The magnitude condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside. In combination with the angle condition, these two mathematical expressions fully determine the root locus.
Let the characteristic equation of a system be
1
+
G
(
s
)
=
0
, where
G
(
s
)
=
P
(
s
)
Q
(
s
)
. Rewriting the equation in polar form is useful.
e
j
2
π
+
G
(
s
)
=
0
G
(
s
)
=
−
1
=
e
j
(
π
+
2
k
π
)
where
(
k
=
0
,
1
,
2
,
.
.
.
)
are the only solutions to this equation. Rewriting
G
(
s
)
in factored form,
G
(
s
)
=
P
(
s
)
Q
(
s
)
=
K
(
s
−
a
1
)
(
s
−
a
2
)
⋯
(
s
−
a
n
)
(
s
−
b
1
)
(
s
−
b
2
)
⋯
(
s
−
b
m
)
,
and representing each factor
(
s
−
a
p
)
and
(
s
−
b
q
)
by their vector equivalents,
A
p
e
j
θ
p
and
B
q
e
j
ϕ
q
, respectively,
G
(
s
)
may be rewritten.
G
(
s
)
=
K
A
1
A
2
⋯
A
n
e
j
(
θ
1
+
θ
2
+
⋯
+
θ
n
)
B
1
B
2
⋯
B
m
e
j
(
ϕ
1
+
ϕ
2
+
⋯
+
ϕ
m
)
Simplifying the characteristic equation,
e
j
(
π
+
2
k
π
)
=
K
A
1
A
2
⋯
A
n
e
j
(
θ
1
+
θ
2
+
⋯
+
θ
n
)
B
1
B
2
⋯
B
m
e
j
(
ϕ
1
+
ϕ
2
+
⋯
+
ϕ
m
)
=
K
A
1
A
2
⋯
A
n
B
1
B
2
⋯
B
m
e
j
(
θ
1
+
θ
2
+
⋯
+
θ
n
−
(
ϕ
1
+
ϕ
2
+
⋯
+
ϕ
m
)
)
,
from which we derive the magnitude condition:
1
=
K
A
1
A
2
⋯
A
n
B
1
B
2
⋯
B
m
.
The angle condition is derived similarly.
