ECE 202 Test 1 Equation Sheet

Inductors:

v = L[di/dt]

  i = [1/L]ò^t_tovdt+i(t₀)

  p = [dw/dt] = vi = Li[di/dt]

  w = [1/2]Li²

Series: L_eq = L₁+L₂+L₃...

Capacitors:

i = C[dv/dt]

  v = [1/C]ò^t_toidt+v(t₀)

  p = [dw/dt] = vi = Cv[dv/dt]

  w = [1/2]Cv²

Series: L_eq = L₁+L₂+L₃...

Parallel: C_eq = C₁+C₂+C₃...

Natural Response:

For RC circuits:

t = RC

For RL circuits:

t = [L/R]

For both: x(t) = X_oe^[(-t)/(t)] ,t>0

Step Response:

x(t) = Finalvalue+(Initial-final)e^[(-t)/(t)]

RLC circuits:

Series: a = [R/2L] Parallel: a = [1/2RC]

Both: w = [1/([ÖLC])] ; w_d = Ö{w²-a²}

s_1,2 = -a+-Ö{a²-w²}

Overdamped(both roots real and distinct): x(t) = A₁e^s₁t+A₂e^s₂t

To find Constants: x(0) = A₁+A₂

[dx(0)/dt] = s₁A₁+s₂A₂

Underdamped(complex roots): x(t) = B₁e^-atcosw_dt+B₂e^-atsinw_dt

x(0) = B₁

[dx(0)/dt] = w_dB₂-aB₁

Critically Damped(roots the same): x(t) = D₁te^-at+D₂e^-at

x(0) = D₂

[dx(0)/dt] = D₁-aD₂

Step response of all RLC: add an X_f to each of the response equations, and to each equation for determining constants that do NOT include derivatives!