Kirchhoff’s Voltage Law (KVL)
• The algebraic sum of the voltages around
a closed loop (CW/CCW) is zero
( voltage rises =  voltage drops)
– Applied to both instantaneous
voltages or voltage phasors
• Loop 24312 (BCDA, CW):
E24+E43+E31+E12=0 or
E31
3
e24+e43+e31+e12=0
• Loop 2342 (ECF, CCW):
E34
E23+E34+E42=0 or
e23+e34+e42=0
E43
4
1
E12
E23
E42
E24
2
12
Kirchhoff’s Current Law (KCL)
• The algebraic sum of the currents arriving at a node is equal to 0.
( currents in =  currents out)
• Node A:
I1+I3=I2+I4+I5
or i1+i3=i2+i4+i5
I1+I3+(-I2)+(-I4)+(-I5)=0
or i1+i3+(-i2)+(-i4)+(-i5)=0
A
13
+
Kirchhoff’s Laws and AC Circuits
• KVL
– Loop 24312, CW:
E24 +E43 +E31 +E12 =0
I2Z2 -I3Z3 +Eb -I1Z1 =0
– Loop 2342, CCW:
E23 +E34 +E42 =0
I4Z4 +I3Z3 -I2Z2 =0
– Loop 242, CW:
E24 + E42 =0
Ea - I2Z2 =0
Ea
Eb
+
• KCL
– Node 2:
I5 -I2 -I4 -I1=0
– Node 3:
I4 +I1 – I3=0
14
Examples 2‐14 ~ 2‐16
15
Instantaneous Power
i ( t )  I m cos( t   i )
+
Load
e ( t )  E m co s( t   e )
  t
p(t )  e(t )i(t )  Em I m cos(  e ) cos(  i )
Using trigonometric identity
cos Acos B 
Z
1
2
cos( A - B)  cos( A  B)
Impedance angle:   e  i
>0 for inductive load and
<0 for capacitive load
| E | Em / 2
| I | I m / 2
1
Em I m  cos(e  i )  cos(2  e  i )
2
1
 Em I m cos(e  i )  cos[2(  e )  (e  i )]
2
1
 Em I m cos   cos[2(  e )   ]
2
1
 Em I m  cos   cos  cos 2(  e )  sin  sin 2(  e )
2

p( )  E  I cos  [1  cos 2(   e )]  E  I sin  sin 2(   e )
 
pR
pX
16
Example:
100
 0
2
80
  60 
I 
2
E
e  100 cos 
i  80 cos(  60o )
Z  1.2560  0.625  j1.0825 Ω
p ( )  E  I cos  [1  cos 2(   e )]  E  I sin  sin 2(   e )
 
pR
pX
 2000(1
 cos 2 )  3464
sin 2


pR
e
pX
i
3464
2000
17
p (t )  E  I cos  [1  cos 2(   e )]  E  I sin  sin 2(   e )
 
pR ( t )
pX (t )
 20
00(1  cos 2)  3
464si
n 2


pR ( t )
pX (t )
Observations:
• The frequency of p, pR and pX is twice of e and i, which is 602=120Hz
• pR(t)
– changes between 0 and 4000 (=2|E||I|cos )
– always 0, and has an average value of 2000 (=|E||I|cos )
– is the power consumed by the load
• pX(t)
– changes between 3464 (=  |E||I|sin)
– has an average value of 0
– is the power borrowed & returned by the load.
18
Apparent, Active (Real) and Reactive Powers
p  E  I cos  [1  cos 2(   e )]  E  I sin  sin 2(   e )
 
pR ( t )
 P[1  cos 2(   e )]
pX (t )
 Q sin 2(   e )
def
| S |  | E || I |
• Apparent power
• Unit ~ volt ampere or VA (kVA or MVA)
Q
def
P  | E |  | I | cos
• Real or active power (average power)
• Unit ~ watt or W
• Power factor (PF): cos= cos(e-i)
– The PF is said to be lagging if  =e-i >0 or leading if  <0
P
def
Q  | E |  | I | sin 
• Reactive power
• Unit ~ var (volt-ampere reactive). Some people use Var, VAR or VAr
• Q>0 if =e-i >0 (inductive) or Q<0 if <0 (capacitive)
19