I think flalign* and \intertext{} are very good for you.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{flalign*}
& & T_{Loop Filter}(s)&=K_{p}+\frac{K_{i}}{s} & & \\
\intertext{The overall transfer function of given PLL can be given as}
& & \frac{{\phi}_{c}(s)}{{\phi}(s)}&=\frac{T_{Loop Filter}(s)*\frac{1}{s}}{1+\frac{T_{Loop Filter}}(s)*\frac{1}{s}} & & \\
& & \frac{\phi_{c}(s)}{\phi(s)}&=\frac{K_{p}(s)+K_{i}}{s^{2}+K_{p}(s)+K_{i}} & &
\intertext{In general transfer function for a second order closed loop control system is given as}
& & T_{second order}&=\frac{{\omega}^2+{2{\delta}}{\omega}s}{s_{2}+2{\delta}{\omega}s+{\omega}^{2}} & & \\
& \rlap{\text{On comparing}} & {\omega}&={sqrt{K_i}} {\delta}=\frac{K_p}{2\sqrt{K_{i}}} & &
\end{flalign*}
\end{document}
This is a case where visual formatting seems necessary, so it should be dealt with only in the very last stage of document preparation.
You can pretend that the very last part of the formula “is not there”:
\documentclass[12pt, oneside]{scrbook}
\usepackage{amsmath,mathtools}
\DeclareMathOperator{\E}{E}
\newcommand{\with}{\,;}
\newcommand{\dx}{\mathop{}\!\mathrm{d}}
\newcommand{\expect}[2][]{\E\if\relax\detokenize{#1}\relax\else_{#1}\fi \left[#2\right]}
\begin{document}
\begin{equation}
\begin{aligned}
\mu_{D_-}
&= \expect{f'(S_-) \cdot \sum_{i=1}^{N} W D} \\
&= N \mu_W \mu_D \int_{-\infty}^{\infty} f'(s) p_\mathcal{N}(s \with \mu_{S_-}, \sigma_{S_-}) \dx{s} \\
\sigma_{D_-}^2
&= \expect{f'(S_-)^2 \cdot \left(\sum_{i=1}^{N} W D\right)^2} - \mu_{D_-}^2 \\
&= N(\sigma_W^2 \sigma_D^2 + \mu_W^2 \sigma_D^2 + \sigma_W^2 \mu_D^2 + N \mu_W^2 \mu_D^2)
\int_{-\infty}^{\infty} f'(s)^2 p_\mathcal{N}(s \with \mu_{S_-}, \sigma_{S_-}) \dx{s}
\mathrlap{{} - \mu_{D_-}^2 \,.}
\end{aligned}
\end{equation}
\end{document}

I amended your definition of \dx
, because it would give wrong spacing on the right. Also \;
seems too big for \with
(to be honest, I wouldn't add a space before the semicolon).
Your definition of \expect
with automatic \left
and \right
produces too big delimiters in many cases (see the third line, for instance).
Here's an amended version that uses xparse
and mathtools
features; the syntax for \expect
is
\expect{formula}
\expect[\big]{formula}
\expect*{formula}
\expect_{sub}{formula}
\expect_{sub}[\big]{formula}
\expect_{sub}*{formula}
as shown in the added equation. In the optional argument one can use \big
, \Big
, \bigg
or \Bigg
; the *-version uses \left
and \right
.
Note also the addition of \,
and \!
in a few places.

Best Answer
Use the
amsmath
package and the commands\text{...}
for text in the formula or\intertext{...}
for text between the lines of multi-line formulas. For example:One advantage of
align*
to\[ ... \]
is that you can align the equations on relation symbols.If you wish to put or in the same line, you could use
\text
andflalign*
: