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Wien bridge oscillator
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==Wien bridge== {{Main article|Wien bridge}} <!-- need diagram with A B C D arms --> <!-- I do not see much point in this section here since there is an article dedicated to Wien bridge. --> Bridge circuits were a common way of measuring component values by comparing them to known values. Often an unknown component would be put in one arm of a bridge, and then the bridge would be nulled by adjusting the other arms or changing the frequency of the voltage source (see, for example, the [[Wheatstone bridge]]). The Wien bridge is one of many common bridges.<ref>{{Harvnb|Terman|1943|p=904}}</ref> Wien's bridge is used for precision measurement of capacitance in terms of resistance and frequency.<ref>{{Harvnb|Terman|1943|p=904}} citing {{Harvnb|Ferguson|Bartlett|1928}}</ref> It was also used to measure audio frequencies. The Wien bridge does not require equal values of ''R'' or ''C''. The phase of the signal at V<sub>p</sub> relative to the signal at V<sub>out</sub> varies from almost 90Β° leading at low frequency to almost 90Β° lagging at high frequency. At some intermediate frequency, the phase shift will be zero. At that frequency the ratio of Z<sub>1</sub> to Z<sub>2</sub> will be purely real (zero imaginary part). If the ratio of ''R<sub>b</sub>'' to ''R<sub>f</sub>'' is adjusted to the same ratio, then the bridge is balanced and the circuit can sustain oscillation. The circuit will oscillate even if ''R<sub>b</sub>'' / ''R<sub>f</sub>'' has a small phase shift and even if the inverting and non-inverting inputs of the amplifier have different phase shifts. There will always be a frequency at which the total phase shift of each branch of the bridge will be equal. If ''R<sub>b</sub>'' / ''R<sub>f</sub>'' has no phase shift and the phase shifts of the amplifiers inputs are zero then the bridge is balanced when:<ref>{{Harvnb|Terman|1943|p=905}}</ref> :<math>\omega^2 = {1 \over R_1 R_2 C_1 C_2}</math> and <math> {R_f \over R_b} = {C_1 \over C_2} + {R_2 \over R_1} </math> where Ο is the radian frequency. If one chooses ''R<sub>1</sub>'' = ''R<sub>2</sub>'' and ''C<sub>1</sub>'' = ''C<sub>2</sub>'' then ''R<sub>f</sub>'' = 2 ''R<sub>b</sub>''. In practice, the values of ''R'' and ''C'' will never be exactly equal, but the equations above show that for fixed values in the Z<sub>1</sub> and Z<sub>2</sub> impedances, the bridge will balance at some ''Ο'' and some ratio of ''R<sub>b</sub>''/''R<sub>f</sub>''.
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