Introductory Physics Volume Two

6.8 Phasor Diagrams 127
What we want to find is the ratio of the amplitude of the supply voltage
and the amplitude of the current: V S0 /I 0 .
Kirchhoff’s loop rule gives us that
V S (t) − V C (t) − V R (t) = 0 −→ V S (t) = V C (t) + V R (t)
In the phasor diagram this implies that the sum of the phasors for V C
and V R must be equal to the phasor of the source voltage V S . Since
the resistor phasor is parallel to the current phasor we know that the
phasors for V C must follow the phasor for V R by 90 ◦ . Thus the sum
(also V S ) forms the hypotenuse of a right triangle.
We can find the phasor for the sum of the voltage on the resistor and
capacitor by adding the individual phasors like we add vectors. Also
notice from the circuit diagram that the sum of the voltages on the
resistor and capacitor is equal to the voltage on the supply: V S =
V R + V C .
Since the lengths of the phasors are the amplitudes of the voltages we
can use the pythagorian theorem to find that
V 2 S 0
= V 2 R 0
+ V 2 C 0
= (RI 0 ) 2 + (Z C I 0 ) 2
−→ V S 2 0
I0
2 = R 2 + ZC
2 √
−→ Z eff = V √
S 0
= R
I 2 + ZC 2 = R 2 + 1
0 (ωC) 2
We see that the impedances do not simply add when the components
are in series. In this case the square of the impedance was the sum of
the squares of parts. This is not a general rule. We can also find the
phase angle (φ) between the supply voltage and the current.
tan φ = −V C 0
= −Z CI 0
= −Z C
V R0 RI 0 R = −1
ωRC
There is a very significant difference between the resistance of a
resistor and the impedance of a capacitor or inductor: the impedance
depends on the frequency. In the previous example we see that the