A commercial airliner has sensing devices to indicate to the cockpit crew that each door

Chapter 10, Problem P10.6-5

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A commercial airliner has sensing devices to indicate to the cockpit crew that each door and baggage hatch is closed. A device called a search coil magnetometer, also known as a proximity sensor, provides a signal indicative of the proximity of metal or other conducting material to an inductive sense coil. The inductance of the sense coil changes as the metal gets closer to the sense coil. The sense coil inductance is compared to a reference coil inductance with a circuit called a balanced inductance bridge (see Figure P 10.6-5). In the inductance bridge, a signal indicative of proximity is observed between terminals a and b by subtracting the voltage at b, \(v_b\), from the voltage at a, \(v_a\) (Lenz, 1990).

The bridge circuit is excited by a sinusoidal voltage source \(v_{\mathrm{s}}=\sin (800 \pi t) \mathrm{V}\). The two resistors, \(R=100 \Omega\), are of equal resistance. When the door is open (no metal is present), the sense coil inductance \(L_{\mathrm{S}}\) is equal to the reference coil inductance \(L_{\mathrm{R}}=40 \mathrm{mH}\). In this case, what is the magnitude of the signal \(\mathbf{V}_{\mathrm{a}}-\mathbf{V}_{\mathrm{b}}\)?

When the airliner door is completely closed, \(L_{\mathrm{s}}=60 \mathrm{mH}\). With the door closed, what is the phasor representation of the signal \(\mathbf{V}_{\mathrm{a}}-\mathbf{V}_{\mathrm{b}}\)?