NBPhO 2026

Difficulty medium

Prerequisites

• Phasor representation of sinusoidal signals at one frequency
• Complex impedance: $Z_C = 1/(j\omega C)$, $Z_L = j\omega L$
• Voltage-divider formula for two impedances in series
• Lissajous figure $\to$ line segment iff $V_x$ and $V_y$ are in phase (or anti-phase)
• Potentiometer modelled as a uniform resistor with a tap at fractional length

Learning objectives

• Translate an $RC$/$RL$ filter network into phasors with the source as real reference
• Recognise the Lissajous-line condition as $V_y/V_x \in \mathbb{R}$ and turn it into one real equation
• Identify a real, in-phase tap (the slider $V_P$) as the analogue of the variable arm in a Wheatstone bridge
• Show that a balanced AC bridge gives a frequency-independent relation between $R$, $L$, $C$ — and that the residual slope of the line encodes the frequency
• Read amplitude information off a Lissajous trace by projecting onto the screen axes

Watch out for

• $V_P$ is real (in phase with the source); $V_B$ and $V_D$ are not. The whole bridge trick relies on this asymmetry.
• Slider geometry: with $A$ on the left and $E$ on the right, the ratio $1{:}2$ from left to right means $AP = \tfrac{1}{3}AE$, so $V_P = \tfrac{2}{3}V$ — not $\tfrac{1}{3}V$.
• The slope $\tan\alpha$ is necessarily negative in this circuit; always $-2 < \tan\alpha < -\tfrac{1}{2}$. A positive answer for $\tan\alpha$ signals a sign error somewhere in the phasor algebra.
• Don't confuse the balance condition $L = 2R_1R_2C$ (a property of the components, frequency-independent) with the slope (which does depend on $\omega$).
• $|V_{BP}|$ is the half-extent of the line segment along the $x$-axis, not the full length of the line — the latter would also include the $y$-projection.