The Current Amplitude at Resonance

Q. A series LCR circuit is connected to a variable frequency 230 V source, with L = 5.0 H, C = 80 μF and R = 40 Ω. The source is tuned to the frequency which drives the circuit in resonance. What is the amplitude of current, in amperes? expects 8.13 A

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Q. A series LCR circuit is connected to a variable frequency 230 V $V = 230$ V source, with L = 5.0 H $L = 5.0$ H, C = 80 μF $C = 80$ μF and R = 40 Ω $R = 40$ Ω. The source is tuned to the frequency which drives the circuit in resonance $ν = ν_{r}$ . What is the amplitude of current, in amperes?

The playground

Inputs

supply

V

230 V

inductance

L

5 H

capacitance

C

80 μF

resistance

R

40 Ω

Outputs

amplitude

i_{m}

The solution

Solved with SeriesLcr

Given

$V = 230$ V
$L = 5.0$ H
$C = 80$ μF
$R = 40$ Ω
$ν = ν_{r}$ from “drives the circuit in resonance”

Find $i_{m}$ .

Working

$C = 80$ μF $= 8 \times 10^{−5}$ F
resonance of a series circuit(circuit) $ν_{r} = \frac{1}{2 \cdot π \cdot \sqrt{L \cdot C}}$
$ν_{r} = \frac{1}{2 \cdot 3.14159265 \cdot \sqrt{5 \cdot 8 \times 10^{−5}}}$
$ν_{r} = 7.95774715$ Hz
drives the circuit in resonance $ν = ν_{r}$
$ν = 7.95774715$
$ν = 7.95774715$ Hz
reactance of an inductor(circuit) $X_{L} = 2 \cdot π \cdot ν \cdot L$
$X_{L} = 2 \cdot 3.14159265 \cdot 7.95774715 \cdot 5$
$X_{L} = 250$ Ω
reactance of a capacitor(circuit) $X_{C} = \frac{1}{2 \cdot π \cdot ν \cdot C}$
$X_{C} = \frac{1}{2 \cdot 3.14159265 \cdot 7.95774715 \cdot 8 \times 10^{−5}}$
$X_{C} = 250$ Ω
impedance of a series circuit(circuit) $Z^{2} = R^{2} + {(X_{L} - X_{C})}^{2}$
$Z^{2} = 40^{2} + {(250 - 250)}^{2}$
$Z = 40$ Ω
current through the impedance(circuit) $V = I \cdot Z$
$230 = I \cdot 40$
$I = 5.75$ A
peak current is root two times rms(circuit) $i_{m} = \sqrt{2} \cdot I$
$i_{m} = \sqrt{2} \cdot 5.75$
$i_{m} = 8.132$ A
$i_{m} \approx 8.13$ Ato 3 significant figures

Conclusion

At resonance the impedance falls to the bare resistance, so the current swings as wide as the resistance lets it.

The Ruby

1require_relative "../physics"23question :the_current_amplitude_at_resonance do4  kind :numerical5  describes "A series LCR circuit is connected to a variable frequency %{supply} source, with L = %{inductance}, " \6            "C = %{capacitance} and R = %{resistance}. The source is tuned to the frequency which drives the " \7            "circuit in resonance. %{amplitude}, in amperes?"8  given supply: "230 V"9  given inductance: "5.0 H"10  given capacitance: "80 μF"11  given resistance: "40 Ω"12  asks amplitude: "What is the amplitude of current"13end

1require_relative "../questions/the_current_amplitude_at_resonance"2require_relative "../science/alternating_current/series_lcr"34solve :the_current_amplitude_at_resonance, using: SeriesLcr do5  objects :circuit67  take supply: circuit.rms_voltage8  take inductance: circuit.inductance9  take capacitance: circuit.capacitance10  take resistance: circuit.resistance1112  deduce_from "drives the circuit in resonance", circuit.frequency => circuit.resonant_frequency1314  amplitude = circuit.peak_current15  answer amplitude:, in: "A", figures: 31617  conclusion "At resonance the impedance falls to the bare resistance, so the current swings as wide as the " \18             "resistance lets it."19end

1require_relative "../solutions/the_current_amplitude_at_resonance"23play_with :the_current_amplitude_at_resonance do4  title "The Current Amplitude at Resonance"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)

ν = ν_{r}

drives the circuit in resonance