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Impedance This worksheet and all related les are licensed under the Creative Commons Attribution License, Impedance
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1 Question 1
In this AC circuit, the resistor oers 300 of resistance, and the inductor oers 400 of reactance.
Together, their series opposition to alternating current results in a current of 10 mA from the 5 volt source:
R = 300 X
L
= 400 5 VAC
I = 10 mA
How many ohms of opposition does the series combination of resistor and inductor oer? What name
do we give to this quantity, and how do we symbolize it, being that it is composed of both resistance (R)
and reactance (X)?
le 00584
Answer 1
Z
total
= 500 .
Follow-up question: suppose that the inductor suers a failure in its wire winding, causing it to open.
Explain what eect this would have on circuit current and voltage drops.
Notes 1
Students may experience diculty arriving at the same quantity for impedance shown in the answer. If
this is the case, help them problem-solve by suggesting they simplify the problem: short past one of the
load components and calculate the new circuit current. Soon they will understand the relationship between
total circuit opposition and total circuit current, and be able to apply this concept to the original problem.
Ask your students why the quantities of 300 and 400 do not add up to 700 like they would if
they were both resistors. Does this scenario remind them of another mathematical problem where 3 + 4 = 5?
Where have we seen this before, especially in the context of electric circuits?
Once your students make the cognitive connection to trigonometry, ask them the signicance of these
numbers addition. Is it enough that we say a component has an opposition to AC of 400 , or is there more
to this quantity than a single, scalar value? What type of number would be suitable for representing such a
quantity, and how might it be written?
2 Question 2
In this AC circuit, the resistor oers 3 k of resistance, and the capacitor oers 4 k of reactance.
Together, their series opposition to alternating current results in a current of 1 mA from the 5 volt source:
5 VAC
X
C
= 4 k R = 3 k I = 1 mA
How many ohms of opposition does the series combination of resistor and capacitor oer? What name
do we give to this quantity, and how do we symbolize it, being that it is composed of both resistance (R)
and reactance (X)?
le 00585
Answer 2
Z
total
= 5 k.
Notes 2
Students may experience diculty arriving at the same quantity for impedance shown in the answer. If
this is the case, help them problem-solve by suggesting they simplify the problem: short past one of the
load components and calculate the new circuit current. Soon they will understand the relationship between
total circuit opposition and total circuit current, and be able to apply this concept to the original problem.
Ask your students why the quantities of 3 k and 4 k do not add up to 7 k like they would if they
were both resistors. Does this scenario remind them of another mathematical problem where 3 + 4 = 5?
Where have we seen this before, especially in the context of electric circuits?
Once your students make the cognitive connection to trigonometry, ask them the signicance of these
numbers addition. Is it enough that we say a component has an opposition to AC of 4 k, or is there more
to this quantity than a single, scalar value? What type of number would be suitable for representing such a
quantity, and how might it be written?
3 Question 3
While studying DC circuit theory, you learned that resistance was an expression of a components
opposition to electric current. Then, when studying AC circuit theory, you learned that reactance was
another type of opposition to current. Now, a third term is introduced: impedance. Like resistance and
reactance, impedance is also a form of opposition to electric current.
Explain the dierence between these three quantities (resistance, reactance, and impedance) using your
own words.
le 01567
Answer 3
The fundamental distinction between these terms is one of abstraction: impedance is the most general
term, encompassing both resistance and reactance. Here is an explanation given in terms of logical sets
(using a Venn diagram), along with an analogy from animal taxonomy:
Impedance
Resistance
Reactance
Mammal
Horse
Rabbit
R
X
Z
Resistance is a type of impedance, and so is reactance. The dierence between the two has to do with
energy exchange
.
Notes 3
The given answer is far from complete.
Ive shown the semantic relationship between the terms
resistance, reactance, and impedance, but I have only hinted at the conceptual distinctions between them.
Be sure to discuss with your students what the fundamental dierence is between resistance and reactance,
in terms of electrical energy exchange.
4 Question 4
In DC circuits, we have Ohms Law to relate voltage, current, and resistance together:
E
= IR
In AC circuits, we similarly need a formula to relate voltage, current, and impedance together. Write
three equations, one solving for each of these three variables: a set of Ohms Law formulae for AC circuits.
Be prepared to show how you may use algebra to manipulate one of these equations into the other two forms.
le 00590
Answer 4
E
= IZ
I
= E
Z
Z
= E
I
If using phasor quantities (complex numbers) for voltage, current, and impedance, the proper way to
write these equations is as follows:
E
= IZ
I
= E
Z
Z
= E
I
Bold-faced type is a common way of denoting vector quantities in mathematics.
Notes 4
Although the use of phasor quantities for voltage, current, and impedance in the AC form of Ohms
Law yields certain distinct advantages over scalar calculations, this does not mean one cannot use scalar
quantities. Often it is appropriate to express an AC voltage, current, or impedance as a simple scalar number.
5 Question 5
It is often necessary to represent AC circuit quantities as complex numbers rather than as scalar numbers,
because both magnitude and phase angle are necessary to consider in certain calculations.
When representing AC voltages and currents in polar form, the angle given refers to the phase shift
between the given voltage or current, and a reference voltage or current at the same frequency somewhere
else in the circuit. So, a voltage of 3.5 V 45
o
means a voltage of 3.5 volts magnitude, phase-shifted 45
degrees behind (lagging) the reference voltage (or current), which is dened to be at an angle of 0 degrees.
But what about impedance (Z)? Does impedance have a phase angle, too, or is it a simple scalar number
like resistance or reactance?
Calculate the amount of current that would go through a 100 mH inductor with 36 volts RMS applied
to it at a frequency of 400 Hz. Then, based on Ohms Law for AC circuits and what you know of the phase
relationship between voltage and current for an inductor, calculate the impedance of this inductor in polar
form
. Does a denite angle emerge from this calculation for the inductors impedance? Explain why or why
not.
le 00588
Answer 5
Z
L
= 251.33
90
o
Notes 5
This is a challenging question, because it asks the student to defend the application of phase angles to a
type of quantity that does not really possess a wave-shape like AC voltages and currents do. Conceptually,
this is dicult to grasp. However, the answer is quite clear through the Ohms Law calculation (Z =
E
I
).
Although it is natural to assign a phase angle of 0
o
to the 36 volt supply, making it the reference
waveform, this is not actually necessary. Work through this calculation with your students, assuming dierent
angles for the voltage in each instance. You should nd that the impedance computes to be the same exact
quantity every time.
6 Question 6
Express the impedance (Z) in both polar and rectangular forms for each of the following components:
A resistor with 500 of resistance
An inductor with 1.2 k of reactance
A capacitor with 950 of reactance
A resistor with 22 k of resistance
A capacitor with 50 k of reactance
An inductor with 133 of reactance
le 00591
Answer 6
A resistor with 500 of resistance: 500
0
o
or 500 + j0
An inductor with 1.2 k of reactance: 1.2 k
90
o
or 0 + j1.2k
A capacitor with 950 of reactance: 950
-90
o
or 0 - j950
A resistor with 22 k of resistance: 22 k
0
o
or 22k + j0
A capacitor with 50 k of reactance: 50 k
-90
o
or 0 - j50k
An inductor with 133 of reactance: 133
90
o
or 0 + j133
Follow-up question: what would the phasors look like for resistive, inductive, and capacitive impedances?
Notes 6
In your discussion with students, emphasize the consistent nature of phase angles for impedances of
pure components.
7 Question 7
Real inductors and capacitors are never purely reactive. There will inevitably be some resistance intrinsic
to these devices as well.
Suppose an inductor has 57 of winding re