# In-phase and quadrature components

Jump to navigation Jump to search Graphic example of the formula  ${\color {Green}\cos(2\pi ft+\phi (t))}\ =$ ${\color {Blue}\cos(2\pi ft)\cos(\phi (t))}\ +\ {\color {Red}\cos(2\pi ft+\pi /2)\sin(\phi (t))}.$ The phase modulation (φ(t), not shown) is a non-linearly increasing function from 0 to π/2 over the interval 0 < t < 16. The two amplitude-modulated components are known as the in-phase component (I, thin blue, decreasing) and the quadrature component (Q, thin red, increasing).

In electrical engineering, a sinusoid with angle modulation can be decomposed into, or synthesized from, two amplitude-modulated sinusoids that are offset in phase by one-quarter cycle (90 degrees or π/2 radians). All three functions have the same center frequency. Such amplitude modulated sinusoids are known as the in-phase and quadrature components.  In some contexts it is more convenient to refer to only the amplitude modulation (baseband) itself by those terms.

## Concept

In vector analysis, a vector with polar coordinates A,φ and Cartesian coordinates x = A cos(φ), y = A sin(φ), can be represented as the sum of orthogonal components: [x,0] + [0,y]. Similarly in trigonometry, the angle sum identity expresses:

sin(x + φ)=sin(x) cos(φ) + sin(x + π/2) sin(φ).

And in functional analysis, when x is a linear function of some variable, such as time, these components are sinusoids, and they are orthogonal functions. A phase-shift of xx + π/2 changes the identity to:

cos(x + φ) = cos(x) cos(φ) + cos(x + π/2) sin(φ),

in which case cos(x) cos(φ) is the in-phase component.  In both conventions cos(φ) is the in-phase amplitude modulation, which explains why some authors refer to it as the actual in-phase component. When a sinusoidal voltage is applied to either a simple capacitor or inductor, the resultant current that flows is "in quadrature" with the voltage.

### Alternating current (AC) circuits

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 1f  seconds. Note that the term in quadrature only implies that two sinusoids are orthogonal, not that they are components of another sinusoid.

### Narrowband signal model

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

$A(t)\cdot \sin[2\pi ft+\phi (t)]\ =\ \underbrace {A(t)\cdot \sin(2\pi ft)\cdot \cos[\phi (t)]} _{\text{in-phase}}\,+\,\underbrace {A(t)\cdot \overbrace {\sin \left(2\pi ft+{\tfrac {\pi }{2}}\right)} ^{\cos(2\pi ft)}\cdot \sin[\phi (t)]} _{\text{quadrature}}.$ 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 ft, the assumption of orthogonality is a common one.[A] Authors often call it a narrowband assumption, or a narrowband signal model.

### IQ phase convention

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 $\phi (t)$ :

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