Natural And Step Response for RCL Circuit in Series

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

$α$ and $ω$

$α = \frac{R}{2 L}$ this is also known as neper frequency

$ω = \frac{1}{\sqrt{L C}}$ also known as Resonant frequency.

We have 3 types of responses:

Over-damped: $ω_{0} < α$
Under-damped: $ω_{o} > α$
Critically damped: $ω_{0} = α$

Natural Response for RCL Series Circuits

Over-damped

When you have $ω_{0} < α$ , the general formula is give by:

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

in this case we can calculate $s_{1}$ and $s_{2}$ using the equations:

$s_{1} = - α + \sqrt{α^{2} - ω_{0}^{2}}$

$s_{2} = - α - \sqrt{α^{2} - ω_{0}^{2}}$

The two equations we use to solve for over-damped are:

1. $i (0) = A_{1} + A_{2} = I_{0}$ or the initial current.

2. $S_{1} A_{1} + S_{2} A_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

Using these two equations, we can find values for A_1 and A_2 and plug those into the general form equation.

To find voltage in terms of t we can use the equation

$V (t) = I_{0} + 1 / C \int_{0}^{t} i (t) d t$

Under-Damped

When you have $ω_{0} > α$ , the general formula is give by:

$ω_{d} = \sqrt{ω_{0}^{2} - α^{2}}$ and is known as the Damped Frequency

$i (t) = B_{1} e^{- α t} \cos (ω_{d} t) + B_{2} e^{- α t} s i n (ω_{d} t)$ for $t \geq o$

Our two equations to solve are:

1. $B_{1} = I_{0}$

2. $- α B_{1} + ω_{d} B_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

Critically Damped

When you have $ω_{0} = α$ , the general formula is give by:

$i (t) = D_{1} + e^{- α t} + D_{2} e^{- α t}$

The two equations are:

1. $D_{2} = I_{0}$

2. $D_{1} - α D_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

Stepped Response for RCL Series Circuits

$α$ and $ω$

$α = \frac{R}{2 L}$ this is also known as neper frequency

$ω = \frac{1}{\sqrt{L C}}$ also known as Resonant frequency.

We have 3 types of responses:

Over-damped: $ω_{0} < α$
Under-damped: $ω_{o} > α$
Critically damped: $ω_{0} = α$

Over-Damped Response

When you have $ω_{0} < α$ , the general formula is give by:

$V_{c} (t) = V_{f} + A_{1} e^{s_{1} t} + A_{2} e^{s_{2} t}$ for $t \geq 0$

$s_{1} = - α + \sqrt{α^{2} - ω_{0}^{2}}$

$s_{2} = - α - \sqrt{α^{2} - ω_{0}^{2}}$

our two equations:

1. $V_{f} + A_{1} + A_{2} = V_{0}$

2. $S_{1} A_{1} + S_{2} A_{2} = \frac{I_{0}}{C}$

Under-Damped

When you have $ω_{0} > α$ , the general formula is give by:

$ω_{d} = \sqrt{ω_{0}^{2} - α^{2}}$ and is known as the Damped Frequency

$V_{c} (t) = V_{f} + B_{1} e^{- α t} \cos (ω_{d} t) + B_{2} e^{- α t} s i n (ω_{d} t)$ for $t \geq 0$

our two equations are:

1. $V_{f} + B_{1} = V_{0}$

2. $- α B_{1} + ω_{d} B_{2} = \frac{I_{0}}{C}$

Critically Damped

When you have $ω_{0} = α$ , the general formula is give by:

$V_{c} (t) = V_{f} + D_{1} t e^{- α t} + D_{2} e^{- α t}$ for $t \geq 0$

our two equations are:

Natural And Step Response for RCL Circuit in Series

α and ω

α=R2L this is also known as neper frequency

ω=1LC also known as Resonant frequency.

We have 3 types of responses:

Natural Response for RCL Series Circuits

Over-damped

i(t)=A1es1t+A2es2t

s1=−α+α2−ω02

s2=−α−α2−ω02

1. i(0)=A1+A2=I0 or the initial current.

