which is the impedance of an
RLC series AC circuit. For circuits without a resistor, take
; for those without an inductor, take
; and for those without a capacitor, take
.
Calculating impedance and current
An
RLC series circuit has a
resistor, a 3.00 mH inductor, and a
capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for
and
are the same as in
[link] and
[link] . (b) If the voltage source has
, what is
at each frequency?
Strategy
For each frequency, we use
to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.
Solution for (a)
At 60.0 Hz, the values of the reactances were found in
[link] to be
and in
[link] to be
. Entering these and the given
for resistance into
yields
Similarly, at 10.0 kHz,
and
, so that
Discussion for (a)
In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that
dominates at high frequency and
dominates at low frequency.
Solution for (b)
The current
can be found using the AC version of Ohm’s law in Equation
:
at 60.0 Hz
Finally, at 10.0 kHz, we find
at 10.0 kHz
Discussion for (a)
The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in
[link] . The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in
[link] . The inductor dominates at high frequency.
How does an
RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law,
, and the expression for impedance
from
gives
The reactances vary with frequency, with
large at high frequencies and
large at low frequencies, as we have seen in three previous examples. At some intermediate frequency
, the reactances will be equal and cancel, giving
—this is a minimum value for impedance, and a maximum value for
results. We can get an expression for
by taking
Substituting the definitions of
and
,
Solving this expression for
yields
where
is the
resonant frequency of an
RLC series circuit. This is also the
natural frequency at which the circuit would oscillate if not driven by the voltage source. At
, the effects of the inductor and capacitor cancel, so that
, and
is a maximum.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?