Phasor Circuit Analysis
Phasor Diagrams, Voltage and
Current Division
Kevin D. Donohue, University of Kentucky
1
Phasor Diagram
Phasors are denoted by vectors in 2-D space. Phasor diagrams
graphically illustrate the summation of complex values
Consider the following summation of complex numbers
resulting from setting up a KVL or KCL equation:
7∠0 = 9∠ − 45° + 6.4∠84.29°
7 = 9 cos( −π 45 / 180) + j 9 sin( −π 45 / 180) + 6.4 cos(π 84.29 / 180) + j 6.4 sin(π 84.29 / 180)
7 = 6.36 − j 6.36 + 0.64 + j 6.36
7∠0
RE
6.4∠84.29°
IM
9∠ − 45°
Kevin D. Donohue, University of Kentucky
2
Phasor Diagram Example
Draw Currents is, ir, il, ic in a phasor diagram to show is = ir + il + ic
where is(t) = sin(1000t) A
is
ir
il
ic
1Ω
1mF
0.5mH
Kevin D. Donohue, University of Kentucky
3
Current Division
By substituting impedance in for resistance (or admittance for
conductance), the current division formula can be generalize to:
…
…
Iˆk
Iˆs
…
Ẑ 2
Ẑ1
…
Ẑ k
1
Zˆ k
Iˆk = Iˆs
1
1
+
+L+
Zˆ1 Zˆ 2
1
+L+
Zˆ k
1
Zˆ N
= Iˆs
…
Ẑ N
…
Yˆk
Yˆ1 + Yˆ2 + L + Yˆk + L + YˆN
Kevin D. Donohue, University of Kentucky
4
Voltage Division
By substituting impedance in for
resistance, the voltage division
formula can be generalize to:
Ẑ1
Ẑ 2
Vˆs
.
.
.
+
Ẑ k Vˆk
.
.
.
Zˆ k
Vˆk = Vˆs
Zˆ1 + Zˆ 2 + L + Zˆ k + L + Zˆ N
−
Ẑ N
Kevin D. Donohue, University of Kentucky
5
In-Phase and Out-of-Phase
Voltages and currents are considered in phase if the phase angle
between their phasor quantities is zero, otherwise they are out of phase
by and amount equal to their phase difference.
Which quantities are in phase?
Vˆ1∠45°
Vˆ2 ∠0°
Vˆ2 ∠ − 90°
Iˆ1∠270°
Iˆ2 ∠ − 45°
Kevin D. Donohue, University of Kentucky
Iˆ2 ∠45°
6
Phase Lead and Phase Lab
Out of phase voltages and currents are considered to either lead or lag
each other. If the phase of quantity 1 is subtracted from the phase of
quantity 2 and the result is positive, then quantity 1 leads quantity 2 or
equivalently quantity 2 lags quantity 1.
Given
Vˆ1∠45°
Vˆ2 ∠0°
Vˆ3∠ − 90°
Iˆ1∠270°
Iˆ2 ∠ − 45°
Iˆ3∠45°
Describe the phase difference between
Î
Î
Î
v1 and i1
v2 and i2
v3 and i3
Kevin D. Donohue, University of Kentucky
7
Eample
Find R such that the v1 leads i1 by 15
degrees, if v1 = cos(400t) V
20 Ω
i1
312.5µ F
R
v1
Show R = 15 Ω
Kevin D. Donohue, University of Kentucky
8