# In-phase and quadrature components

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 (π/2 radians). All three functions have the same frequency. The amplitude modulated sinusoids are known as **in-phase** and **quadrature** components.^{[1]} In some contexts it is more convenient to refer to only the amplitude modulation (*baseband*) itself by those terms.^{[2]}

### 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 expression **sin( x + φ)** can be represented by

**sin(**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. When

*x*) cos(*φ*) + sin(*x*+ π/2) sin(*φ*).*φ*= 0, sin(

*x*+

*φ*) reduces to just the

**in-phase**component, sin(

*x*) cos(

*φ*), and the

**quadrature**component, sin(

*x*+ π/2) sin(

*φ*), is zero. A phase-shift of

*x*→

*x*+ π/2 changes the identity to

**cos(**, in which case cos(

*x*+*φ*) = cos(*x*) cos(*φ*) + cos(*x*+ π/2) sin(*φ*)*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. We can also observe that in both conventions the quadrature component

*leads*the in-phase component by one-quarter cycle.

#### 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 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(2π*ft* + φ), whose orthogonal components are sin(2π*ft*) cos(φ) and sin(2π*ft* + 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.

#### Narrowband signal model

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

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 2π*ft*, the assumption of orthogonality is a common one. Authors often call it a *narrowband assumption*, or a **narrowband signal model**.^{[3]}^{[4]} Orthogonality is important in many applications, including demodulation, direction-finding, and bandpass sampling.

### See also

- IQ imbalance
- Constellation diagram
- Phasor
- Polar modulation
- Quadrature amplitude modulation
- Single-sideband modulation

### References

**^**Gast, Matthew (2005-05-02).*802.11 Wireless Networks: The Definitive Guide*.**1**(2 ed.). Sebastopol,CA: O'Reilly Media. p. 284. ISBN 0596100523.**^**Franks, L.E. (September 1969).*Signal Theory*. Information theory. Englewood Cliffs, NJ: Prentice Hall. p. 82. ISBN 0138100772.**^**Wade, Graham (1994-09-30).*Signal Coding and Processing*.**1**(2 ed.). Cambridge University Press. p. 10. ISBN 0521412307.**^**Naidu, Prabhakar S. (November 2003).*Modern Digital Signal Processing: An Introduction*. Pangbourne RG8 8UT, UK: Alpha Science Intl Ltd. pp. 29–31. ISBN 1842651331.

### Further reading

- Steinmetz, Charles Proteus (2003-02-20).
*Lectures on Electrical Engineering*.**3**(1 ed.). Mineola,NY: Dover Publications. ISBN 0486495388. - Steinmetz, Charles Proteus (1917).
*Theory and Calculations of Electrical Apparatus***6**(1 ed.). New York: McGraw-Hill Book Company. B004G3ZGTM.

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