Five Millihenries Twenty Microfarads And A Hundred Twenty Ohms Resonating

Q. An L-C-R series circuit with L = 5 mH, C = 20 μF, R = 120 Ω is connected to an AC source. The resonant frequency of the circuit is (take π² = 10), in hertz? expects 500 Hz

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Q. An L-C-R series circuit with L = 5 mH $L = 5$ mH, C = 20 μF $C = 20$ μF, R = 120 Ω $R = 120$ Ω is connected to an AC source. The resonant frequency of the circuit is (take π² = 10 $π = 3.16227766$ ), in hertz?

The playground

Inputs

inductance

L

5 mH

capacitance

C

20 μF

resistance

R

120 Ω

Outputs

resonance

ν_{r}

The solution

Solved with SeriesLcr

Given

$π = 3.16227766$ from “take π² = 10”
$L = 5$ mH
$C = 20$ μF
$R = 120$ Ω

Find $ν_{r}$ .

Working

$L = 5$ mH $= 0.005$ H
$C = 20$ μF $= 2 \times 10^{−5}$ F
resonance of a series circuit(circuit) $ν_{r} = \frac{1}{2 \cdot π \cdot \sqrt{L \cdot C}}$
$ν_{r} = \frac{1}{2 \cdot 3.16227766 \cdot \sqrt{0.005 \cdot 2 \times 10^{−5}}}$
$ν_{r} = 500$ Hz
$ν_{r} \approx 500$ Hzto 3 significant figures

Conclusion

The reactances cancel where 2πfL = 1/(2πfC), at f = 1/(2π√(LC)). With π² taken as 10, 2π√(LC) = 2√(10 × 5 × 10⁻³ × 20 × 10⁻⁶) = 2 × 10⁻³ s, so f = 500 Hz.

The Ruby

1require_relative "../physics"23question :five_millihenries_twenty_microfarads_and_a_hundred_twenty_ohms_resonating do4  kind :numerical5  describes "An L-C-R series circuit with L = %{inductance}, C = %{capacitance}, R = %{resistance} is connected to " \6            "an AC source. %{resonance} (take π² = 10), in hertz?"7  given inductance: "5 mH"8  given capacitance: "20 μF"9  given resistance: "120 Ω"10  asks resonance: "The resonant frequency of the circuit is"11end

1require_relative "../questions/five_millihenries_twenty_microfarads_and_a_hundred_twenty_ohms_resonating"2require_relative "../science/alternating_current/series_lcr"34solve :five_millihenries_twenty_microfarads_and_a_hundred_twenty_ohms_resonating, using: SeriesLcr do5  objects :circuit67  deduce_from "take π² = 10", pi**2 == 1089  take inductance: circuit.inductance, capacitance: circuit.capacitance, resistance: circuit.resistance1011  resonance = circuit.resonant_frequency12  answer resonance:, in: "Hz", figures: 31314  conclusion "The reactances cancel where 2πfL = 1/(2πfC), at f = 1/(2π√(LC)). With π² taken as 10, " \15             "2π√(LC) = 2√(10 × 5 × 10⁻³ × 20 × 10⁻⁶) = 2 × 10⁻³ s, so f = 500 Hz."16end

1require_relative "../solutions/five_millihenries_twenty_microfarads_and_a_hundred_twenty_ohms_resonating"23play_with :five_millihenries_twenty_microfarads_and_a_hundred_twenty_ohms_resonating do4  title "Five Millihenries Twenty Microfarads And A Hundred Twenty Ohms Resonating"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	Reactance
rms voltage	$V$	V	0 … ∞	PeakAndRms
peak voltage	$v_{m}$	V	—	PeakAndRms
rms current	$I$	A	0 … ∞	PeakAndRms
peak current	$i_{m}$	A	—	PeakAndRms
frequency	$ν$	Hz	0 … ∞	Reactance
angular frequency	$ω$	rad/s	—	Reactance
inductance	$L$	H	—	Reactance
capacitance	$C$	F	0 … ∞	Reactance
inductive reactance	$X_{L}$	Ω	0 … ∞	Reactance
capacitive reactance	$X_{C}$	Ω	0 … ∞	Reactance
voltage across the inductor	$V_{L}$	V	—	Reactance
voltage across the capacitor	$V_{C}$	V	—	Reactance
resistance	$R$	Ω	0 … ∞	SeriesLcr
impedance	$Z$	Ω	0 … 1e+09	SeriesLcr
power drawn	$P$	W	—	SeriesLcr
resonant frequency	$ν_{r}$	Hz	—	SeriesLcr
phase angle	$φ$	—	-3.142 … 3.142	SeriesLcr
resonant angular frequency	$ω_{r}$	rad/s	—	SeriesLcr
peak voltage of ground	$v_{m,g}$	V	—
rms voltage of ground	$V_{g}$	V	0 … ∞
peak voltage of circuit	$v_{m}$	V	—
rms voltage of circuit	$V$	V	0 … ∞
peak current of ground	$i_{m,g}$	A	—
rms current of ground	$I_{g}$	A	0 … ∞
peak current of circuit	$i_{m}$	A	—
rms current of circuit	$I$	A	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of circuit	$ω$	rad/s	—
frequency of circuit	$ν$	Hz	0 … ∞
inductive reactance of ground	$X_{L,g}$	Ω	0 … ∞
inductance of ground	$L_{g}$	H	—
inductive reactance of circuit	$X_{L}$	Ω	0 … ∞
inductance of circuit	$L$	H	—
capacitive reactance of ground	$X_{C,g}$	Ω	0 … ∞
capacitance of ground	$C_{g}$	F	0 … ∞
capacitive reactance of circuit	$X_{C}$	Ω	0 … ∞
capacitance of circuit	$C$	F	0 … ∞
voltage across the inductor of ground	$V_{L,g}$	V	—
voltage across the inductor of circuit	$V_{L}$	V	—
voltage across the capacitor of ground	$V_{C,g}$	V	—
voltage across the capacitor of circuit	$V_{C}$	V	—
impedance of ground	$Z_{g}$	Ω	0 … 1e+09
resistance of ground	$R_{g}$	Ω	0 … ∞
impedance of circuit	$Z$	Ω	0 … 1e+09
resistance of circuit	$R$	Ω	0 … ∞
power drawn of ground	$P_{g}$	W	—
power drawn of circuit	$P$	W	—
phase angle of ground	$φ_{g}$	—	-3.142 … 3.142
phase angle of circuit	$φ$	—	-3.142 … 3.142
resonant frequency of ground	$ν_{r,g}$	Hz	—
resonant frequency of circuit	$ν_{r}$	Hz	—
resonant angular frequency of ground	$ω_{r,g}$	rad/s	—
resonant angular frequency of circuit	$ω_{r}$	rad/s	—

