5.3. Noise characteristics LECTURE 6. Contents 5. Sources of errors 5.1. Impedance matching 5.4.1. Non-energetic matching 5.4.2. Energetic matching 5.4.3. Non-reflective matching 5.4.4. To match or not to match? 5.2. Noise types 5.2.1. Thermal noise 5.2.2. Shot noise 5.2.3. 1/f noise 5.3. Noise characteristics 5.3.1. Signal-to-noise ratio, SNR 5.3.2. Noise factor, F, and noise figure, NF 5.3.3. Calculating SNR and input noise voltage from NF 5.3.4. Vn-In noise model 5.4. Noise matching 5.4.1. Optimum source resistance 5.4.2. Methods for the increasing of SNR 5.4.3. SNR of cascaded noisy amplifiers

5. SOURCES OF ERRORS 5. SOURCES OF ERRORS Measurement errors can occur due to the undesirable interaction between the measurement system and: the object under test, the environment, observer. E n v i r o n m e n t Disturbance +Dy +D x x y 1 Measurement Object Matching Measurement System Matching Observer Influence Influence

5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Non-energetic matching 5.1. Impedance matching Systematic measurement errors can occur due to the undesirable interaction between the measurement system and: the object under test. +D x x Measurement Object Measurement System Matching Influence

5.1.2. Non-energetic matching 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Non-energetic matching There are three types of impedance matching: non-energetic, energetic, and non-reflective. 5.1.2. Non-energetic matching Non-energetic matching is used to minimize the transfer of energy between the measurement object and the measurement system. After matching, measurement system will not supply any appreciable energy to, or receive from the measurement object. Non-energetic matching is usually used in active measurement systems, which do possess internal power amplification. Reference: [1]

Example: Non-energetic matching 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.1. Non-energetic matching Example: Non-energetic matching Measurement object Measurement system Rin >> RS  vin  vS the power supplied by the object is small most part of it is dissipated in Rin RS vS vin Rin Rin << RS  iin  iS the power supplied by the object is small most part of it is dissipated in Rin Measurement object Measurement system iin RS iS Rin

5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energetic matching The aim of energetic matching is to extract the maximum available power from the measurement object, so that the required power gain in the measurements system can be as small as possible. Energetic matching is especially important for passive measurement systems, which do not possess internal power amplification. Reference: [1]

 VS2 Rin (RS+Rin)2 + (XS+Xin)2 Pin = Iin2 Rin = . 2 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energetic matching To optimize the energetic matching, let us consider the following equivalent circuits of the measurement object and the measurement system. Measurement object Measurement system iin ZS= RS + XS Zin= Rin + Xin vS vin The average power delivered to the measurement system can be found as: VS2 Rin (RS+Rin)2 + (XS+Xin)2 Pin = Iin2 Rin = .  2 Iin and VS are measured in A rms and V rms.

Rin = RS and Xin = - XS or Zin= ZS* . Pin = = . V S2 4 Rin 4 RS 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energetic matching For a fixed non-zero RS, this power is maximal if the following optimal matching is obtained: Rin = RS and Xin = - XS or Zin= ZS* . Therefore, the maximum power that a measurement object with a fixed non-zero RS can deliver to a measurement system is: Pin = = . V S2 4 Rin 4 RS Reference: [1]

Rin = 0 and Xin = - XS . V S2 Pin = . Rin 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energetic matching If RS = 0, then the optimal matching is obtained when Rin = 0 and Xin = - XS . In this case, the maximum power a measurement object can deliver to a measurement system is : V S2 Pin = . Rin Reference: [1]

5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.2. Energetic matching Available power (H. T. Friis, 1944) is defined as the maximum power that can be delivered to a load from a source having fixed nonzero resistance Vin 2 Pa  Pin = . 4 RS RS0 NB: The maximum power matching usually causes greater measurement errors, since the input and output impedances of the chain affect the measurement. For this reason, the measurement systems almost always are active ones (with built-in power gain). Reference: [1]

 5.1.3. Non-reflective matching L Z0 = R0 = . C 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching 5.1.3. Non-reflective matching Non-reflective or characteristic matching is used for transporting high-frequency measurement signals along transmission lines. If a transmission line is not terminated characteristically, reflections off the ends of the line will cause standing waves on the line; the line output signal is no longer a good measure for the line input signal. The characteristic impedance, Z0, of a transmission line equals its input impedance if the transmission line length were infinite. For a lossless transmission line with the series inductance per meter L and the parallel capacitance per meter C,  L Z0 = R0 = . C Reference: [1]

Illustration: Non-reflective matching: ZS = R0 = Zin 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching Illustration: Non-reflective matching: ZS = R0 = Zin Measurement object Measurement system Z0 Z0 ZS vS Zin vin = 0.5vS NB: When ZS = R0 = Zin holds, energetic matching is also achieved simultaneously, since ZS = RS and Zin = Rin. R0 is an apparent resistance that does not dissipate energy; half of the energy delivered by vs is dissipated in RS and the other half in Rin. Reference: [1]

Example: The characteristic impedances of different connections 5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.3. Non-reflective matching Example: The characteristic impedances of different connections Type of connection Characteristic impedance DEFINITION Coaxial cable 50 - 75 W Printed circuit board traces 50 - 150 W Twisted wire pairs 100 - 120 W Ribbon cable 200 - 300 W Free space 376 W Reference: [1]

5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.4. When to match and when not? 5.1.5. To match or not to match? Do match by adjusting impedances, by adding voltage buffers or by adding matching transformers: To transfer maximum power to the load. The source must be capable. To minimise reflections from the load. Important in audio, fast (high frequency) systems, to avoid ringing or multiple pulses (e.g. in counting systems). To transmit fast pulses. Pulse properties can contain important information. Note that the same physics is encountered in other areas, e.g. optical coatings, gel in ultrasound scans, optical grease, etc. Reference: www.hep.ph.ic.ac.uk/Instrumentation/

5. SOURCES OF ERRORS. 5.1. Influencing the measurement object: matching. 5.1.4. When to match and when not? Do not match: High impedance source with small current signals. Typical for many photodiode sensors, or other sensors that must drive high impedance load. Short cables are required to avoid difficulties. Weak voltage source. Drawing power from source would affect the result, e.g. bridge circuits. If you need to change properties of a fast pulse, e.g. pulse widening for ease of detection. Electronics with limited drive capabilities, e.g. logic circuits, many are designed to drive other logic, not long lines, CMOS circuits, even with follower, are an example. Reference: www.hep.ph.ic.ac.uk/Instrumentation/

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