State three commonly used detectors for a.c bridge measurements. With the aid of a labelled diagram, derive the balance equation for an inductive impedance a.c bridge in terms of rectangular co-ordinates.

Kwa daraja la Wein lililosawazishwa, tunatumia masharti ya usawa wa daraja la a.c.
Vizuizi vya mikono ni:

• Mkono AD: $Z_{AD} = R_1 + \frac{1}{j\omega C_1}$
• Mkono AB: $Z_{AB} = \frac{R_2 \cdot \frac{1}{j\omega C_2}}{R_2 + \frac{1}{j\omega C_2}} = \frac{R_2}{1 + j\omega R_2 C_2}$
• Mkono BC: $Z_{BC} = R_4$
• Mkono DC: $Z_{DC} = R_3$

Panua na panga upya:

Ili kutenganisha sehemu halisi na dhahania, zidisha upande wa kulia kwa kiambishi tamati (conjugate) cha denomineta:

Kulinganisha sehemu halisi:

Kulinganisha sehemu dhahania:

Kutoka Eq. 1, tunaweza kuandika:

Badilisha Eq. 3 kwenye Eq. 2:

Gawanya pande zote mbili kwa $R_3$:

Hii inatoa usemi wa $C_1$ unaotegemea $\omega$:

Ili kupata usemi wa $C_1$ usiotegemea $\omega$, tunaweza kutumia Eq. 1 na Eq. 2 tena.
Badilisha $\omega^2 = \frac{1}{R_1 R_2 C_1 C_2}$ (kutoka hapo juu) kwenye Eq. 1:

Panga upya ili kupata $C_1$:

Usemi wa capacitor $C_1$ ni:

Kutoka hatua ya 4 ya sehemu (a), tuligundua usemi wa $\omega^2$:

Masafa $f$ yanatolewa na $f = \frac{\omega}{2\pi}$, hivyo:

$R_1 = 100 \, \Omega$
$C_1 = 1 \, \mu F = 1 \times 10^{-6} \, F$
$R_2 = 200 \, \Omega$
$C_2 = 2 \, \mu F = 2 \times 10^{-6} \, F$

Kukokotoa thamani ya namba:

Masafa katika usawa ni: