Natural And Step Responses For RCL Parallel Circuits

Any time we are dealing with parallel RCL circuits we must first find two values

$\alpha$ and $\omega$

$\alpha =\frac{1}{2RC}$ this is also known as neper frequency

$\omega =\frac{1}{\sqrt{LC}}$ also known as Resonant frequency.

We have 3 types of responses:

• Over-damped: ${\omega }_0<\alpha$
• Under-damped: ${\omega }_o>\alpha$
• Critically damped: ${\omega }_0=\alpha$

Natural Response for RCL Parallel Circuits

Over-damped

When you have ${\omega }_0<\alpha$, the general formula is give by:

$V(t)=A_1{e}^{s_{1}t}+A_2{e}^{s_{2}t}$

in this case we can calculate $s_1$ and $s_2$ using the equations:

$s_1=-\alpha +\sqrt{{\alpha }^2-{\omega }_0^2}$

$s_2=-\alpha -\sqrt{{\alpha }^2-{\omega }_0^2}$

We can then use two equations to solve for our $A_1$ and $A_2$ Values.

1. $V_0=A_1+A_2$ where $V_0$ is the initial voltage

2. $s_1{A}_1+s_2{A}_2=\frac{1}{C}(-I_0-\frac{V_0}{R})$

We can plug our initial voltage in for $V_0$ and our $s_1$ and $s_2$ values in and solve the equation using substitution ($A_2=-V_0-A_1$)

We then plug the values we derived back into the initial equation.

Under-Damped

${\omega }_0>\alpha$
The general equation of the under damped circuit is give by:

$V(t)=B_1{e}^{-\alpha t}\cos ({\omega }_{d}t)+B_2{e}^{-\alpha t}\sin ({\omega }_{d}t)$

To solve for under damped...first we must find ${\omega }_d$

${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$ and is known as the Damped Frequency

To solve, have two equations:

1. $V_0=B_1$ where $V_0 is the initial voltage.

2. $-\alpha B_1+{\omega }_d{B}_2=\frac{1}{C}(-I_0-\frac{V_0}{R})$

Once we solve for $B_2$ we plug all of our values back into the general equation for $V(t)$.

Critically Damped Response

${\omega }_0=\alpha$

For critically damped, our general form equation is given as:

$V(t)=D_{1}t{e}^{-\alpha t}+D_2{e}^{-\alpha t}$

To solve we use the two following equations:

1. $D_2=V_0$ where $V_0$ is the initial voltage (and what you would get if you plugged 0 into $V(t)$)

2. $D_1-\alpha D_2=\frac{1}{C}(-I_0-\frac{V_0}{R})$

Solving for these gets us $D_1$ and $D_2$ which we plug back into the original equation.

Stepped Response of RLC Parallel Circuit

The same over or under-damped takes place for stepped response.

We have 3 types of responses:

• Over-damped: ${\omega }_0<\alpha$
• Under-damped: ${\omega }_o>\alpha$
• Critically damped: ${\omega }_0=\alpha$

Over-damped response

Over-damped: ${\omega }_0<\alpha$
The general form for the stepped response is given by the equation:

$I_L(t)=I_f+A_1{e}_{s_{1}t}+A_2{e}^{s_{2}t}$

Our two equations for this are:

1. $I_f+A_1+A_2=I_0$

2. $A_1{S}_1+A_2{S}_2=\frac{V_0}{L}$

Under-Damped

Under-damped: ${\omega }_o>\alpha$

${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$

Our general equation is given by:

$I_L(t)=I_f+B_1{e}^{-\alpha t}\cos ({\omega }_{d}t)+B_2{e}^{-\alpha t}\sin ({\omega }_{d}t)$

We solve this formula using these two equations:

1. $I_f+B_1=I_0$ where $I_0$ is our initial current.

2. $-\alpha B_1+{\omega }_d{B}_2=\frac{V_0}{L}$

We can then plug these values back into our initial equation.

Critically Damped

${\omega }_0=\alpha$

Our main equation for a critically damped circuit is given by:

$I_L(t)=I_f+D_{1}t{e}^{-\alpha t}+D_2{e}^{-\alpha t}$

We solve for the missing values using these two equations:

1. $I_f+D_2=I_0$ where $I_f$ is the final current and $I_0$ is the initial current

2. $D_1-\alpha D_2=\frac{V_0}{L}$

Once we have these values we can plug them back into the equation.