SME
Circuit Analysis II · Problem type
Coupling coefficient and mutual voltage
Use this for “Coils of 4 mH and 9 mH have a mutual inductance of 3 mH. Find k, and v₂ when i₁ rises 1 A in 1 ms.”
Table
Coupling coefficient (k)
Voltage in coil 2 (v₂)
(V)
This is an example. Type your own values to start a new problem.
k
=
M (mH)
L₁ (mH)
×
L₂ (mH)
v₂ (V)
=
M (mH)
×
Δi₁ (A)
Δt (ms)
L₁ and L₂ are the two coils’ own inductances and M the part they share. The table holds L₁, L₂, Δi₁ and Δt and steps k, so M changes.
Formulas
k
=
M
÷
√
L₁
×
L₂
0.5
=
3
÷
√
4 × 9
v₂
=
M
×
Δi₁
Δt
3
=
3
×
1
1
Assumptions
•
Part of the flux from coil 1 links coil 2; k is the fraction shared, from 0 (none) to 1 (all).
•
The current in coil 1 changes at a steady rate, di₁/dt = Δi₁ ÷ Δt, so v₂ is steady.
•
Coil 2 is open: no current flows in it, so only M induces its voltage.
Step-by-step
WE KNOW
L₁ = 4 mH, L₂ = 9 mH, M = 3 mH, Δi₁ = 1 A, Δt = 1 ms
FIND
coupling coefficient (k), voltage in coil 2 (v₂)
Step 1 · Find coupling coefficient (k)
Use k = M ÷ √(L₁ × L₂)
M
can
be
at
most
√
L₁L₂
,
when
all
the
flux
is
shared;
k
is
the
fraction
of
that
reached.
k
=
3
÷
√
4 × 9
k
=
3
÷
√
36
k
=
3
6
k
=
0.5
Step 2 · Find voltage in coil 2 (v₂)
Use v₂ = M × Δi₁ ÷ Δt
The
flux
coil
1
sends
through
coil
2
follows
i₁,
so
v₂
is
M
times
the
rate
i₁
changes.
v₂
=
3
×
1
1
v₂
=
3
V
Check
Put the numbers back in. Both sides should match.
0.5
=
3
÷
√
4 × 9
✓
3
=
3
×
1
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.