AC bridge

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\textbf{Final Exam EE1}\\* February \newcommand{\ts}{\textsuperscript}2\ts{nd}, 2012\\*
Prof. Dr. rer. nat. Rasmus Rettig
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\section{AC bridge}
The voltage measured between the points A and B in the included
schematics is UAB=0V. The RMS voltage of the source is 10V, its
frequency is 1kHz.\\*
\includegraphics[width=0.7\linewidth]{AC_circuit.png}

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\textbf{Final Exam EE1}\\* February \newcommand{\ts}{\textsuperscript}2\ts{nd}, 2012\\*
Prof. Dr. rer. nat. Rasmus Rettig
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a)What is the generic form of the balancing condition for an AC bridge
circuit? Start with this condition, add the respective impedances as
indicated in the schematic and derive the equations for R1 and C1.
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Balancing Condition:
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Calculation:
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Formula R1=...\\*
Formula R2=...
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\textbf{Final Exam EE1}\\* February \newcommand{\ts}{\textsuperscript}2\ts{nd}, 2012\\*
Prof. Dr. rer. nat. Rasmus Rettig
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b) Calculate the values for R1 and C1 based on the parameters in the
schematic.\\*
Calculation:
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R1=$\rule{3cm}{0.15mm}$\\*
C1=$\rule{3cm}{0.15mm}$\\*
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\section*{16.1.  A.C. Bridges\cite{AC bridge}}
Resistances can be measured by direct-current Wheatstone bridge, shown in Fig. 16.1 (a) for which the condition of balance is that
\begin{center}
\[
\frac{R_1}{R_2} = {R_4}{R_3} \textnormal{ or } R_1R_3 \hspace{1cm}R_2R_4
\]
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Inductances and capacitances can also be measured by a similar four-arm bridge, as shown in Fig. 16.1 (b); instead of using a source of direct current, alternating current is employed and galvanometer is replaced by a vibration galvanometer (for commercial frequencies or by telephone detector if frequencies are higher (500 to 2000 Hz))
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\includegraphics[width=0.7\linewidth]{cir5.PNG}\\*
\textbf{Fig. 16.1}
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The condition for balance is the same as before but instead of resistances, impedances are used i.e.
\begin{center}
\[
\frac{Z_1}{Z_2} = \frac{Z_4}{Z_3} \textnormal{ or } Z_1Z_3=Z_2Z_4
\]
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But there is one important difference i.e. not only should there be balance for the magnitudes of the impedances but also a phase balance. Writing the impedances in their polar form, the above condition becomes
\[
Z_1<\phi_1Z_3<\phi_3=Z_2<\phi_2Z_4<\phi_4 \textnormal{ or } Z_1Z_3<\phi_1 + \phi_3 = Z_2Z_4<\phi_2 + \phi_4
\]
Hence, we see that, in fact, there are two balance conditions which must be satisfied simultaneously in a four-arm a.c. impedance bridge.
\[
(i)\hspace{0.1cm}Z_1Z_3 = Z_2Z_4 \hspace{4cm}\textnormal{...for magnitude balance}
\]
\[
(ii) \hspace{0.1cm}\phi_1 + \phi_3 = \phi_2 + \phi_4\hspace{3.15cm}\textnormal{...for phase angle balance}
\]
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\section*{7.8 AC bridges\cite{EE1 script}}
If we replace the resistors of a Wheatstone bridge by capacitors and inductors we get an AC bridge
that can be treated in a similar way to the DC bridge. The following bridges are designed to measure
inductances, capacity or frequency.
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\[
\textnormal{Voltage across the bridge: }\underline{U_a}\textnormal{\hspace{10cm}}
\]
\[
\underline{U_a} = \underline{U} \left(\frac{\underline{Z_2}}{\underline{Z_1} + \underline{Z_2}} - \frac{\underline{Z_4}}{\underline{Z_3} + \underline{Z_4}}\right)
\textnormal{\hspace{5cm}}
\]
\[
\textnormal{Balance condition:\hspace{5cm}}
\]
\[
\frac{\underline{Z_1}}{\underline{Z_2}} = \frac{\underline{Z_3}}{\underline{Z_4}}\textnormal{\hspace{5cm}}
\]
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\textbf{Maxwell-Wien bridge}\\*
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\[
\textnormal{To be measured: inductor }\textnormal{\hspace{10cm}}
\]
\[
\textnormal{inductance } L_1 \textnormal{ with}\textnormal{\hspace{10cm}}
\]
\[
\textnormal{dissipative element } R_1 \textnormal{\hspace{10cm}}
\]
\[
\textnormal{Balance condition:\hspace{5cm}}
\]
\[
R_1 = \frac{R_2 R_3}{R_4}\textnormal{\hspace{10cm}}
\]
\[
L_1 = R_2 R_3 C_4\textnormal{\hspace{10cm}}
\]
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\textbf{Capacity bridge}\\*
\includegraphics[width=0.5\linewidth]{cir3.PNG}
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\[
\textnormal{To be measured: capacitor }\textnormal{\hspace{10cm}}
\]
\[
\textnormal{capacity } C_1 \textnormal {with dissipative element } R_1 \textnormal{ }\textnormal{\hspace{10cm}}
\]
\[
\textnormal{Balance condition:\hspace{5cm}}
\]
\[
R_1 = \frac{R_2 R_3}{R_4}\textnormal{\hspace{10cm}}
\]
\[
C_1 = \frac{C_2 R_4}{R_3}\textnormal{\hspace{10cm}}
\]
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\textbf{Wien-Robinson bridge}\\*
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\[
\textnormal{To be measured: frequency }\omega = 2\pi \textnormal{f} \textnormal{\hspace{10cm}}
\]
\[
\textnormal{Balance condition:\hspace{5cm}}
\]
\[
1 = \omega^2R_1R_2C_1C_2 \textnormal{ and }\frac{C_1}{C_2} + \frac{R_2}{R_1} = \frac{R_4}{R_3}\textnormal{\hspace{10cm}}
\]
\[
\textnormal{Let } C_1 = C_2 = C, R_1 = R_2 = \textnormal{Rand}R_4 = 2R_3\textnormal{\hspace{10cm}}
\]
\[
\textnormal{Hence : } \omega = \frac{1}{RC}\textnormal{\hspace{10cm}}
\]
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\begin{thebibliography}{9}
\bibitem{AC bridge}
A.C. Bridges.
\\\texttt{http://www.nct-tech.edu.lk/Download/Technology\%{}20Zone/AC\%{}20Bridges.pdf}

\bibitem{EE1 script}
Prof. Dr.-Ing. Jörg Dahlkemper.
\textit{Department Informations- und Elektrotechnik}.\\
Electrical Engineering 1, 22.09.2017.
\end{thebibliography}
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