SME
Circuit Analysis II · Problem type
Two-wattmeter method
Use this for “Two wattmeters on a balanced three-phase load read 1000 W and 2000 W. Find the total power and the power factor.”
Diagram
P = ⟨3,000, 0⟩
|P| = 3,000.
Q = ⟨0, 1,732⟩
|Q| = 1,732.
Tip to tail: S = ⟨3,000, 1,732⟩
|S| = 3,464 at 30°.
This is an example. Type your own values to start a new problem.
P (W)
=
W₁ (W)
+
W₂ (W)
Q (var)
=
3
×
(
W₂ (W)
−
W₁ (W)
)
S
Apparent power
calculated
θ
Power factor angle
calculated · °
pf
Power factor
calculated
Formulas
P
=
W₁
+
W₂
3,000 = 1,000 + 2,000
Q
=
√
3
×
(
W₂
−
W₁
)
1,732.0508
=
√
3
×
(2,000
−
1,000)
θ
=
tan⁻¹(
Q
P
)
30
=
tan⁻¹(
1,732.0508
3,000
)
S
=
√
P
2
+
Q
2
3,464.1016
=
√
3,
000
2
+ 1,
732.0508
2
pf
=
cos(
θ
)
0.866 = cos(30°)
Assumptions
•
A three-wire load measured by two wattmeters, each voltage coil returning to the third line; P = W₁ + W₂ for any load.
•
For a balanced load Q = √3(W₂ − W₁), so tan θ = √3(W₂ − W₁) ÷ (W₁ + W₂).
•
W₁ reads less on a lagging load and goes negative below pf 0.5; θ < 0 means a leading load.
Step-by-step
WE KNOW
W₁ = 1,000 W, W₂ = 2,000 W
FIND
real power (P), reactive power (Q), power factor angle (θ), apparent power (S), power factor (pf)
Step 1 · Find real power (P)
Use P = W₁ + W₂
The two readings add to the total real power.
P
=
1,000
+
2,000
P
=
3,000
W
Step 2 · Find reactive power (Q)
Use Q = √3 × (W₂ − W₁)
In
a
balanced
load
the
readings
differ
by
VL
IL
sin
θ
,
and
Q
is
√
3
times
that.
Q
=
√
3
×
(2,000
−
1,000)
Q
=
√
3
×
1,000
Q
=
1.732
×
1,000
Q
=
1,732.051
var
Step 3 · Find power factor angle (θ)
Use θ = tan⁻¹(Q ÷ P)
The
power
triangle:
θ
’s
tangent
is
reactive
over
real
power.
θ
=
tan⁻¹(
1,732.051
3,000
)
θ
=
30°
Step 4 · Find apparent power (S)
Use S = √(P² + Q²)
The power triangle’s long side, by Pythagoras.
S
=
√
3,
000
2
+ 1,
732.051
2
S
=
√
9,000,000 + 3,000,000.6666
S
=
√
12
,000,000
S
=
3,464.102
VA
Step 5 · Find power factor (pf)
Use pf = cos(θ)
The power factor is the cosine of the power factor angle.
pf = cos(30°)
pf = 0.866
Check
Put the numbers back in. Both sides should match.
3,000 = 1,000 + 2,000 ✓
1,732.051
=
√
3
×
(2,000
−
1,000)
✓
30
=
tan⁻¹(
1,732.051
3,000
)
✓
3,464.102
=
√
3,
000
2
+ 1,
732.051
2
✓
0.866 = cos(30°) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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