The algebraic and trigonometric machinery the rest of the FE leans on: quadratics, logs, partial fractions, triangle laws, identities, and polar conversion.
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B. Complex numbers
Complex Numbers and Phasors
The three equivalent forms of a complex number, Euler's identity, complex arithmetic and roots, and the phasor representation that turns AC circuits into algebra.
Complex numbers are the single most leveraged idea on the FE-ECE: master them and AC circuits, impedance, transfer functions, and the Fourier and Laplace transforms all collapse from calculus into algebra. The whole trick is that a sinusoid
carries only two pieces of information at a fixed frequency, an amplitude and a phase, and a complex number stores exactly two numbers. The FE Reference Handbook's Algebra of Complex Numbers and Euler's Identity sections give you the forms and operations; this concept welds them to the phasor so you can read an AC problem and immediately write an impedance. Note the handbook uses
j
for
−1
, since
i
denotes current.
The three forms
A complex number wears three interchangeable outfits. Rectangular form z=a+jb exposes the real part a and imaginary part b, and is the form for addition and subtraction. Polar form z=c∠θ gives magnitude c and angle θ, and is the form for multiplication, division, and powers. Exponential form z=cejθ is polar written so the algebra of exponents does the work for you. They describe the same point in the complex plane; fluency is the ability to switch on demand.
z=a+jb=c∠θ=cejθ,c=a2+b2,θ=arctanab
Euler's identity
Euler's identity is the hinge between the trigonometric and exponential worlds and the reason phasors work at all. From ejθ=cosθ+jsinθ you recover the real and imaginary parts of any unit phasor, and adding the conjugate relation e−jθ=cosθ−jsinθ lets you write cosθ and sinθ themselves as sums of exponentials. This is exactly how a cosine becomes the real part of a rotating phasor Re{ejωt}.
ejθ=cosθ+jsinθ,cosθ=2ejθ+e−jθ,sinθ=2jejθ−e−jθ
Arithmetic: add in rectangular, multiply in polar
Pick the form that makes the operation trivial. Add and subtract in rectangular form, combining real with real and imaginary with imaginary. Multiply and divide in polar form, where magnitudes multiply or divide and angles add or subtract. Trying to multiply two numbers in rectangular form is legal but slow and error-prone; trying to add in polar form is worse. The conjugate z∗=a−jb reflects across the real axis, and zz∗=a2+b2=c2 is the squared magnitude, the move that rationalizes a division.
z1z2=c1c2∠(θ1+θ2),z2z1=c2c1∠(θ1−θ2)
Powers and roots
Raising to a power is effortless in polar form by De Moivre's theorem: magnitude to the power, angle times the power. Roots run the same machine in reverse, and here the FE likes to catch you, because every nonzero complex number has exactly k distinct k-th roots spaced evenly by 360∘/k around a circle of radius c1/k. You get them by adding 360∘n to the angle before dividing, for n=0,1,…,k−1.
z1/k=c1/k∠kθ+360∘n,n=0,1,…,k−1
From sinusoid to phasor
A phasor is the complex amplitude of a sinusoid at a known frequency: strip off the cos(ωt) carrier and keep the magnitude and phase. The universal convention on the FE is the cosine reference, so v(t)=Vmcos(ωt+ϕ) maps to the phasor V=Vm∠ϕ (or Vrms∠ϕ when working in RMS). A sine must first be converted with sinθ=cos(θ−90∘). The frequency ω does not appear in the phasor; it returns only when you map back to the time domain.
v(t)=Vmcos(ωt+ϕ)⟷V=Vm∠ϕ
Impedance: why phasors earn their keep
Once voltages and currents are phasors, the calculus of capacitors and inductors becomes multiplication by a complex impedance. A resistor is ZR=R, an inductor ZL=jωL, and a capacitor ZC=1/(jωC)=−j/(ωC). Ohm's law then reads V=IZ in the complex plane, so series and parallel combinations, dividers, and Thevenin reductions all work exactly as in DC, just with complex numbers. The angle of Z is the phase by which voltage leads current.
ZR=R,ZL=jωL,ZC=jωC1=−ωCj
Exam strategy
Decide the form before you compute: rectangular for sums, polar for products, quotients, powers, and roots, and convert once rather than fighting the wrong form. Keep angle mode and the j-versus-i convention straight. For phasors, convert every source to the cosine reference up front and carry RMS or peak consistently through the whole problem. When a question asks for all roots, remember there are exactly k of them spaced 360∘/k apart, and the most common wrong answer reports only one.
Key equations
Three equivalent formsz=a+jb=c∠θ=cejθ
Rectangular (a real, b imag), polar (c magnitude, θ angle), exponential. Same point in the complex plane.
Magnitude and anglec=a2+b2,θ=arctan(ab)(quadrant-corrected)
Rectangular componentsa=ccosθ,b=csinθ
Polar to rectangular: real and imaginary parts of c∠θ.
Euler's identitye±jθ=cosθ±jsinθ
Links exponential and trigonometric forms; the basis of phasors and Fourier analysis.
Conjugate and magnitude productz∗=a−jb,zz∗=a2+b2=c2
De Moivre (powers)zn=cn∠(nθ)=cn(cosnθ+jsinnθ)
k-th rootsz1/k=c1/k∠kθ+360∘n,n=0,…,k−1
Sinusoid to phasor (cosine reference)Vmcos(ωt+ϕ)↔V=Vm∠ϕ
Element impedancesZR=R,ZL=jωL,ZC=jωC1
Phasor Ohm's lawV=IZ
Complex Ohm's law. ∠Z is the phase by which voltage leads current.