2. S1A1+S2A2=1L(−V0−RI0)

V(t)=I0+1/C∫0ti(t)dt

Under-Damped

ωd=ω02−α2 and is known as the Damped Frequency

i(t)=B1e−αtcos⁡(ωdt)+B2e−αtsin(ωdt) for t≥o

1. B1=I0

2. −αB1+ωdB2=1L(−V0−RI0)

Critically Damped

i(t)=D1+e−αt+D2e−αt

1. D2=I0

2. D1−αD2=1L(−V0−RI0)

Stepped Response for RCL Series Circuits

α and ω

α=R2L this is also known as neper frequency

ω=1LC also known as Resonant frequency.

We have 3 types of responses:

Over-Damped Response

Vc(t)=Vf+A1es1t+A2es2t for t≥0

s1=−α+α2−ω02

s2=−α−α2−ω02

1. Vf+A1+A2=V0

2. S1A1+S2A2=I0C

Under-Damped

ωd=ω02−α2 and is known as the Damped Frequency

Vc(t)=Vf+B1e−αtcos⁡(ωdt)+B2e−αtsin(ωdt) for t≥0

1. Vf+B1=V0

2. −αB1+ωdB2=I0C

Critically Damped

Vc(t)=Vf+D1te−αt+D2e−αt for t≥0

1. Vf+D2=V0

2. D1−αD2=I0C

$α$ and $ω$

$α = \frac{R}{2 L}$ this is also known as neper frequency

$ω = \frac{1}{\sqrt{L C}}$ also known as Resonant frequency.

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

$s_{1} = - α + \sqrt{α^{2} - ω_{0}^{2}}$

$s_{2} = - α - \sqrt{α^{2} - ω_{0}^{2}}$

1. $i (0) = A_{1} + A_{2} = I_{0}$ or the initial current.

2. $S_{1} A_{1} + S_{2} A_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

$V (t) = I_{0} + 1 / C \int_{0}^{t} i (t) d t$

$ω_{d} = \sqrt{ω_{0}^{2} - α^{2}}$ and is known as the Damped Frequency

$i (t) = B_{1} e^{- α t} \cos (ω_{d} t) + B_{2} e^{- α t} s i n (ω_{d} t)$ for $t \geq o$

1. $B_{1} = I_{0}$

2. $- α B_{1} + ω_{d} B_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

$i (t) = D_{1} + e^{- α t} + D_{2} e^{- α t}$

1. $D_{2} = I_{0}$

2. $D_{1} - α D_{2} = \frac{1}{L} (- V_{0} - R I_{0})$

$α$ and $ω$

$α = \frac{R}{2 L}$ this is also known as neper frequency

$ω = \frac{1}{\sqrt{L C}}$ also known as Resonant frequency.

$V_{c} (t) = V_{f} + A_{1} e^{s_{1} t} + A_{2} e^{s_{2} t}$ for $t \geq 0$

$s_{1} = - α + \sqrt{α^{2} - ω_{0}^{2}}$

$s_{2} = - α - \sqrt{α^{2} - ω_{0}^{2}}$

1. $V_{f} + A_{1} + A_{2} = V_{0}$

2. $S_{1} A_{1} + S_{2} A_{2} = \frac{I_{0}}{C}$

$ω_{d} = \sqrt{ω_{0}^{2} - α^{2}}$ and is known as the Damped Frequency

$V_{c} (t) = V_{f} + B_{1} e^{- α t} \cos (ω_{d} t) + B_{2} e^{- α t} s i n (ω_{d} t)$ for $t \geq 0$

1. $V_{f} + B_{1} = V_{0}$

2. $- α B_{1} + ω_{d} B_{2} = \frac{I_{0}}{C}$

$V_{c} (t) = V_{f} + D_{1} t e^{- α t} + D_{2} e^{- α t}$ for $t \geq 0$

1. $V_{f} + D_{2} = V_{0}$

2. $D_{1} - α D_{2} = \frac{I_{0}}{C}$