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Professor Jeffrey Miller
ES309 Final Formula Sheet
ES309 Final Formula Sheet
This sheet will be distributed with the exam.
Quantity Unit Name Units Frequency hertz (Hz) s- Force newton (N) kg m / s^2 Energy joule (J) N m Power watt (W) J / s Electric charge coulomb (C) A s Electric potential volt (V) J / C Electric resistance ohm (Ω) V / A Electric conductance siemens (S) A / V Electric capacitance farad (F) C / V Magnetic flux weber (Wb) V s Inductance henry (H) Wb / A
Ohm’s Law v = i R = dw / dq where v is the voltage, i is the current, R is the resistance, w is the energy, and q is the charge Electric Current i = dq / dt where i is the current, q is the charge, and t is the time
Power p = v i = dw/dt where p is the power, v is the voltage, i is the current, w is the energy, t is the time Energy w = ∫ p dt where w is the energy, p is the power, t is the time, integrated from t 0 to t
Conductance G = 1 / R where G is the conductance and R is the resistance
Equivalent resistance in series Req = ∑Ri = R 1 + R 2 + … + Rk Equivalent resistance in parallel 1 / Req = ∑ (1 / Ri) = 1 / R 1 + 1 / R 2 + … + 1 / Rk With two resistors in parallel Req = R 1 * R 2 / (R 1 + R 2 )
Voltage-Division Equation – to find the voltage vj across a resistor Rj with resistors connected in series from a voltage source distributing v volts: vj = i Rj = v * Rj / Req
Current-Division Equation – to find the current ij across a resistor Rj with resistors connected in parallel from a current source distributing i amps: ij = v / Rj = i * Req / Rj
Operational Amplifier – vo is the output voltage, vp is the non-inverting input, vn is the inverting input, A is the op amp gain, Vcc is the positive power supply, -Vcc is the negative power supply, ic+ is the current at the positive power supply, ic- is the current at the negative output supply, io is the output current, ip is the current at the non-inverting input, in is the current at the inverting input
-Vcc if A * (vp – vn) < -Vcc vo = A * (vp – vn) if -Vcc <= A * (vp – vn) <= +Vcc +Vcc if A * (vp – vn) > +Vcc
vp = vn ip = in = 0
Professor Jeffrey Miller
ES309 Final Formula Sheet
io = -(ic+ + ic-) Inverting-Amplifier
vo = -vs * Rf / Rs in = is + if = 0
Summing-Amplifier
vo = -(va * Rf / Ra + vb * Rf / Rb + vc * Rf / Rc) vn = vp = 0 in = 0
Noninverting-Amplifier
vo = vg * (Rs + Rf) / Rs vn = vg
Difference-Amplifier
vo = (vb – va) * Rb / Ra in = ip = 0 vn = vp = vb * (Rd / (Rc + Rd))
Inductor v = L di/dt i(t) = (1/L) ∫ v dt + i(t 0 ) from t 0 to t p = dw/dt = L i di/dt w = ∫ p dt = ½ L i
Capacitor i = C dv/dt v(t) = (1/C) ∫ i dt + v(t 0 ) from t 0 to t p = vi = Cv dv/dt w = ½ Cv^2
