The Hong Kong Polytechnic University
Per Unit
Why per unit?
•
•
•
•
•
•
•
Multiple voltage levels: 400kV, 275kV, 132kV, 11kV, 400V
Makes circuit analysis rather confusing
Transformer impedance (Ω) depends on prim/sec referral
Normalize all quantities to help understanding
Avoid confusion due to transformers
Ideal transformer winding can be eliminated (assumes proper specification of base values)
Voltages, currents and impedances expressed in per unit do not change when referred
from primary to secondary
• Per unit impedances of equipment of similar type are usually similar if equipment ratings
are used as base values
• Manufacturer usually specify impedance of item of plant in per unit (or percent) using
nameplate rating as base
• Per unit values result in more meaningful data
1
The Hong Kong Polytechnic University
Per Unit
Basic Units
• The 4 basic electrical quantities are:
‒ Voltage V
(volt)
‒ Current I
(amp)
‒ Impedance Z
(ohm)
‒ Power S
(VA)
• For single‐phase circuits,
‒ V(volt) = Z(ohm) × I(amp);
‒ S (VA) = V(volt) × I(amp)*
2
The Hong Kong Polytechnic University
Per Unit
Per unit notation
• In per unit notation, the physical quantity is expressed as a fraction of
the reference value, i.e.
per unit value = actual value/base value in the same unit.
e.g. V(in per unit) = V(in kV)/V base (in kV)
where the base value is a reference value for magnitude.
• In per unit notation we would like to keep the basic relations:
Spu = Vpu Ipu*
Vpu = Zpu Ipu;
• Hence the base quantities should be chosen such that:
Base voltage (VB) = base impedance (ZB) × base current (IB)
Base power (SB) = base voltage (VB) × base current(IB)
3
The Hong Kong Polytechnic University
Per Unit
Base Values
• Thus only two of the base quantities can be arbitrarily chosen, the
other two will follow directly.
• It is common practice to specify base power (SB) and base voltage (VB).
• Then it follows
base current
base impedance
IB = SB/VB
ZB = VB/IB =VB2/SB
4
The Hong Kong Polytechnic University
Per Unit
Base values for 3‐phase systems
• For 3‐phase systems it is common practice to describe system
operation with:
total 3‐phase power S = S3‐φ
line voltage V = Vline
line current I = Iline
equivalent impedance/phase Z = Zph
with (in magnitude)
Vline = √3ZphIline
S3‐φ = √3VlineIline
5
The Hong Kong Polytechnic University
Per Unit
• Hence if base values are chosen for:
total 3‐phase power SB
line voltage VB
Then
base line current:
IB = SB/ √3VB
base impedance:
ZB = VB/ √3IB = VB2/SB
6
The Hong Kong Polytechnic University
Per Unit
• Supply: 400 V, 50 Hz, 3‐phase
• Load: 3 identical coils with Z = 20+j15 Ω in star connection
• Find: line current, power supplied, power factor
Take (for example)
Base power (total 3‐phase) SB = 10 kVA
Base voltage (line‐to‐line) VB = 400 V
Then
Base current IB = SB/√3VB = 14.43 A
Base impedance ZB = VB2/SB = 16 Ω
Given
V = 400 V = 1.0 p.u
Z = 25∠36.9o Ω = 1.5625∠36.9o p.u.
7
The Hong Kong Polytechnic University
Per Unit
• Current
I = V/Z = 1.0 /1.5625 p.u. = 0.64 p.u.
= 0.64×14.43 A = 9.235 A
• Apparent power
S = VI = 1.0×0.64 = 0.64 p.u. = 6.4 kVA
• Power factor
p.f. = cos 36.9o = 0.8
• Active power
P = VI × pf = 0.64 × 0.8 = 0.512 p.u.
= 0.512 × 10 = 5.12 kW.
8
The Hong Kong Polytechnic University
Per Unit
Base values for a transformer
• In a transformer, two circuits are not directly connected but magnetically
coupled. The voltages of the windings are in the ratio of turns and
currents in inverse ratio.
• For the coupled circuit, we should then choose
‒ The same base power
‒ Base voltages in the ratio of turns
• This will ensure Spu, Vpu, Ipu, to remain unchanged when passing through
an ideal transformer.
9
The Hong Kong Polytechnic University
Per Unit
Let n1,n2 be the number of turns in primary and secondary winding.
Z1, Z2 be the primary and secondary winding impedance.
Total impedance referred to primary:
ZT1 = Z1 + (n1/n2) 2 Z2
Total impedance referred to secondary:
ZT2 = Z2 + (n2/n1)2 Z1 = (n2/n1)2 ZT1
If base values were chosen for the transformer:
VB1 = (n1/n2)VB2.
SB1 = SB2;
Then
IB1 = (n2/n1)IB2 ;
ZB1 = (n1/n2)2 ZB2
Thus per unit impedance of transformer
Zpu = ZT1/ZB1 = ZT2/ZB2
is the same whether we use the total impedance referred to primary or secondary
10
The Hong Kong Polytechnic University
Per Unit
Equivalent circuit for transformer
• In the per unit representation, the equivalent circuit of a transformer is a
simple winding impedance Zpu (with excitation branch ignored).
(Ipu)1
(Vpu)1
(Zpu)
(Ipu)2
(Vpu)2
11
The Hong Kong Polytechnic University
Per Unit
Base Conversion
• If the per unit values are given based on SB1 and VB1 which are different
from the chosen base SB2 and VB2 for analysis, the given per unit values
must be modified before they can be used. Thus
(Vpu)2 = V/VB2 = (Vpu)2 × VB1/VB2
(Spu)2 = S/SB2 = (Spu)1 × SB1/SB2
(Ipu)2 = I/IB2 = (Ipu)1 × IB1/IB2
= (Ipu)1 × VB2/VB1 x SB1/SB2
(Zpu)2 = Z/ZB2 = (Zpu)1×ZB1/ZB2
= (Zpu)1× (VB1/VB2)2 x SB2/SB1
12
13
14
The Hong Kong Polytechnic University
Per Unit
Advantages
• Normally we are dealing with numbers near unity rather than over a wide
range.
• Provides a more meaningful comparison of parameters of machines with
different ratings.
• As the per unit values of parameters of a machine of a given design normally
falls within a certain range, a typical value can be used if such parameters are
not provided.
15