Worked examples
Convert, multiply, and divide
Problem. Given z1=3+j4 and z2=1−j2, find z1 and z2 in polar form, then compute z1z2 and z1/z2.
Solution. Polar: z1=32+42∠arctan(4/3)=5∠53.1∘
z1z2=11.2∠−10.3∘,z2z1=2.24∠116.6∘
Cube roots of a complex number
Problem. Find all cube roots of z=8∠60∘.
Solution. Magnitude of each root: 81/3=2.00. Angles: 360∘+360∘n
Sinusoid to phasor and impedance
Problem. A source v(t)=170cos(377t+30∘)V drives a series R=10Ω
Common pitfalls
•Adding or subtracting in polar form, or multiplying/dividing in rectangular form: pick the form that makes the operation a one-liner.
•Reporting only one root when k are asked for. A k-th root has exactly k values spaced 360∘/k apart.
•Forgetting to convert a sine source to the cosine reference (sinθ=cos(θ−90∘)) before reading off a phasor angle.
•Mixing RMS and peak in the same problem; the phasor magnitude must be one or the other consistently throughout.
•Sign slip on capacitive impedance: ZC=1/(jωC)=−j/(ωC) is negative imaginary, so current leads voltage.
•Quadrant error: arctan(b/a) from a calculator can land in the wrong half-plane; check the signs of a and b.
•Treating ω as part of the phasor; the phasor holds only magnitude and phase, and ω re-enters only on the return to time domain.
References
NCEES FE Reference Handbook — Mathematics: Algebra of Complex Numbers and Polar Coordinate System
NCEES FE Reference Handbook — Mathematics: Euler's Identity and Roots
NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Circuits
C. Discrete mathematics
Discrete Mathematics
Sets, relations and functions, propositional logic and truth tables, graphs and digraphs, counting, and recurrences at the depth the FE-ECE actually tests.
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D. Analytic geometry
Analytic Geometry
Lines and distance, the four conic sections and how to classify them, and conversion among polar, cylindrical, and spherical coordinate systems.
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E. Calculus
Calculus
Limits, derivatives (including partials), optimization, integrals, the fundamental theorem, and Taylor/L'Hopital tools the FE math block leans on every form.
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F. Ordinary differential equations
Ordinary Differential Equations
Solve first-order linear/separable ODEs and second-order constant-coefficient equations, classify over/critical/under-damped responses, and fit initial conditions.
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G. Linear algebra
Linear Algebra
Matrix algebra, determinants, inverses, Cramer's rule and Gaussian elimination, rank and singularity, and the eigenvalue/eigenvector problem behind FE systems work.
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H. Vector analysis
Vector Analysis
Dot, cross, and triple products; gradient, divergence, and curl; line and surface integrals; plus an intro to the divergence and Stokes theorems used across fields.
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Rectangular to polar. Quadrant-correct θ from the signs of a and b (or use atan2(b,a)).
Use rectangular form. Combine real with real, imaginary with imaginary.
Use polar form. Multiply/divide magnitudes, add/subtract angles.
Conjugate reflects across the real axis; product gives squared magnitude. Used to rationalize division.
Raise to integer power n in polar form: magnitude to the power, angle times the power.
Every nonzero z has exactly k distinct roots, evenly spaced by 360∘/k.
Phasor = complex amplitude. Convert sines via sinθ=cos(θ−90∘) first.
Phasor-domain impedances; ω=2πf in rad/s, R in Ω, L in H, C in F.
;
z2=12+22∠arctan(−2/1)=2.24∠−63.4∘
. Product (add angles):
z1z2=(5)(2.24)∠(53.1∘+(−63.4∘))=11.2∠−10.3∘
. Check in rectangular:
(3+j4)(1−j2)=3−j6+j4+8=11−j2
, whose magnitude
121+4=11.2
and angle
arctan(−2/11)=−10.3∘
agree. Quotient (subtract angles):
z1/z2=2.245∠(53.1∘−(−63.4∘))=2.24∠116.6∘
. Note the magnitude is
5/5=5=2.24
, which only coincidentally equals
∣z2∣
. Final:
z1z2=11.2∠−10.3∘
,
z1/z2=2.24∠116.6∘
.
for
n=0,1,2
, giving
20∘
,
140∘
, and
260∘
. So the three roots are
2∠20∘
,
2∠140∘
,
2∠260∘
. Sanity check: the angles are spaced exactly
120∘=360∘/3
apart, as a complete set of cube roots must be. Cubing the first:
23∠(3⋅20∘)=8∠60∘
, recovering
z
. Final:
2∠20∘,2∠140∘,2∠260∘
.
z1/3=2∠{20∘,140∘,260∘}
and
L=20mH
. Find the phasor voltage (peak), the impedance, and the current magnitude.
Solution. Phasor (peak): V=170∠30∘V. Inductive reactance: XL=ωL=377(0.020)=7.54Ω, so Z=10+j7.54=12.5∠37.0∘Ω. Current: I=V/Z=12.5∠37.0∘170∠30∘=13.6∠−7.0∘A, so ∣I∣=13.6A peak. Sanity check: the current lags the voltage by 37∘ (the impedance angle), correct for an inductive load. Final: V=170∠30∘V, Z=12.5∠37.0∘Ω, ∣I∣=13.6A peak.