Equivalent inductance in series Leq = ∑Li = L 1 + L 2 + … + Lk Equivalent inductance in parallel 1 / Leq = ∑ (1 / Li) = 1 / L 1 + 1 / L 2 + … + 1 / Lk Equivalent inductance initial current i(t 0 ) = i 1 (t 0 ) + i 2 (t 0 ) + … + ik(t 0 )
Equivalent capacitance in series 1 / Ceq = ∑ (1 / Ci) = 1 / C 1 + 1 / C 2 + … + 1 / Ck Equivalent capacitance in parallel Ceq = ∑Ci = C 1 + C 2 + … + Ck Equivalent capacitance initial voltage v(t 0 ) = v 1 (t 0 ) + v 2 (t 0 ) + … + vk(t 0 )
Professor Jeffrey Miller
ES309 Final Formula Sheet
Parallel RLC Natural Response
d^2 v/dt^2 + (1/RC) dv/dt + v/LC = 0 s1,2 = (-1/2RC) ± √((1/(2RC))^2 – (1/(LC))) s1,2 = -α ± √(α^2 - ω 02 ) α = 1 / (2RC) ω 0 = 1 / √(LC)
Overdamped Parallel RLC Natural Response (ω 02 < α^2 ) v(t) = A 1 es1t^ + A 2 es2t v(0+) = A 1 + A 2 dv(0+)/dt = ic(0+)/C = A 1 s 1 + A 2 s 2
Underdamped Parallel RLC Natural Response (ω 02 > α^2 ) v(t) = B 1 e-αt^ cos(ωdt) + B 2 e-αt^ sin(ωdt) ωd = √(ω 02 - α^2 ) v(0+) = B 1 dv(0+)/dt = ic(0+)/C = -αB 1 + ωdB 2
Critically-damped Parallel RLC Natural Response (ω 02 = α^2 ) v(t) = D 1 te-αt^ + D 2 e-αt v(0+) = D 2 dv(0+)/dt = ic(0+)/C = D 1 – αD 2
Parallel RLC Step Response
i = iL + v/R + C dv/dt 0 = d^2 v/dt^2 + 1/(RC) * dv/dt + v/(LC)
Overdamped Parallel RLC Step Response (ω 02 < α^2 ) iL(t) = i + A 1 ’ es1t^ + A 2 ’ es2t
Underdamped Parallel RLC Step Response (ω 02 > α^2 ) iL(t) = i + B 1 ’ e-αt^ cos(ωdt) + B 2 ’ e-αt^ sin(ωdt) ωd = √(ω 02 - α^2 )
Critically-damped Parallel RLC Step Response (ω 02 = α^2 ) iL(t) = i + D 1 ’ te-αt^ + D 2 ’e-αt v(0+) = D 2 dv(0+)/dt = ic(0+)/C = D 1 – αD 2
Professor Jeffrey Miller
ES309 Final Formula Sheet
Series RLC Natural Response
d^2 i/dt^2 + (R/L) di/dt + i/(LC) = 0 s1,2 = (-R/(2L)) ± √((R/(2L))^2 – (1/(LC))) s1,2 = -α ± √(α^2 - ω 02 ) α = R / (2L) ω 0 = 1 / √(LC)
Overdamped Series RLC Natural Response (ω 02 < α^2 ) i(t) = A 1 es1t^ + A 2 es2t
Underdamped Series RLC Natural Response (ω 02 > α^2 ) i(t) = B 1 e-αt^ cos(ωdt) + B 2 e-αt^ sin(ωdt) ωd = √(ω 02 - α^2 )
Critically-damped Series RLC Natural Response (ω 02 = α^2 ) i(t) = D 1 te-αt^ + D 2 e-αt
Series RLC Step Response
Overdamped Series RLC Step Response (ω 02 < α^2 ) vc(t) = v + A 1 ’ es1t^ + A 2 ’ es2t
Underdamped Series RLC Step Response (ω 02 > α^2 ) vc(t) = v + B 1 ’ e-αt^ cos(ωdt) + B 2 ’ e-αt^ sin(ωdt) ωd = √(ω 02 - α^2 )
Critically-damped Series RLC Step Response (ω 02 = α^2 ) vc(t) = v + D 1 ’ te-αt^ + D 2 ’e-αt
Sinusoidal Functions v = vm cos(ωt + φ) v = vm sin(ωt + φ) i = im cos(ωt + φ) i = im sin(ωt + φ)
ω = 2πf = angular frequency T = 1/f = period vm = amplitude φ = phase angle
Phasors cosθ = R{ejθ} sinθ = I{ejθ}
P-1{vm ejθ} = R{vm ejθ^ ejωt^ }
V = vm ejθ^ = P{vm cos(ωt + φ)} = vm cosφ + jvm sinφ
