Impedance
Impedance is the combination of all the resistance, capacitive
reactance and inductive reactance in a circuit. It is represented by
the letter Z. As noted above, capacitive reactance and inductive
reactance cancel each other in a series circuit. For example, assume
you have 30 ohms of inductive reactance and 10 ohms of capacitive
reactance. The 10 ohms of capacitive reactance will cancel 10 of the
ohms of inductive reactance. This leaves 20 ohms of inductive reactance
for the circuit total.
Resistance does not combine directly with either inductive reactance or
capacitive reactance. Resistance and reactance can be represented by
vectors placed at right angles to each other. You can represent
resistance (R) as the base of a right triangle and reactance (X) as the
adjacent of the triangle. Impedance (Z) is then represented by the
hypotenuse of the triangle.
Impedance can be calculated by plotting the resistance and reactance on
graph paper then measuring the length of the hypotenuse. Mathematically
is can be calculated by using the Pythagorean theorem. This is
expressed in the following formula.
h^{2} = s1^{2} + s2^{2}

h 
=

The hypotenuse (impedance in ohms) 

s1 
= 
The base (resistance in ohms) 

s2 
= 
The adjacent (reactance in ohms) 
The above formula for the Pythagorean theorem can be reduced to the following formula for calculating impedance.
The steps to solve this formula are:
 Subtract capacitive reactance from inductive reactance
 Square this value and set it aside
 Square the resistance
 Add this value to the value set aside above
 Take the squareroot
For example, if a circuit has 100 ohms of resistance, 30 ohms of
capacitive reactance and 50 ohms of inductive reactance, the impedance
can be calculated with the follow keystrokes on a typical scientific
calculator:
5 

0





3 

0


=


X^{2}


MS


1


0


0


X^{2}


M+


MR


√ 

101

Following the above steps this is what these keystrokes are doing (your calculator may be different):

Subtract X_{C} from X_{L} (50 – 30 =)

Square the result (X^{2})

Store the above in memory (MS)

Square the resistance (100 X^{2})
 Add this square to memory (M+)
 Retrieve the sum of the squares from memory (MR)
 Get the square root (
√ )
This gives an impedance of 101 ohms.
The result is a positive number, meaning that the reactance in the
circuit is predominantly inductive. However, after capacitive
reactance is subtracted from inductive reactance, the remaining
reactance is much less than the resistance. Since resistance is the
dominant factor of the impedance, the circuit is a resistive circuit.
If, the reactance were the dominant factor, the circuit would be
reactive. At a given frequency, a circuit is either resistive,
capacitive or inductive, depending on which is dominant after
subtracting the capacitive reactance from the inductive reactance.
Phase angle
As explained above, voltage and current do not change together in
capacitors and inductors. In a capacitor, the voltage always lags
behind the current; in an inductor the current always lags behind the
voltage. This delay is called phase angle. In capacitors and inductors,
whatever the frequency this lag is always 90 degrees. This essentially
means that, in a capacitor, when the current is at the peak the
voltage is at zero, when the voltage reaches the peak the current is at
zero. By the time the voltage also drops to zero the current
has changed direction and has reached its peak in the opposite
direction. When the voltage also reaches its opposite peak, the current
is at zero again. Voltage will always chase the current in this manner.
Current and voltage with a capacitor

In a resistor the voltage and current are always in phase; when the
current peaks so does the voltage. In a series circuit the
current is always the same. When you have a resistor and a
capacitor in a series AC circuit, the voltage across the capacitor will
be 90 degrees outofphase with the voltage across the resistor. Keep
in mind that Kirchhoff's voltage law still applies. At any moment in
time the voltages across the resistor and capacitor will add up to the
voltage at the AC source (generator, oscillator, etc.) at that moment.
With the voltages outofphase the total opposition to current flow (the impedance) is not the direct
product of voltage divided by the current. The phase angles have to be
factored in. The total phase angle factor is a trigonometric function
as follows:
phase angle = arctan_{ }

reactance

—————————— 
resistance 
For example, if a circuit has a resistance of 200 ohms and at a
particular frequency a capacitive reactance of 50 ohms, the phase angle
would be calculated as follows on a typical calculator:
5


0


÷


2


0


0


=


tan^{1}


14

Which gives a phase angle of 14 degrees.
Impedance plot with 50 ohms of capacitive reactance
and 200 ohms of resistance.

Rectangular notation
The above example uses polar notation. It represents the impedance as a
vector—a quantity and a phase angle. Impedance can also be
expressed in rectangular notation. Here is the above impedance expressed in rectangular
notation:
200  j50
This denotes 200 ohms of resistance and 50 ohms of reactance. The 
j
(negative number) indicates capacitive reactance. Inductive reactance
uses + j (positive number). When worked together, inductive reactance
and capacitive
reactance cancel each other.
The Pythagorean theorem is used to convert form rectangular notation to
polar notation along with the previous phase angle formula. To convert
polar notation to rectangular notation, the impedance is multiplied by
the sine of the angle to get the reactance and by the cosine of the
angle to get the resistance.
sin 14 X 206 = 50 (the reactance) cos 14 X 206 = 200 (the resistance)