= Electrical impedance phase measurement
{title2=$|Z|=R V_1/V_R,\quad\phi=2\pi\Delta t/T$}
An actual teaching experiment turns the <phasor> definition into readings. Leipzig University's laboratory handout Measurements with an oscilloscope, experiment E 3e, uses a <function generator>, a two-channel <oscilloscope>, a known <resistor> and a <capacitor>. Figure 5 places the <capacitor> above the <resistor>, with a common return below the <resistor>. Channel 1 measures the applied terminal <electric potential difference> $v_1$; channel 2 measures $v_R$. The handout asks for peak-to-peak values and relative timing, then comparison of measured and calculated <electrical impedances> with $R=3\,\mathrm{k\Omega}$, $C=0.47\,\mu\mathrm F$ at $100$ and $300\,\mathrm{Hz}$. It also asks students to check the oscilloscope time base against a digitally counted generator <frequency>. These are apparatus and procedural specifications, not reported measured results.<Measurements with an oscilloscope, experiment E 3e, Physikalisches Grundpraktikum, Universität Leipzig>[Measurements with an oscilloscope, E 3e, p. 1 tasks 1 and 4; pp. 5–6 Measurement of impedances, figure 5 and experimental comparison]{ref}
For this arrangement, <Ohm's law> gives $i(t)=v_R(t)/R$ if probe loading and the sensing <resistor>'s <electrical reactance> are negligible. Record both voltage magnitudes with the same convention: peak, peak-to-peak, or <root mean square>. After the transient settles, compare matched rising zero crossings $t_1$ of $v_1$ and $t_R$ of $v_R$, with <period> $T=1/f$. Then
$$
|Z|=R\frac{V_1}{V_R},\qquad
\Delta t=t_R-t_1,\qquad
\phi=\frac{2\pi\Delta t}{T},\qquad
Z=|Z|(\cos\phi+\mathrm i\sin\phi).
$$
The recovered <electrical impedance> includes the known <resistor>. For a separate unknown component in series with that <resistor>, subtract $R$ from the inferred total <electrical impedance>. A current crossing earlier than the applied voltage crossing gives $\Delta t<0$, hence negative $\phi$. For the stipulated <series RC impedance> example at $1000\,\mathrm{Hz}$, this predicts $\Delta t=-0.160717\,\mathrm{ms}$; it is a calculated timing, not a lab observation. Timing supplies the <complex argument> that an amplitude ratio alone cannot recover.
As additional measurement controls, first feed the same <sinusoid> to both channels to check relative delay and polarity; then replace the reactive chain with a known <resistor>, expecting nearly zero <phase difference>. Keep both probe returns at the common return shown in figure 5. A reversed voltage polarity adds half a turn to its inferred <wave phase>. Match crossings with the same slope and choose $-T/2<\Delta t\leq T/2$; adjacent cycles otherwise change $\phi$ by a full turn. A sweep continuous in <frequency> helps track that branch. These controls test the signed inference rather than assuming that the scope's delay readout has the desired sign.
Measure the loaded terminal <electric potential difference> on channel 1 at every <frequency>. The generator's <output impedance> can change the applied magnitude; its front-panel setting is not the measured terminal value. The generator's internal <electrical impedance> is outside the terminal ratio above, while the known sensing <resistor> is inside it. Probe loading, finite timing resolution, component tolerances and distorted waveforms limit the inference. A repeat at lower drive <amplitude> tests whether one linear <electrical impedance> still fits. <Electrical reactance bench measurement> develops these loading and reference-direction issues further.
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