- Capacitor stores energy in the electric field.
-
Formula
$$\epsilon = \epsilon_0\epsilon_r$$ $$C = \epsilon \frac{A}{d}$$ $$C = \epsilon_0\epsilon_r \frac{A}{d}$$ -
Parameters
C = Capacitance (F)
ε = Permittivity (F/m)
εᵣ = Relative Permittivity (unitless)
ε₀ = Vacuum Permittivity (F/m)
A = Plate Area (m²)
d = Distance (m) -
Energy Stored
$$E = \frac12 C V^2$$ - Inductor stores energy in the magnetic field.
-
Basic Formula
$$v = L \frac{di}{dt}$$ -
Inductance Parameters
L = Inductance (Henries, H)
N = Number of turns (unitless)
μ = Permeability (H/m)
A = Cross-section area (m²)
l = Coil length (m) -
Ideal Solenoid
$$L = \frac{\mu N^2 A}{l}$$ -
Energy Stored
$$E = \frac12 L I^2$$ -
Time Constant
$$\tau = RC$$ -
Charging
$$V_C(t) = V_s\left(1 - e^{-t/RC}\right)$$ $$I(t) = \frac{V_s}{R} e^{-t/RC}$$ -
Discharging
$$V_C(t) = V_0 e^{-t/RC}$$ $$I(t) = -\frac{V_0}{R} e^{-t/RC}$$ -
Time Constant
$$\tau = \frac{L}{R}$$ -
Rising Current (Switch ON)
$$I(t) = \frac{V_s}{R}\left(1 - e^{-tR/L}\right)$$ $$V_L(t) = V_s e^{-tR/L}$$ -
Falling Current (Switch OFF)
$$I(t) = I_0 e^{-tR/L}$$ $$V_L(t) = -I_0 R e^{-tR/L}$$ - Convert each element to Z (or Y) using ω = 2πf.
- If series: add Z values. If parallel: add Y values and invert to find Z.
- Compute magnitude and phase: |Z| = √(Re² + Im²), ∠Z = atan2(Im, Re) (convert to degrees).
- Use I = V/Z for currents or V = I·Z for voltages (phasor algebra).
Resistors
$$R_s = R_1 + R_2 + \ldots$$ $$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$$Capacitors
$$C_s = \frac{1}{\frac{1}{C_1} + \frac{1}{C_2} + \ldots}$$ $$C_p = C_1 + C_2 + \ldots$$Inductors
$$L_s = L_1 + L_2 + \ldots$$ $$\frac{1}{L_p} = \frac{1}{L_1} + \frac{1}{L_2} + \ldots$$Key Trick
- Capacitor acts like Resistor formula but reversed (series ↔ parallel).- Inductor acts like resistor (same formulas as R).
Element Impedance & Admittance (AC: ω = 2πf)
Resistor:
ZR = R |
YR = 1/R
Inductor:
ZL = jωL |
YL = 1/(jωL) = -j/(ωL)
Capacitor:
ZC = 1/(jωC) |
YC = jωC
Series: add impedances. Parallel: add admittances and invert to get Z.
Schematic sketches
Series (example: R — L — C)
Parallel (example: R ∥ C ∥ L)
Calculator
Results
ω = —
Ztotal = —
|Z| = —
∠Z = —
Ytotal = —
|Y| = —
∠Y = —
Phase is degrees. Z shown as (Re + jIm).
Worked examples (step-by-step)
Example 1 — Series: R = 100 Ω, C = 10 μF, f = 50
Hz, Vs = 10 V
ω = 2πf = 2π·50 = 314.1592653589793 rad/s
XC = 1/(ωC) = 1/(314.1592653589793·10×10⁻⁶) =
318.30988618379064 Ω
Z = R − jXC = 100 − j318.30988618379064 Ω
|Z| = √(100² + 318.3099²) = 333.6482933304736 Ω
∠Z = atan2(−318.3099, 100) = −72.55940550948813°
Current magnitude: I = V/|Z| = 10 / 333.6482933304736 =
0.029971680358919596 A
So I = 0.02997 A ∠ +72.5594° (current leads voltage because net
reactance is capacitive)
Example 2 — Parallel: R = 1 kΩ, C = 0.1 μF, f = 1
kHz
ω = 2π·1000 = 6283.185307179586 rad/s
Y>R = 1/R = 0.001 S
YC = jωC = j·6283.1853·0.1×10⁻⁶ = j0.0006283185307179585
S
Y = 0.001 + j0.0006283185307179585 S
|Y| = 0.0011810098120013967 S, ∠Y = 32.14190763534205°
Z = 1/Y = 846.7330159648304 Ω ∠ −32.1419°
Waveforms & key quantities
Sine (sinusoidal)
v(t) = Vp·sin(ωt + φ)
Peak Vp, Peak-to-peak Vpp = 2Vp, RMS Vrms = Vp/√2, Average over full cycle = 0.
v(t) = Vp·sin(ωt + φ)
Peak Vp, Peak-to-peak Vpp = 2Vp, RMS Vrms = Vp/√2, Average over full cycle = 0.
Square wave
Peak Vp, Vpp=2Vp, Vrms=Vp (for symmetric ±), average = 0 (symmetric).
Peak Vp, Vpp=2Vp, Vrms=Vp (for symmetric ±), average = 0 (symmetric).
Triangular
Peak Vp, Vrms = Vp/√3, average (symmetric) = 0.
Peak Vp, Vrms = Vp/√3, average (symmetric) = 0.
Angles: radians ↔ degrees: θ(deg) = θ(rad)·180/π, θ(rad) =
θ(deg)·π/180.
Phase relation: if current leads voltage, circuit is
capacitive; if current lags, inductive.
Sinusoid
Square
Triangular
Oscilloscope (grid & axes)
Typical oscilloscope view shows horizontal divisions (time) and vertical
divisions (voltage). Label each division value from the controls (time/div and
V/div).
Short method (reminder)
Horizontal axis = time. Vertical axis = voltage/current.
Use time/div & V/div on your scope to scale measurements.