📐 Formula Bank
Searchable, categorized formulas with dimensional analysis
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Dimensions: [V] = [ML²T⁻³A⁻¹]
Dimensions: [V_rms] = [ML²T⁻³A⁻¹]
Household "220V" means V_rms = 220V
Peak voltage = 220√2 = 311V
Dimensions: [X_L] = [ML²T⁻³A⁻²] (same as resistance)
X_L ∝ f → Higher frequency → More opposition
At f = 0 (DC): X_L = 0 (inductor is short circuit)
At f → ∞: X_L → ∞ (inductor blocks AC)
Dimensions: [X_C] = [ML²T⁻³A⁻²]
X_C ∝ 1/f → Higher frequency → Less opposition
At f = 0 (DC): X_C → ∞ (capacitor blocks DC)
At f → ∞: X_C → 0 (capacitor is short circuit)
Dimensions: [Z] = [ML²T⁻³A⁻²] = Ω
Wrong: Z = R + X_L + X_C
Correct: Z = √[R² + (X_L - X_C)²]
Because voltage phasors add vectorially, not algebraically
Dimensions: [φ] = dimensionless (angle)
φ > 0: X_L > X_C → Inductive → Current lags voltage
φ < 0: X_C > X_L → Capacitive → Current leads voltage
φ = 0: X_L = X_C → Resonance → In phase
Dimensions: [ω₀] = [T⁻¹] ✓
Dimensions: Check [1/√(LC)] = [1/√(ML²T⁻⁴A⁻²)] = [T⁻¹] ✓
If asked to double resonant frequency:
f₀ ∝ 1/√(LC)
To double f₀: Make LC → LC/4
Options: L/4, C/4, or L/2 AND C/2
Dimensions: [Q] = dimensionless ✓
Dimensions: [P] = [ML²T⁻³] = W
Never forget cos φ!
P = VI only works for DC or pure resistive AC
For general AC: P = VI cos φ
| Circuit | cos φ | Power |
|---|---|---|
| Pure R | 1 | Maximum |
| Pure L or C | 0 | Zero (wattless) |
| RLC | Between 0 and 1 | Depends on R/Z |
Basic AC Formulas
Reactance Formulas
LCR Circuit Formulas
Power Formulas
Resonance Formulas
Dimensional Analysis Guide
Use it to:
- Check if your final answer can be correct
- Eliminate wrong options in MCQs
- Remember formula structure
- Catch algebraic mistakes instantly
| Quantity | SI Unit | Dimensions |
|---|---|---|
| Voltage (V) | Volt (V) | [ML²T⁻³A⁻¹] |
| Current (I) | Ampere (A) | [A] |
| Resistance (R) | Ohm (Ω) | [ML²T⁻³A⁻²] |
| Inductance (L) | Henry (H) | [ML²T⁻²A⁻²] |
| Capacitance (C) | Farad (F) | [M⁻¹L⁻²T⁴A²] |
| Reactance (X_L, X_C) | Ohm (Ω) | [ML²T⁻³A⁻²] |
| Impedance (Z) | Ohm (Ω) | [ML²T⁻³A⁻²] |
| Frequency (f) | Hertz (Hz) | [T⁻¹] |
| Angular frequency (ω) | rad/s | [T⁻¹] |
| Power (P) | Watt (W) | [ML²T⁻³] |
Example 1: Check f₀ = 1/(2π√(LC))
[f₀] = 1/√([L][C]) = 1/√([ML²T⁻²A⁻²][M⁻¹L⁻²T⁴A²])
= 1/√[T²] = [T⁻¹] ✓ Correct!
Example 2: Check X_L = ωL
[X_L] = [ω][L] = [T⁻¹][ML²T⁻²A⁻²] = [ML²T⁻³A⁻²] ✓ Same as resistance!
Example 3: Check Q = (1/R)√(L/C)
[Q] = [R⁻¹]√([L]/[C]) = [M⁻¹L⁻²T³A²]√([ML²T⁻²A⁻²]/[M⁻¹L⁻²T⁴A²])
= [M⁻¹L⁻²T³A²]√[M²L⁴T⁻⁶A⁻⁴] = [M⁻¹L⁻²T³A²][ML²T⁻³A⁻²] = dimensionless ✓