Electrical Circuits Crib Sheet
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\(\tau_L = \frac{L}{R_T}\) \(\tau_R = R_T C\) To find \(R_T\) turn all current sources into opens and voltage sources to shorts and open the energy-storing component, then simplify the resistors.
Powers of Ten Prefixes \(T = 10^{12} \) \(G = 10^9 \) \(M = 10^6 \) \(k = 10^3\) \(1 = 10^0\) \(m = 10^{-3} \) \(\mu = 10^{-6} \) \(n = 10^{-9} \) \(p = 10^{-12} \)
Things about transient circuits \(i_L\) can’t jump. \(v_C\) can’t jump. \(L \frac{di}{dt} = v_L\) \(C \frac{dv}{dt} = i_L\) \(CV^2/2\) Energy stored in capacitor Initial conditions are the following: \(i_L(0^-), v_C(0^-)\). \(u(t) = 1, \)for\( t>0 \lor 0,\) for \(t<0\) look up the characteristic equations for series and parallel
Things about any RLC circuits R or L in parallel \(\frac{product}{sum}\) R or L in series just sum C is backwards VDP \(v_x = v_s \frac{R_x}{\Sigma R}\) CDP \(i_x = i_s \frac{R_{other(s)}}{\Sigma R}\)
Nortons and Thevenins Norton | current source and resistor in parallel Thevenin | voltage source and resistor in series \(v_T = i_N R \land R_T = R_N\) Note: transformations change inner voltage, current, and power values, but not outside To transform ANY circuit between two nodes: \(R_T\) is equal to the simplified version of all resistors after voltage sources are replaced with shorts and current sources are replaced with opens. \(V_T\) then becomes the voltage between the two nodes. Max Power occurs when \(R_T = R_N = R \)
General Equations \(V = IR\) \(P = IV\)
General Solution for First-Order Switching Transient DC Circuits: Don’t forget, current can’t jump for inductors and voltage can’t jump for capacitors. \(y(t) = y(0)e^{-{t}/{\tau}}+y(\infty)(1- e^{-{t}/{\tau}})\) Transforming cos to sin:
30 60 90 triangles too 45 45 triangles
Second-order Switching Transient DC Circuits: NEED QUADRATIC EQUATION NEED DERIVATIVE RULEs FOR PRODUCTS
Choose Characteristic EQ
Parallel: \(s^2+\frac{1}{RC}s+\frac{1}{LC} = 0\)
Series: \(s^2+\frac{R}{L}s+\frac{1}{LC} = 0\)
Solve for s
Only 1 real \(s\) \(v(t) = A_1e^{st}+A_2te^{st}+A_3\)
Real and distinct \(s_1, s_2\) \(v(t) = B_1e^{s_1t}+B_2e^{s_2t}+B_3\)
Complex \(\alpha \pm \hat \jmath \beta\) \(v(t) = C_1e^{\alpha t}sin(\beta t)+C_2e^{\alpha t}cos(\beta t)+C_3\)














