SME
Circuit Analysis II · Problem type
Series RLC band-pass: resonance, Q and bandwidth
Use this for “A series RLC circuit has R = 10 Ω, L = 10 mH and C = 1 μF. Find the resonant frequency, Q and the bandwidth.”
Diagram
Corner at f₀ = 1,592 Hz.
At f₀ = 1,592 Hz: |H| = 1.00 = 0 dB, phase 0°.
This is an example. Type your own values to start a new problem.
f₀ (Hz)
=
1
2π
L (mH)
×
C (μF)
Q
=
1
R (Ω)
×
L (mH)
C (μF)
B (Hz)
=
f₀ (Hz)
Q
Formulas
f₀
=
1
2π ×
√
L
×
C
1,591.5494
=
1
÷
(2π
×
√(0.01
×
(1
×
10
-6
)))
Q
=
(
1
R
)
×
√(
L
C
)
10
=
(
1
10
)
×
√(
0.01
1 ×
10
-6
)
B
=
f₀
Q
159.1549
=
1,591.5494
10
Formulas are worked in H, F. The step-by-step shows the conversions.
Assumptions
•
The output is taken across R: at f₀ the reactances of L and C cancel, so the gain is 1 (0 dB).
•
The band B runs between the two −3 dB frequencies, and B = f₀ ÷ Q for this circuit.
•
The inductor and capacitor are ideal: R is the only resistance in the loop.
Step-by-step
WE KNOW
R = 10 Ω, L = 10 mH, C = 1 μF
FIND
resonant frequency (f₀), quality factor (Q), bandwidth (B)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to H, F).
L = 10 mH = 0.01 H (1 H = 1000 mH)
C = 1 μF = 1 × 10⁻⁶ F (1 F = 10⁶ μF)
Step 2 · Find resonant frequency (f₀)
Use f₀ = 1 ÷ (2π × √(L × C))
At
resonance
2πfL
=
1
2πfC
,
so
(2πf)
2
=
1
LC
.
f₀
=
1
2π ×
√
0.01 ×
10
-6
f₀
=
1,591.549
Hz
Step 3 · Find quality factor (Q)
Use Q = (1 ÷ R) × √(L ÷ C)
Q
compares
the
reactance
at
resonance,
√(
L
C
),
with
R
:
the
smaller
R
,
the
sharper
the
peak.
Q
=
(
1
10
)
×
√(
0.01
10
-6
)
Q
=
1
10
×
√
10
,000
Q
=
1
10
×
100
Q
=
0.1
×
100
Q
=
10
Step 4 · Find bandwidth (B)
Use B = f₀ ÷ Q
The sharper the resonance, the narrower the band.
B
=
1,591.549
10
B
=
159.1549
Hz
Check
Put the numbers back in. Both sides should match.
1,591.549
=
1
2π ×
√
0.01 ×
10
-6
✓
10
=
(
1
10
)
×
√(
0.01
10
-6
)
✓
159.1549
=
1,591.549
10
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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