2nd Order Circuits Analysis PDF

## 2nd Order Circuits ### Circuit 1 - Image: A circuit with a voltage source *Vg*, a resistor with a resistance of 8 ohms, an inductor with an inductance of 2H, and another resistor with a resistance of 4 ohms. The inductor has a 1H inductor in parallel with it. - 2 energy storing elements (a...

## 2nd Order Circuits ### Circuit 1 - Image: A circuit with a voltage source *Vg*, a resistor with a resistance of 8 ohms, an inductor with an inductance of 2H, and another resistor with a resistance of 4 ohms. The inductor has a 1H inductor in parallel with it. - 2 energy storing elements (an inductor and a capacitor) - It could be a 2nd order circuit ### Circuit 2 - Image: A circuit with a voltage source *Vg*, two resistors *R1* and *R2*, and two capacitors *C1* and *C2*. It is noted that *R1* is in parallel with the capacitor *C1* and *R2* is in parallel with the capacitor *C2*. - This circuit is not a 2nd order circuit. - There is no decoupling of differential equations needed. - This circuit has two first order differential equations. ### Circuit 1 Analyzed - Image: A circuit with a voltage source *Vg*, a resistor with a resistance of 8 ohms, an inductor with an inductance of 2H, and another resistor with a resistance of 4 ohms. The inductor has a 1H inductor in parallel with it. - *i(t) = ?* where *t > 0* - The circuit is described by the following equations: * _Vg - 8i1 - 2 di1/dt - 4(i1 - i2) = 0_ * _4(i1 - i2) - di2/dt = 0_ - **Decoupling:** * _i1 = 1/4 *(di2/dt + 4i2)_ * _di1/dt = 1/4 *(d^2i2/dt^2 + 4 di2/dt)_ - Substituting and simplifying the equations: * _d^2i2/dt^2 + 10 di2/dt + 16i2 = 2Vg_ - This governs the equation for a 2nd order differential equation. ### Natural Response - *Vg = 0* - _i = i2_ - _d^2i/dt^2 + 10 di/dt + 16 i = 0_ - _i = Ae^st_ - _di/dt = sAe^st_ - _d^2i/dt^2 = s^2Ae^st_ - _s^2 + 10s + 16 = 0_ - _s = (- 10 ± √102 - 4 * 1 * 16) / 2_ - _s1 = - 8_, _s2 = - 2_ (poles) ### Recall From High School - _Ax^2 + Bx + C = 0_ - _x = (-B ± √B^2 - 4AC) / 2A_ - _△ = B^2 - 4AC_ (Discriminant) - _△ > 0 ⇒_ 2 distinct real roots - _△ = 0 ⇒_ 2 identical real roots - _△ < 0 ⇒_ 2 complex conjugate roots - _x + jy ⇒_ x - jy - _j = √-1_ (Complex Conjugation) ### Final Analysis - _i(t) = Ae^{- 8t} + Aze^{- 2t}_ (Superposition) ### Time Constants - _T1 = 1/|S1| = 1/|- 8| = 1/8 = 0.125s = 125ms_ - _T2 = 1/|S2| = 1/|- 2| = 1/2 = 0.5s = 500ms_ ### Time Constant Implications - The natural response of a circuit is dominated (controlled, set) by the longest time constant. - The lowest natural frequency (pole) determines the natural response (transient response) of the system.