In many measurement techniques with time varying perturbations of a sample like impedance spectroscopy, IMPS/IMVS or Kelvin probes the notion of in-phase and quadrature components is present.
Why bother with adding more names to the problem?
It turns out they do simplify the analysis of the sample response to the applied perturbation.
Suppose your instrument generates the perturbation $g(t)$ (that could be an electric voltage or current, light intensity, temperature and so on)
\begin{equation}
g(t) = \sum_{k=1}^K a_k\cos(\omega_k t + \phi_k),
\label{gener}
\end{equation}
which is then applied to a sample. The sample, in general, responds in similar fashion but with a slightly modified signal
$$
s(t) = \sum_{k=1}^K b_k \cos(\omega_k t + \phi_k + \Delta\phi_k),
$$
which has different amplitudes $b_k$ and phase shifts $\Delta \phi_k$ relative to the original stimulus $g(t)$. The frequencies $\omega_k$ are the same$^*$.
Using the trigonometric formula $\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)$, the response above can be rewritten in the more convenient form$^{**}$ as
\begin{equation}
s(t) = \sum_{k=1}^K ( \textcolor[rgb]{0, 0.73, 0.93}{X_k} \cos(\omega_k t + \phi_k) + \textcolor[rgb]{0, 0.73, 0.93}{Y_k}\sin(\omega_k t + \phi_k )),
\label{XY}
\end{equation}
where quantities from the stimulus in Eq.$\ref{gener}$ are explicitly used.
Now the meaning of the in-phase $X_k$ components and the quadrature $Y_k$ is clearer.
- The $X_k$ tells immediately how much of the original stimulus has survived in the sample’s response.
- The $Y_k$ gives the amount of the stimulus pushed to the orthogonal$^{***}$ subspace by the sample.
For example, a sample that has the ability to shift some of the stimulus into the $Y_k$ component must be able to store the energy and release it later.
$^*$ In the case of nonlinear samples there might be more frequencies created by the sample, but the original frequency components will be present anyway.
$^{**}$ It is assumed that the signal is available on a time interval $T$ long enough so that $\cos(\omega_k t)$ functions are orthogonal for diffrent $k$.
$^{***}$ The $\sin()$ function is orthogonal to the $\cos()$ function on the interval equal to their period, i.e.,
$\int_0^T \cos(\omega t)\sin(\omega t)\mbox{d}t = 0$, for $T = \frac{2\pi}{\omega}$.
