Phasors in engineering
A phasor represents a sinusoidal quantity by its magnitude and phase angle. In AC circuits,
voltages and currents with the same frequency can be combined using complex numbers instead of
time-domain sine waves.
\[
Z=M\angle\theta.
\]
The magnitude \(M\) may represent RMS voltage, RMS current, peak amplitude, impedance magnitude,
or a force magnitude depending on the engineering context.
Polar form
Polar form gives a complex quantity by magnitude and angle.
\[
Z=M\angle\theta.
\]
This form is useful when a sinusoidal signal has a clear amplitude and phase shift.
Rectangular form
Rectangular form gives a complex quantity by real and imaginary components.
\[
Z=a+ib.
\]
In a phasor diagram, \(a\) is the horizontal component and \(b\) is the vertical component.
Polar to rectangular conversion
To convert a phasor from polar form to rectangular form, use cosine for the real component
and sine for the imaginary component.
\[
Z
=
M(\cos\theta+i\sin\theta).
\]
\[
a=M\cos\theta,
\qquad
b=M\sin\theta.
\]
Rectangular to polar conversion
To convert from rectangular form back to polar form, use the magnitude and the two-argument arctangent.
\[
M=\sqrt{a^2+b^2}.
\]
\[
\theta=\operatorname{atan2}(b,a).
\]
The function \(\operatorname{atan2}\) is preferred because it places the angle in the correct quadrant.
Adding phasors
Phasors are added by components. First convert each phasor to rectangular form, then add real parts
and imaginary parts separately.
\[
Z_R
=
\sum_{k=1}^{n}Z_k.
\]
\[
\operatorname{Re}(Z_R)=\sum_k \operatorname{Re}(Z_k),
\qquad
\operatorname{Im}(Z_R)=\sum_k \operatorname{Im}(Z_k).
\]
Subtracting phasors
Subtraction is also done component by component.
\[
Z_1-Z_2
=
\left(a_1-a_2\right)
+
i\left(b_1-b_2\right).
\]
In mechanics, this is the same idea as finding the difference between two vectors.
Multiplying phasors
Multiplication is especially simple in polar form: multiply magnitudes and add angles.
\[
\left(M_1\angle\theta_1\right)
\left(M_2\angle\theta_2\right)
=
M_1M_2\angle\left(\theta_1+\theta_2\right).
\]
This is useful for complex impedance and transfer-function calculations.
Dividing phasors
Division in polar form divides magnitudes and subtracts angles.
\[
\frac{M_1\angle\theta_1}{M_2\angle\theta_2}
=
\frac{M_1}{M_2}
\angle
\left(\theta_1-\theta_2\right).
\]
This operation appears in AC circuit formulas such as Ohm's law in phasor form.
\[
\tilde{I}=\frac{\tilde{V}}{Z}.
\]
Phasor diagrams
A phasor diagram places the real component on the horizontal axis and the imaginary component
on the vertical axis. The resultant phasor is the vector from the origin to the final combined point.
\[
Z_R=a_R+ib_R.
\]
For a sum of phasors, a head-to-tail diagram shows how each phasor contributes to the final resultant.
Engineering triangle convention
In triangle solving, side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\),
and side \(c\) is opposite angle \(C\).
\[
A+B+C=180^\circ.
\]
This convention is used in surveying, mechanics, statics, and geometry-based design problems.
Law of cosines
The law of cosines is used when three sides are known or when two sides and the included angle are known.
\[
c^2=a^2+b^2-2ab\cos C.
\]
Similar formulas can be written for sides \(a\) and \(b\).
\[
a^2=b^2+c^2-2bc\cos A.
\]
Law of sines
The law of sines is useful when two angles and one side are known.
\[
\frac{a}{\sin A}
=
\frac{b}{\sin B}
=
\frac{c}{\sin C}.
\]
Once two angles are known, the third angle follows from the angle-sum rule.
Right triangles
In a right triangle, the hypotenuse is found from the Pythagorean theorem.
\[
c=\sqrt{a^2+b^2}.
\]
The acute angles can be found using trigonometric ratios.
\[
\tan A=\frac{a}{b}.
\]
Triangle area
When all three sides are known, Heron's formula gives the area.
\[
s=\frac{a+b+c}{2}.
\]
\[
K=\sqrt{s(s-a)(s-b)(s-c)}.
\]
The same area can also be used to find a height, such as \(h_c=2K/c\).
Common mistakes
- Adding phasor magnitudes directly instead of adding rectangular components.
- Using degrees when the calculator expects radians, or radians when it expects degrees.
- Forgetting that AC RMS and peak phasor magnitudes are different conventions.
- Using \(\tan^{-1}(b/a)\) instead of \(\operatorname{atan2}(b,a)\), which can put the angle in the wrong quadrant.
- Mixing impedance, voltage, current, and force units in the same phasor sum.
- Using the law of sines in an ambiguous triangle case without checking for possible second solutions.
- Forgetting that side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
