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 RLC circuit can generate a sinusoidal wave.

There are primarily two types of second order circuits:

• Parallel RLC circuits
• Series RLC circuits

This page will analyze them and derive some useful equations.

Series RLC Circuits

Natural Response

An unforced series RLC circuit

Consider an un-forced RLC circuit. We want to find $V_{C}$.

First, we can use KVL and KCL

$V_{R}+V_{L}+V_{C}=0$

$iR+L{\frac {di}{dt}}+V_{C}=0$

Next, we can use $i=C{\frac {dV_{C}}{dt}}$ and substitution to get

$RC{\frac {dV_{C}}{dt}}+L{\frac {d}{dt}}{\frac {CV_{C}}{dt}}V_{C}=0$

Changing the order and moving the constants,

$LC{\frac {d^{2}V}{dt^{2}}}+RC{\frac {dV_{C}}{dt}}+V_{C}=0$

Moving constants away from the first term to get a second-order differential equation,

${\frac {d^{2}V_{C}}{dt^{2}}}+{\frac {R}{L}}{\frac {dV_{C}}{dt}}+{\frac {1}{LC}}V_{C}=0$

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{1}{2RC} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \frac{1}{\sqrt{LC}} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = A_1 e^{s_1 t} + A_2 e^{s_2 t} }

Forms

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = A_1 e^{s_1 t} + A_2 e^{s_2 t} }

The A's can be solved by substituting in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(0)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dv(0) / dt = i_C / C}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(0) = A_1 + A_2 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{1,2} = - \alpha \pm j \omega_d }

where there is damped radian frequency

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_d = \sqrt{w_0^2 - \alpha^2} }

From Euler's identity, the natural response comes to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1 = V_0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_d} , which is why it is called the damped radian frequency.

The oscillation amplitude decreases exponentially at a rate determined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} , so it is called the damping factor.

Notice that when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \neq 0} , there is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha > 0} and the circuit is damped.