Complex Impedance
Impedance, Z  R  jX
| Z |  R2  X 2
 R
θ  ArcTan X
where X  X L  X C
1
XC 
  90
2πfC
X L  2fL 90
Tips
Plotting Impedance
Admittance , Y = 1/Z •
Y=G+jB
Conductance, G = 1/R
•
Susceptance, B = 1/X
|Y| =  G2 + B2
•
/ Y = ArcTan(B/G)
= ArcTan (-R/X)
Z = 1/Y
•
Resistance (R) along
positive x-axis (right).
Inductive reactance
(XL) along positive yaxis (up).
Capacitive reactance
(XC) along negative yaxis (down).
Negative x-axis (left)
not used.
1. ELI the ICE man lives in SARL PARC
a) Inductive Circuits: Voltage leads current (phase angle is positive)
b) Capacitive Circuits: Voltage lags current (phase angle is negative)
c) SARL: Series Above Resonance appears Inductive
d) PARC: Parallel Above Resonance appears Capacitive
2. If reactance & resistance have same magnitude, phase angle = ±45°
a) If reactance is less than resistance, phase angle is less than ±45°
b) If reactance is greater than resistance, phase angle is greater than ±45°
3. If the reactance = 0, phase angle = 0. Circuit is purely resistive., voltage & current in phase.
4. For series circuits, add impedances
5. For parallel circuits, add admittances
Time Constants
6. When taking the reciprocal of an angle, flip the sign
For RC Ckt, τ = R x C
7. In parallel circuits, the resultant impedance is smaller than the resistance
or reactance.
For RL Ckt, τ = L/R
Power Equations
• Apparent Power (v-a) P = I x E
• Real Power (watts)  P = I x E x cos(θ)
• Power Factor (PF) = Real Pwr/Apparent Pwr = cos(θ)
Resonant Freq , Quality Factor, Band Width
• Resonant Frequency (fR) = 1 / (2π √LC )
• Quality factor (Q) = PS / PD = X / R
• Half-power bandwidth (BW) = f2-f1 = Δf = f0 / Q = fR / Q
Powdered Iron Cores
• N = 100 x √L / AL
• L = Inductance in μH
• AL = Inductance Index in μH/(100 turns).
• N = Number of Turns
Ferrite Cores
• N = 1000 x √L / AL
• L = Inductance in mH
• AL = Inductance Index in mH/(1000 turns)
• N = Number of Turns
RC and RL Time Constants
Time
Constants
1
2
3
4
5
Charging
Discharging
% of Applied
Voltage
63.20%
86.50%
95.00%
98.20%
99.30%
% of Starting
Voltage
36.80%
13.50%
5.00%
1.80%
0.70%
Units Conversion
giga
G
mega
M
kilo
K
basic unit
milli
m
micro
μ
pico
p
109
106
103
100
10-3
10-6
10-12
1,000,000,000
1,000,000
1,000
1
0.001
0.0000001
0.000000000001