# Characteristic Impedance of Transmission Line Formula

When do we need to understand the characteristic impedance of transmission line? We need to understand this topic when we are faced with a very long parallel conductor wire that carries an electric current.

In an ideal condition, an open circuit won’t have an electrical current flowing in the circuit. But here we will talk about the actual field, so anything can happen without us knowing without telling.

Contents

## Infinite Length of Conductor Wires in Parallel

Assume that we have a simple circuit consists of:

• A voltage source
• A switch
• A very long conductor wire

Well, we won’t use a load this time to implement what we need right now, an open circuit with a very long conductor wire.

Observe the illustration below. Looking from the illustration above, you would say that it is an open circuit therefore there is no electrical energy generated in that circuit.

While this may be true in an ideal condition, it will produce different answers in an actual field. This circuit definitely produces some electrical energy, may it voltage or current.

Looking closer to the circuit, it quite represents what we know as a capacitor. A capacitor is basically a pair of conductor plates separated by an insulating layer. An open circuit may not have a current flowing through it but it still has a voltage between two points.

Not only capacitance, we will also find inductance in this kind of circuit.

## Capacitance and Inductance in a Parallel Conductor Wires

As said above, a pair of conductor wires in a parallel also represents a capacitor which is a pair of conductor plates separated by an insulating layer or medium. Observe the illustration below. Since we have a pair of conductor plates and an insulating medium (dielectric layer), we can conclude that we have a lot of capacitors here. Thus, our circuit will produce a lot of what we know as capacitances.

We need to understand that a capacitance is the ability of a material or element to store energy in the form of electric charges when there is a difference in electric potential measured at two points.

Capacitance is measured by the change of charge caused by the difference of electric potentials.

We can measure the current drawn by a capacitor based on the the change of voltage rate over time mathematically by: Where dv(t)/dt is the instantaneous rate of change of voltage.

This is why when the switch is closed, the capacitors will charge up by the sudden change of voltage and draw the current from the source.

The instantaneous rate of change of voltage will cause the charging current rises infinitely.

But if there is only a pair of conductor wires, then it should be only one capacitor formed right? How do we get so many capacitors at once?

While it is true that our parallel wires produce capacitance, our “series” wires will produce its friend, inductance. We will learn this right away.

Observe the illustration below. If a pair of conductor wires in parallel represents a capacitor, then a series conductor wire represents an inductor. If you observe carefully, an inductor is only a conductor wire wound to a core such as ferrite or air.

These series wires form a series impedance until the end of the circuit.

An inductor is capable of storing energy in the form of a magnetic field. The energy is the opposition against the change of electric current flowing through it. The generated magnetic field’s strength is proportional to the magnitude of the current.

Based on Faraday’s law, the change in magnetic field induces an electromotive force (EMF) and can be seen as a voltage in the conductor. The change of current creates the induced voltage and this voltage opposes the change in current.

We can calculate the induced voltage by using a simple equation below: Where di/dt is the change of current.

So what happens if we have both capacitance and inductance in a circuit?

Of course we will have an electric field produced by capacitance and magnetic field produced by inductance as shown below. The electric charge in the wires transfer to and from each segment with the speed of light (nearly), the voltage and current charges will propagate down the length of the wires with the same velocity. This propagation results in charging the capacitance and inductance respectively to their full voltage and current, respectively.

When the switch is open, there is no electrical activity in the circuit. The moment the switch is closed, the voltage and current are charged. The wave propagation begins. It continues indefinitely. ## Characteristic Impedance of Transmission Line

The characteristic impedance of a transmission line is the same as the natural impedance of a transmission line or equivalent resistance of a transmission line if it is infinitely long.

Since the wire is very long, the capacitor won’t be charged fully equal to the voltage source, the inductor won’t be charged by unlimited current. Both capacitor and inductor will only charge as long as the switch is closed.

We can see them as a common constant load of an electric circuit. They are not considered as a mere conductor wire anymore but a circuit component with a unique characteristic. It is what we call a “transmission line” not a “conductor wire”.

Our transmission line is considered as a resistive load rather than reactive load even though we have inductance and capacitance. The transmission line only absorbs energy just as a resistor dissipates energy. Thus, we can assume they are no different.

The impedance (resistance) of our transmission line is measured in Ohm (Ω) and known as “characteristic impedance”.

Observe the illustration below. We have a pair of infinitely long conductor wires in parallel with air insulation. We can calculate its characteristic impedance using Where:

Z0 = characteristic impedance
d = distance between the center of conductors
k = relative insulation permittivity between conductors

If the transmission line is built with coaxial cables, we modify the characteristic impedance equation a bit. The equation will be Where

Z0 = characteristic impedance
d1 = inside diameter of outer conductor
d2 = outside diameter of inner conductor
k = relative insulation permittivity between conductors

The fraction terms in both equations must be expressed in the same units of measurement. Both the characteristic impedance and the propagation velocity will be impacted if the insulating substance is something other than air (or a vacuum).

The velocity factor of a transmission line is defined as the difference between the true propagation velocity of the line and the speed of light in a vacuum.

Velocity factor, which is the ratio of a material’s electric field permittivity to that of a pure vacuum, is solely dependent on the relative permittivity of the insulating material (also known as its dielectric constant).

Any cable type, coaxial or otherwise, can have its velocity factor easily computed using the following formula: Where:

v = velocity of the wave propagation
c = velocity of light in a vacuum
k = relative insulation permittivity between conductors

The characteristic impedance (Z0) of a transmission line increases as the conductor spacing increases, as can be observed in either of the first two equations. When the conductors are separated from one another, the distributed capacitance decreases (capacitor “plates” are spaced more apart), while the distributed inductance rises (less cancellation of the two opposing magnetic fields).

Less parallel capacitance and more series inductance cause the line to draw less current for a given applied voltage, resulting in a higher impedance. The parallel capacitance rises when the two conductors are separated, but the series inductance falls. Both modifications cause a greater current to be drawn for a specific applied voltage, which equates to a lower impedance.

The characteristic impedance of a transmission line is equal to the square root of the ratio of the line’s inductance per unit length divided by the line’s capacitance per unit length, provided that there are no dissipative effects such dielectric “leakage” and conductor resistance: Where:

Z0 = characteristic impedance
L = line’s inductance per unit length
C = line’s capacitance per unit length

## Summary

After reading a lot of information here, we can make some summaries to wrap everything here.

1. A transmission line is a pair of parallel wires with dispersed capacitance and inductance over its length, which gives it specific properties.
2. Both a voltage “wave” and a current “wave” travel over a transmission line at almost the speed of light when a voltage is rapidly introduced to one end of the line.
3. An infinitely long transmission line will draw current from the DC source as if it were a constant resistance if a DC voltage is placed to one end of the line.
4. The resistance that a transmission line would display if its length were infinite is its typical impedance (Z0). This is completely distinct from the metallic resistance of the wires themselves and the leakage resistance of the dielectric separating the two conductors. Even if the dielectric were flawless (infinite parallel resistance) and the wires superconducting, characteristic impedance would still exist since it is solely a function of the capacitance and inductance dispersed over the length of the line (zero series resistance).
5. The velocity factor is a fractional figure that compares the propagation speed of a transmission line to the speed of light in a vacuum. Coaxial cables and ordinary two-wire lines have values between 0.66 and 0.80. It is the reciprocal (1/x) of the square root of the relative permittivity of the cable’s insulation for any type of cable.