Measurement of Alternating Electrical Signals

Abstract

The most comprehensive information on alternating signals is obtained by observing their temporal evolution, i.e., by measuring continuously in time the value of the signal (current or voltage as required by the problem under consideration). This kind of measurement is generally carried out with devices capable of displaying the time development of the waveform as the oscilloscopes, instruments that we will deal with extensively in the following chapter.

Problems

Problem 1

An ADC digitizes a sinusoidal voltage signal taking N samples at constant time intervals covering a span equal to the signal period. Sampled values are affected by quantization errors, uncorrelated and identically distributed with variance equal to $\sigma ^2$. Data are used to identify their maximum and minimum from which the peak-to-peak Xpp is computed. Evaluate the standard uncertainty on this last quantity. [A. $u_{pp}=\sqrt{2}\sigma$.]

Problem 2

An ADC digitizes a sinusoidal voltage signal taking N samples at constant time intervals covering a span equal to the signal period. Sampled values are affected by quantization errors, uncorrelated and identically distributed with variance equal to $\sigma ^2$. Show that the estimate of the effective value, obtained as the root mean square of the sampled values, is affected by a bias related to the variance of quantization errors. [A. $\hat{V}_{rms}=\sqrt{V^2_{rms}+\sigma ^2}$.]

Problem 3

An ADC digitizes a sinusoidal voltage signal taking N samples at constant time intervals covering a span equal to the signal period. Sampled values are affected by quantization errors, uncorrelated and identically distributed with variance equal to $\sigma ^2$. Show that the estimate of the effective value, obtained as the root mean square of the sampled values, besides being biased (see previous problem) is affected by a statistical uncertainty related to the variance of quantization errors. [A. $u_V=\sigma /\sqrt{N}$.]

Problem 4

An ADC, with negligible quantization error, digitizes a sinusoidal voltage signal taking N samples at constant time intervals covering a span equal to the signal period. Sampled values are affected by an ambient noise, uncorrelated and identically distributed with variance equal to $\sigma ^2_n$. Show that the estimate of the effective value, obtained as the root mean square of the sampled values, is affected by a bias related to the variance of ambient noise. [A. $\hat{V}_{rms}=\sqrt{V^2_{rms}+\sigma ^2_n}$.]

Problem 5

An ADC, with negligible quantization error, digitizes a sinusoidal voltage signal taking N samples at constant time intervals covering a span equal to the signal period. Sampled values are affected by an ambient noise, uncorrelated and identically distributed with variance equal to $\sigma ^2_n$. Show that the estimate of the effective value, obtained as the root mean square of the sampled values, besides being biased (see previous problem) is affected by a statistical uncertainty related to the noise variance. [A. $u_V=\sigma _n/\sqrt{N}$.]

Problem 6

A resistor R is heated by Joule dissipation. To compute its temperature, assume that the resistor is cooled by thermal conduction and that the power lost to the environment is given by $Q=-k(T-T_0)$. The resistor is in equilibrium with the environment at temperature $T_0$ before being connected to an ideal voltage generator with output equal to $V_0$. Denoting with C its thermal capacity, calculate the time evolution of the resistor temperature after the connection to the generator. [A. $T=T_0+(P\tau /C)(1-\text{ e }^{-t/\tau }$ where $P=V_0^2/R$ and $\tau =C/k$.]