5.3.3  The resistance/impedance relationship
Resistance, measured in ohms, is the property
of a conductor to limit the flow of current through it when
a voltage is applied.

I
= 
U 


R 
where 
I
= 
current, A 

U
= 
applied voltage. V 

R
= 
circuit resistance, Ohms 
Thus, a voltage of one volt
applied to a one ohm resistance results in a current of
one ampere.
When the supply voltage is alternating,
a second effect, known as reactance (symbol X) is to be
considered. It applies only when the circuit includes inductance
and/or capacitance, and its value, measured in ohms, depends
on the frequency of the supply as well as on the values
of the inductance and/or the capacitance concerned. For
almost all installation work the frequency is constant at
50 Hz. Thus, inductive reactance is directly proportional
to inductance and capacitive reactance is inversely proportional
to capacitance.


Xl = 2(pi)fL and Xc
=

1 



2(pi)fC 
where

Xl =

inductive reactance (Ohms) 


Xc =

capacitive reactance
(Ohms) 


(pi) =

the mathematical constant
(3.142) 


f =

the supply frequency
(Hz) 


L =

circuit inductance (H) 


C =

circuit capacitance (F) 

Resistance (R) and reactance (Xl or Xc)
in series add together to produce the circuit impedance
(symbol z), but not in a simple arithmetic manner. Impedance
is the effect which limits alternating current in a circuit
containing reactance as well as resistance.

Z =

U 


I 
where

Z
=

impedance (Ohms) 

U =

applied voltage (V) 

I
=

current (A) 
It follows that a one volt supply connected
across a one ohm impedance results in a current of one ampere.
When resistance and reactance are added
this is done as if they were at right angles, because the
current in a purely reactive circuit is 90° out of phase
with that in a purely resistive circuit. The relationships
between resistance, reactance and impedance are:
a) resistive and capacitive
circuit  b) resistive and inductive circuit
Fig 5.8 Impedance diagrams
These relationships can be shown in the
form of a diagram applying Pythagoras' theorem as shown
in {Fig 5.8}. The two diagrams are needed because current
lags voltage in the inductive circuit, but leads it in the
capacitive. The angle between the resistance R and the impedance
Z is called the circuit phase angle, given the symbol a
(Greek 'phi'). If voltage and current are both sinusoidal,
the cosine of this angle, cos a, is the circuit power factor,
which is said to be lagging for the inductive circuit, and
leading for the capacitive.
In practice, all circuits have some inductance
and some capacitance associated with them. However, the
inductance of cables only becomes significant when they
have a crosssectional area of 25 mm² and greater. Remember
that the higher the earth fault loop impedance the smaller
the fault current will be. Thus, if simple arithmetic is
used to add resistance and reactance, and the resulting
impedance is low enough to open the protective device quickly
enough, the circuit will be safe. This is because the Pythagorean
addition will always give lower values of impedance than
simple addition.
For example, if resistance is 2 Ohms and
reactance 1 Ohm, simple arithmetic addition gives
Z

= R + X – 2 + 1 = 3 Ohms 
and correct addition gives


Z

= Ö(R²
+ X²) 

= Ö(2² + 1²) =
Ö 5 = 2.24 Ohms 
If 3 Ohms is acceptable, 2.24
Ohms will allow a larger fault current to flow which will
operate the protective device more quickly and is thus even
more acceptable.