Rectangular ↔ Polar Conversion
📐 Complex Impedance Forms
Positive X = inductive (current lags), Negative X = capacitive (current leads).
Rectangular Form
Z = R + jX
R = Re(Z), X = Im(Z)
X > 0 → inductive
X < 0 → capacitive
Polar Form
Z = |Z|∠φ
|Z| = √(R²+X²)
φ = arctan(X/R)
φ in range (−90°, +90°)
Conversion
R = |Z|·cos(φ)
X = |Z|·sin(φ)
|Z| = √(R²+X²)
φ = atan2(X, R)
Admittance Y = 1/Z
Y = G + jB
G = R / |Z|²
B = −X / |Z|²
|Y| = 1/|Z|, φ_Y = −φ
Series Impedance: Z_total = Z₁ + Z₂ + …
ℹ️ Series Impedance Rules
Z_total = (R₁+R₂+…) + j(X₁+X₂+…)
Enter each impedance as rectangular (R+jX) or polar (|Z|∠φ°).
Parallel Impedance: 1/Z_total = 1/Z₁ + 1/Z₂ + …
ℹ️ Parallel Impedance Rules
Y_total = Y₁ + Y₂ + … → Z_total = 1/Y_total
For two impedances: Z = (Z₁ × Z₂) / (Z₁ + Z₂)
Calculate Z from R, L, C + Frequency
📐 Series vs Parallel RLC
Series RLC
Z = R + j·(XL − XC)
XL = 2πfL
XC = 1/(2πfC)
X = XL − XC
Parallel RLC
Y = G + j·(BC − BL)
G = 1/R
BL = 1/(2πfL)
BC = 2πfC
Magnitude & Phase
|Z| = √(R² + X²)
φ = arctan(X/R)
inductive: φ > 0°
capacitive: φ < 0°
Resonance
At f₀: XL = XC
f₀ = 1/(2π√LC)
Series: Z = R (min)
Parallel: Z = R (max)
❓ FAQ
What is complex impedance?
Impedance Z is the total opposition to AC current flow, combining resistance (R, real part) and reactance (X, imaginary part): Z = R + jX. The j indicates a 90° phase shift. Impedance is measured in ohms (Ω).
What is the difference between rectangular and polar form?
Rectangular form Z = R + jX directly shows the resistive and reactive components. Polar form Z = |Z|∠φ shows the magnitude (total impedance) and phase angle. Rectangular is easier for addition; polar is easier for multiplication.
How do I combine impedances in series?
In series, add the real parts and imaginary parts separately: Z_total = (R₁+R₂) + j(X₁+X₂). This is straightforward in rectangular form.
How do I combine impedances in parallel?
In parallel, add the reciprocals: 1/Z_total = 1/Z₁ + 1/Z₂ + … It's easier to work with admittances (Y = 1/Z): Y_total = Y₁ + Y₂, then Z_total = 1/Y_total.
What does a positive vs negative phase angle mean?
A positive phase angle (φ > 0°) means the impedance is inductive — voltage leads current. A negative phase angle (φ < 0°) means the impedance is capacitive — current leads voltage.
What is admittance?
Admittance Y = 1/Z is the reciprocal of impedance. Its real part is conductance G = R/|Z|² and imaginary part is susceptance B = −X/|Z|². Admittance is measured in siemens (S).