} \cdot \frac{m_{4} +1}{R}-V_{B} \cdot m_{4}  
-\label{eq:​Vouttotal_imbalance} 
-\end{equation} 
-which can no longer be simplifed. However, we can bring it to the form of 
-\begin{equation} 
-V_{O}= A \left ( V_{A} - V_{B}  \right )+ \frac{B}{2}\left ( V_{A} + V_{B}  \right )  
-\label{eq:​Vout_as_sum} 
-\end{equation} 
-The factors $A$ and $B$ (the differential gain and common gain respectively) are 
- 
-\begin{equation} 
-\begin{aligned} 
-A &= \frac{m_{2}+2m_{2}m_{4}+m_{4}}{2 \left ( m_{2}+1 \right )}  \\ 
-B &= \frac{m_{2}-m_{4}}{ m_{2}+1 }  
-\end{aligned} 
-\end{equation} 
- 
-Here is $A$ the desired differential gain and $B$ the residu due to $m_{2} \neq m_{4}$. This means 
-\begin{equation} 
-CMRR = \frac{A}{\left |B  \right |} = \frac{m_{2}+2m_{2}m_{4}+m_{4}}{\left | m_{2}-m_{4} \right |} 
-\label{eq:​CMRR_OpAmp} 
-\end{equation} 
-where we can see that if $m_{2}$ becomes equal to $m_{4}$, the CMRR becomes infinite. 
- 
-====== Sensor Technology TOC ====== 
- 
-These are the chapters for the Sensor Technology course: 
-  * Chapter 1: [[theory:​sensor_technology:​st1_measurement_theory|Measurement Theory]] 
-  * Chapter 2: [[theory:​sensor_technology:​st2_measurement_errors|Measurement Errors]] 
-  * Chapter 3: [[theory:​sensor_technology:​st3_measurement_technology|Measurement Technology]] 
-  * Chapter 4: [[theory:​sensor_technology:​stb_conventions_for_good_graphs_and_tables|Circuits,​ Graphs, Tables, Pictures and Code]] 
-  * Chapter 5: [[theory:​sensor_technology:​st4_sensor_theory|Basic Sensor Theory]] 
-  * Chapter 6: [[theory:​sensor_technology:​st6_sensoractuatorsystems|Sensor-Actuator Systems]] ​ 
-  * Chapter 7: [[theory:​sensor_technology:​st7_modelling_main|Modelling]] 
-  * Chapter 8: [[theory:​sensor_technology:​st8_accelerometer_model|Modelling:​ The Accelerometer]] - example of a second order system 
-  * Chapter 9: [[theory:​sensor_technology:​st9_scaling|Modelling:​ Scaling]] - why small things appear to be stiffer 
-  * Chapter 10: [[theory:​sensor_technology:​st10_lumped_element_models|Modelling:​ Lumped Element Models]] 
-  * Chapter 11: [[theory:​sensor_technology:​st11_finite_element_models|Modelling:​ Finite Element Models]] 
-  * Chapter 12: [[theory:​sensor_technology:​st12_impedance_spectroscopy|Modelling:​ Transducer Characterization by Impedance Spectroscopy]] 
-  * Chapter 13: [[theory:​sensor_technology:​st13_lumped_element_models_advanced|Modelling:​ Systems Theory]] 
-  * Chapter 14: [[theory:​sensor_technology:​st14_differential_equation_numerical_models|Modelling:​ Numerical Integration]] 
-  * Chapter 15: [[theory:​sensor_technology:​st15_signal_conditioning_and_sensor_read-out|Signal Conditioning and Sensor Read-out]] 
-  * Chapter 16: [[theory:​sensor_technology:​st16_resistive_sensors|Resistive Sensors]] 
-  * Chapter 17: [[theory:​sensor_technology:​st17_capacitive_sensors|Capacitive Sensors]] 
-  * Chapter 18: [[theory:​sensor_technology:​st18_magnetic_sensors|Magnetic Sensors]] 
-  * Chapter 19: [[theory:​sensor_technology:​st19_optical_sensors|Optical Sensors]] 
-  * Chapter 20: [[theory:​sensor_technology:​st20_actuators|Actuators]] - an example of an electrodynamic motor 
-  * Chapter 21: [[theory:​sensor_technology:​st21_actuator_models|Actuator principles for small speakers]] 
-  * Chapter 22: [[theory:​sensor_technology:​st22_adc_and_dac|ADC and DAC]] 
-  * Chapter 23: [[theory:​sensor_technology:​st23_bus_interfaces|Bus Interfaces]] - SPI, I<​sup>​2</​sup>​C,​ IO-Link, Ethernet based 
-  * Appendix A: [[theory:​sensor_technology:​sta_easyunitconversion|Systematic unit conversion]] 
-  * Appendix B: Common Mode Rejection Ratio (CMRR) 
-  * Appendix C: [[theory:​sensor_technology:​std_schmitt_trigger|A Schmitt Trigger for sensor level detection]] <- Next 
- 
  
theory/sensor_technology/stc_common_mode_rejection_ratio_cmrr.1539747658.txt.gz · Last modified: 2018/10/17 03:40 by 40.77.167.206