Second Order Circuits

Second order circuits are circuits that have two energy storage elements, resulting in second-order differential equations.

One application of second order circuits is in timing computers. As we will see, an underdamped RLC circuit can generate a sinusoidal wave.

There are primarily two types of second order circuits:

• Parallel RLC circuits
• Series RLC circuits

each with two responses:

• Natural response (no power source)
• Forced response (has power source)

This page will analyze them and derive some useful equations.

Parallel RLC Circuits Natural Response

A parallel unforced RLC circuit

By KCL,

${\frac {v}{R}}+{\frac {1}{L}}\int _{0}^{t}vd\tau +I_{0}+C{\frac {dv}{dt}}=0$

By differentiating once with respect to $t$ and rearranging some constants,

${\frac {d^{2}v}{dt^{2}}}+{\frac {1}{RC}}{\frac {dv}{dt}}+{\frac {v}{LC}}=0$

we get a homogeneous second-order differential equation, which has a standard solution that I will not go into detail. Briefly, it is solved by assuming $v=Ae^{st}$ since derivatives of $v$ must take the same form to cancel out to zero.

By applying the standard solution, we have

$Ae^{st}\left(s^{2}+{\frac {s}{RC}}+{\frac {1}{LC}}\right)=0$

Characteristic Equation

The above simplifies to

$s^{2}+{\frac {s}{RC}}+{\frac {1}{LC}}=0$

This is the characteristic equation of the differential equation, as the root of the quadratic determines properties of $v(t)$

$s_{1,2}=-{\frac {1}{2RC}}\pm {\sqrt {\left({\frac {1}{2RC}}\right)^{2}-{\frac {1}{LC}}}}=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}}}$

where

$\alpha ={\frac {1}{2RC}}$

and

$\omega _{0}={\frac {1}{\sqrt {LC}}}$

It can be pretty easily proven that the sum of the two roots is also a solution

$v=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}$

Forms

Equations for Analyzing the Step Response of Parallel RLC Circuits

Depending on the root, there are three forms:

• Overdamped, where there are real, distinct solutions
• Underdamped, where there are complex solutions
• Critically damped, where the solutions are not distinct.

Overdamped

For an overdamped response, we have

$v=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}$

The A's can be solved by substituting in $v(0)$ and $dv(0)/dt=i_{C}/C$

$v(0)=A_{1}+A_{2}$

$dv(0)/dt=A_{1}s_{1}+A_{2}s_{2}={\frac {1}{C}}\left({\frac {-V_{0}}{R}}-I_{0}\right)$

Underdamped

For an underdamped response, we have

$s_{1,2}=-\alpha \pm j\omega _{d}$

where there is damped radian frequency

$\omega _{d}={\sqrt {w_{0}^{2}-\alpha ^{2}}}$

From Euler's identity, the natural response comes to

$v(t)=B_{1}e^{-\alpha t}\cos \omega _{d}t+B_{2}e^{-\alpha t}\sin \omega _{d}t$

The rest is identical to that of overdamped:

$B_{1}=V_{0}$

$-\alpha B_{1}+\omega _{d}B_{2}={\frac {1}{C}}\left({\frac {-V_{0}}{R}}-I_{0}\right)$

Characteristics

Voltage alternates between positive and negative values due to the two types of energy-storage elements. It's like a mass suspended on a spring.

The oscillation rate is fixed by $\omega _{d}$, which is why it is called the damped radian frequency.

The oscillation amplitude decreases exponentially at a rate determined by $\alpha$, so it is called the damping factor.

Notice that when $R\neq 0$, there is $\alpha >0$ and the circuit is damped.

Series RLC Circuits Natural Response

An unforced series RLC circuit

Calculations are very similar to that of parallel circuits, so I'll speed things up.

The second order differential from KVL is

${\frac {d^{2}i}{dt^{2}}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {i}{LC}}=0$

The resulting characteristic equation is

$s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0$

with roots

$s_{1,2}=-{\frac {R}{2L}}\pm {\sqrt {\left({\frac {R}{2L}}\right)^{2}-{\frac {1}{LC}}}}=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}}}$

with neper frequency $\alpha ={\frac {R}{2L}}rad/s$ and resonant radian frequency $\omega _{0}={\frac {1}{\sqrt {LC}}}rad/s$

The form equations are identical to parallel, only with different $\alpha$:

Equations for Analyzing the Step Response of Series RLC Circuits

Overdamped:

$i(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}$

$A_{1}+A_{2}=I_{0}$

$A_{1}s_{1}+A_{2}s_{2}={\frac {1}{L}}(-RI_{0}-V_{0})$

Underdamped:

$i(t)=B_{1}e^{-\alpha t}\cos \omega _{d}t+B_{2}e^{-\alpha t}\sin \omega _{d}t$

$B_{1}=I_{0}$

$-\alpha B_{1}+\omega _{d}B_{2}={\frac {1}{L}}(-RI_{0}-V_{0})$

Critically damped:

$i(t)=D_{1}te^{-\alpha t}+D_{2}te^{-\alpha t}$

$D_{2}=I_{0}$

$D_{1}-\alpha D_{2}={\frac {1}{L}}(-RI_{0}-V_{0})$