In-phase and quadrature components

The term alternating current applies to a voltage vs. time function that is sinusoidal with a frequency f.  When it is applied to a typical (linear) circuit or device, it causes a current that is also sinusoidal. In general there is a constant phase difference, φ, between any two sinusoids. The input sinusoidal voltage is usually defined to have zero phase, meaning that it is arbitrarily chosen as a convenient time reference. So the phase difference is attributed to the current function, e.g. sin(2πft + φ), whose orthogonal components are sin(2πft) cos(φ) and sin(2πft + π/2) sin(φ), as we have seen. When φ happens to be such that the in-phase component is zero, the current and voltage sinusoids are said to be in quadrature, which means they are orthogonal to each other. In that case, no electrical power is consumed. Rather it is temporarily stored by the device and given back, once every ​¹⁄_f  seconds. Note that the term in quadrature only implies that two sinusoids are orthogonal, not that they are components of another sinusoid.

In an angle modulation application, with carrier frequency f, φ is also a time-variant function, giving:

${\displaystyle A(t)\cdot \sin[2\pi ft+\phi (t)]\ =\ \underbrace {A(t)\cdot \sin(2\pi ft)\cdot \cos[\phi (t)]} _{\text{in-phase}}\,+\,\underbrace {A(t)\cdot \overbrace {\sin \left(2\pi ft+{\tfrac {\pi }{2}}\right)} ^{\cos(2\pi ft)}\cdot \sin[\phi (t)]} _{\text{quadrature}}.}$

When all three terms above are multiplied by an optional amplitude function, A(t) > 0, the left-hand side of the equality is known as the amplitude/phase form, and the right-hand side is the quadrature-carrier or IQ form. Because of the modulation, the components are no longer completely orthogonal functions. But when A(t) and φ(t) are slowly varying functions compared to 2πft, the assumption of orthogonality is a common one.[upper-alpha 1] Authors often call it a narrowband assumption, or a narrowband signal model.[3][4]

The terms I-component and Q-component are common ways of referring to the in-phase and quadrature signals. Both signals comprise a high-frequency sinusoid (or carrier) that is amplitude-modulated by a relatively low-frequency function, usually conveying some sort of information. The two carriers are orthogonal, with I lagging Q by ¼ cycle, or equivalently leading Q by ¾ cycle.  The physical distinction can also be characterized in terms of ${\displaystyle \phi (t)}$:

• ${\displaystyle \phi (t)=0}$: The composite signal reduces to just the I-component, which accounts for the term in-phase.
• ${\displaystyle \phi (t)=\pi /2}$: The composite signal reduces to just the Q-component.
• ${\displaystyle \phi (t)=2\pi f_{m}t,\quad f_{m}>0}$: The amplitude modulations are orthogonal sinusoids, I leading Q by ¼ cycle.
• ${\displaystyle \phi (t)=2\pi f_{m}t,\quad f_{m}<0}$: The amplitude modulations are orthogonal sinusoids, Q leading I by ¼ cycle.