The L-R-C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ω . Let the source voltage be given by υ = V cos ω t. (a) Show that each of the instantaneous voltages υ R , υ L , and υ C at any instant is equal to u and that i = i R + i L + i C , where i is the current through the source and i R , i L , and i C are the currents through the resistor, inductor, and capacitor, respectively, (b) What are the phases of i R , i L , and i C with respect to υ ? Use current phasors to represent i , i R , i L , and i C. In a phasor diagram, show the phases of these four currents with respect to υ . (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is I = I R 2 + ( I C − I L ) 2 . (d) Show that the result of part (c) can be written as I = V / Z , with 1/ Z = ( 1 / R 2 ) + [ ω C − ( 1/ ω L ) ] 2 .
The L-R-C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ω . Let the source voltage be given by υ = V cos ω t. (a) Show that each of the instantaneous voltages υ R , υ L , and υ C at any instant is equal to u and that i = i R + i L + i C , where i is the current through the source and i R , i L , and i C are the currents through the resistor, inductor, and capacitor, respectively, (b) What are the phases of i R , i L , and i C with respect to υ ? Use current phasors to represent i , i R , i L , and i C. In a phasor diagram, show the phases of these four currents with respect to υ . (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is I = I R 2 + ( I C − I L ) 2 . (d) Show that the result of part (c) can be written as I = V / Z , with 1/ Z = ( 1 / R 2 ) + [ ω C − ( 1/ ω L ) ] 2 .
The L-R-C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ω. Let the source voltage be given by υ = Vcos ωt. (a) Show that each of the instantaneous voltages υR, υL, and υC at any instant is equal to u and that i = iR + iL + iC, where i is the current through the source and iR, iL, and iC are the currents through the resistor, inductor, and capacitor, respectively, (b) What are the phases of iR, iL, and iC with respect to υ? Use current phasors to represent i, iR, iL, and iC. In a phasor diagram, show the phases of these four currents with respect to υ. (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is
I
=
I
R
2
+
(
I
C
−
I
L
)
2
. (d) Show that the result of part (c) can be written as I = V/Z, with 1/Z =
(
1
/
R
2
)
+
[
ω
C
−
(
1/
ω
L
)
]
2
.
An L-R-C series circuit is constructed using a 175 Ω resistor, a 12.5 μF capacitor, and an 8.00 mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 V.
At the angular frequency in part A, find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value.
What are the potential difference: v, vR, vC, vL
An L-R-C series circuit consists of a 2.70 μF capacitor, a 5.00 mH inductor, and a 50.0 Ω resistor connected across an ac source of voltage amplitude 14.0 V having variable frequency.
A. Under the conditions of part A (ω = 2.09×104 rad/s), what is the average power delivered to each circuit element? (PR,PC,PL)
B. What is the maximum current through the capacitor?
A resistor ( R = 9.00 × 102 Ω), a capacitor (C = 0.250 µF), and an inductor (L = 2.50 H) are connected in series across a 2.40 × 102-Hz AC source for which ΔV max = 1.40 × 102 V. Calculate (a) the impedance of the circuit, (b) the maximum current delivered by the source, and (c) the phase angle between the current and voltage. (d) Is the current leading or lagging the voltage?
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