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31. Alternating Current Circuits
Assignment is due at 2:00am on Wednesday, March 21, 2007
Credit for problems submitted late will decrease to 0% after the deadline has passed.
The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help.
The unopened hint bonus is 2% per part.
You are allowed 4 attempts per answer.
Reactance and Phase
Voltage and Current in AC Circuits
Learning Goal:
To understand the relationship between AC voltage and current in resistors, inductors, and capacitors, especially the phase shift between the voltage and the current.
In this problem, we consider the behavior of resistors, inductors, and capacitors driven individually by a sinusoidally alternating voltage source, for which the voltage is given as a function of time by
. The main challenge is to apply your knowledge of the basic properties of resistors, inductors, and capacitors to these "singleelement" AC circuits to find the current
through
each. The key is to understand the phase difference, also known as the phase angle, between the voltage and the current. It is important to take into account the sign of the current, which will be called
positive when it flows clockwise from the b terminal (which has positive voltage relative to the a terminal) to the a terminal (see figure). The sign is critical in the analysis of circuits containing
combinations of resistors, capacitors, and inductors.
Part A
First, let us consider a resistor with resistance
connected to an AC source (diagram 1). If the AC source provides a voltage
, what is the current
through the resistor as a
function of time?
Hint A.1
Ohm's law
Hint not displayed
Express your answer in terms of
,
,
, and
.
ANSWER:
=
Note that the voltage and the current are in phase; that is, in the expressions for
and
, the arguments of the cosine functions are the same at any moment of time. This will not be the
case for the capacitor and inductor.
Part B
Now consider an inductor with inductance
in an AC circuit (diagram 2). Assuming that the current in the inductor varies as
, find the voltage
that must be driving the
inductor.
Part B.1
Kirchhoff's loop rule
Part not displayed
Part B.2
The derivative of
Part not displayed
Hint B.3
The phase relationship between sine and cosine
Hint not displayed
Express your answer in terms of
,
,
, and
. Use the cosine function, not the sine function, in your answer.
ANSWER:
=
Graphs of
and
are shown below. As you can see, for an inductor, the voltage leads (i.e., reaches its maximum before) the current by
; in other words, the current lags the voltage by
. This can be conceptually understood by thinking of inductance as giving the current inertia: The voltage "tries" to push current through the inductor, but some sort of inertia resists the change
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This note was uploaded on 11/02/2009 for the course MASTERING PHYS taught by Professor All during the Spring '09 term at Kettering.
 Spring '09
 All

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