SME
Engineering Electromagnetics · Problem type
Line constants
Use this for “A cable has 250 nH/m and 100 pF/m. Find Z₀ and the speed of a signal on it.”
Diagram
Z₀
=
√(
L′
C′
)
=
√(
2.5 ×
10
-7
1
×
10
-10
)
Z₀ = 50 Ω
v
=
1
÷
√
L′C′
=
2 ×
10
8
m/s;
VF
=
v
c
=
2 ×
10
8
3
×
10
8
=
0.6667
Each short length stores energy in its L′ and its C′; their ratio sets Z₀ and their product the speed.
This is an example. Type your own values to start a new problem.
Z₀ (Ω)
=
L′ (nH/m)
C′ (pF/m)
v (m/s)
=
1
L′ (nH/m)
×
C′ (pF/m)
VF
Velocity factor
calculated
Formulas
Z₀
=
√(
L′
C′
)
50
=
√(
2.5 ×
10
-7
1 ×
10
-10
)
v
=
1
÷
√
L′ × C′
2 ×
10
8
=
1
÷
√(2.5
×
10
-7
×
(1
×
10
-10
))
L′
=
Z₀
v
2.5 ×
10
-7
=
50
2 ×
10
8
VF
=
v
3.00 ×
10
8
0.6667
=
2 ×
10
8
3.00 ×
10
8
Formulas are worked in H/m, F/m. The step-by-step shows the conversions.
Assumptions
•
A lossless line: its wires have no resistance and its insulation no leakage.
•
L′ and C′ are per meter of line, from the maker’s data or the cable’s shape.
•
The velocity factor compares v with c = 3.00 × 10⁸ m/s; it is 1 ÷ √εᵣ for a line filled with one insulator.
Step-by-step
WE KNOW
L′ = 250 nH/m, C′ = 100 pF/m
FIND
characteristic impedance (Z₀), signal speed (v), velocity factor (VF)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to H/m, F/m).
L′ = 250 nH/m = 2.5 × 10⁻⁷ H/m (1 H/m = 10⁹ nH/m)
C′ = 100 pF/m = 1 × 10⁻¹⁰ F/m (1 F/m = 10¹² pF/m)
Step 2 · Find characteristic impedance (Z₀)
Use Z₀ = √(L′ ÷ C′)
The ratio of voltage to current on the wave is set by L′ against C′.
Z₀
=
√(
2.5 ×
10
-7
10
-10
)
Z₀
=
√
2
,500
Z₀
=
50
Ω
Step 3 · Find signal speed (v)
Use v = 1 ÷ √(L′ × C′)
Each meter of line stores charge in C′ and flux in L′; together they set the speed.
v
=
1
÷
√
2.5 ×
10
-7
×
10
-10
v
=
1
÷
√
2.5 ×
10
-17
v
=
1
5 ×
10
-9
v
=
2 ×
10
8
m/s
Step 4 · Find velocity factor (VF)
Use VF = v ÷ (3.00 × 10⁸)
The speed as a share of the speed of light.
VF
=
2 ×
10
8
3.00 ×
10
8
VF = 0.6667
Check
Put the numbers back in. Both sides should match.
50
=
√(
2.5 ×
10
-7
10
-10
)
✓
2 ×
10
8
=
1
÷
√
2.5 ×
10
-7
×
10
-10
✓
2.5 ×
10
-7
=
50
2 ×
10
8
✓
0.6667
=
2 ×
10
8
3.00 ×
10
8
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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