v_{m,c} = \sqrt{2} \cdot V_{c}

peak is root two times rms

i_{m,c} = \sqrt{2} \cdot I_{c}

peak current is root two times rms

ω_{c} = 2 \cdot π \cdot ν_{c}

angular frequency and frequency

X_{L,c} = 2 \cdot π \cdot ν_{c} \cdot L_{c}

reactance of an inductor

X_{C,c} = \frac{1}{2 \cdot π \cdot ν_{c} \cdot C_{c}}

reactance of a capacitor

V_{L,c} = I_{c} \cdot X_{L,c}

current through the inductor

V_{C,c} = I_{c} \cdot X_{C,c}

current through the capacitor

{Z_{c}}^{2} = {R_{c}}^{2} + {(X_{L,c} - X_{C,c})}^{2}

impedance of a series circuit

V_{c} = I_{c} \cdot Z_{c}

current through the impedance

P_{c} = {I_{c}}^{2} \cdot R_{c}

power spent in the resistance

φ_{c} = atan (\frac{X_{C,c} - X_{L,c}}{R_{c}})

phase of the current

ν_{r,c} = \frac{1}{2 \cdot π \cdot \sqrt{L_{c} \cdot C_{c}}}

resonance of a series circuit

ω_{r,c} = 2 \cdot π \cdot ν_{r,c}

resonance in radians a second

v_{m,g} = \sqrt{2} \cdot V_{g}

peak is root two times rms(ground)

v_{m} = \sqrt{2} \cdot V

peak is root two times rms(circuit)

i_{m,g} = \sqrt{2} \cdot I_{g}

peak current is root two times rms(ground)

i_{m} = \sqrt{2} \cdot I

peak current is root two times rms(circuit)

ω_{g} = 2 \cdot π \cdot ν_{g}

angular frequency and frequency(ground)

ω = 2 \cdot π \cdot ν

angular frequency and frequency(circuit)

X_{L,g} = 2 \cdot π \cdot ν_{g} \cdot L_{g}

reactance of an inductor(ground)

X_{L} = 2 \cdot π \cdot ν \cdot L

reactance of an inductor(circuit)

X_{C,g} = \frac{1}{2 \cdot π \cdot ν_{g} \cdot C_{g}}

reactance of a capacitor(ground)

X_{C} = \frac{1}{2 \cdot π \cdot ν \cdot C}

reactance of a capacitor(circuit)

V_{L,g} = I_{g} \cdot X_{L,g}

current through the inductor(ground)

V_{L} = I \cdot X_{L}

current through the inductor(circuit)

V_{C,g} = I_{g} \cdot X_{C,g}

current through the capacitor(ground)

V_{C} = I \cdot X_{C}

current through the capacitor(circuit)

{Z_{g}}^{2} = {R_{g}}^{2} + {(X_{L,g} - X_{C,g})}^{2}

impedance of a series circuit(ground)

Z^{2} = R^{2} + {(X_{L} - X_{C})}^{2}

impedance of a series circuit(circuit)

V_{g} = I_{g} \cdot Z_{g}

current through the impedance(ground)

V = I \cdot Z

current through the impedance(circuit)

P_{g} = {I_{g}}^{2} \cdot R_{g}

power spent in the resistance(ground)

P = I^{2} \cdot R

power spent in the resistance(circuit)

φ_{g} = atan (\frac{X_{C,g} - X_{L,g}}{R_{g}})

phase of the current(ground)

φ = atan (\frac{X_{C} - X_{L}}{R})

phase of the current(circuit)

ν_{r,g} = \frac{1}{2 \cdot π \cdot \sqrt{L_{g} \cdot C_{g}}}

resonance of a series circuit(ground)

ν_{r} = \frac{1}{2 \cdot π \cdot \sqrt{L \cdot C}}

resonance of a series circuit(circuit)

ω_{r,g} = 2 \cdot π \cdot ν_{r,g}

resonance in radians a second(ground)

ω_{r} = 2 \cdot π \cdot ν_{r}

resonance in radians a second(circuit)