Hertzian Dipole Calculator

A Hertz antenna calculator is a useful tool for analyzing an electrically short Hertzian dipole. It uses the operating frequency and physical antenna length to calculate the wavelength, length-to-wavelength ratio, electrical length, radiation resistance, directivity, estimated gain, and whether the antenna meets the calculator's electrically short criterion.

A Hertzian dipole is an idealized antenna model with a uniform current distribution over a very small length. It is widely used in electromagnetic theory because its fields and radiation characteristics can be analyzed mathematically.

This calculator is designed around two simple inputs:

• Frequency: entered in MHz
• Physical Length: entered in meters

From these values, the calculator determines the antenna's electrical size and uses that information to estimate radiation resistance.

The basic calculation flow is:

Frequency → Wavelength → Length/Wavelength Ratio → Electrical Length → Radiation Resistance

The calculator is especially useful for antenna theory, RF education, preliminary antenna analysis, and understanding why very short antennas can have extremely low radiation resistance.

What Is a Hertzian Dipole Antenna?

A Hertzian dipole is an idealized electrically short dipole used to study electromagnetic radiation. In the ideal model, the dipole is extremely small compared with the wavelength and carries a uniform current distribution.

The term "Hertzian dipole" is sometimes used interchangeably with "short dipole," but there is an important technical distinction.

A practical electrically short dipole normally has a current distribution that is approximately maximum at the center and decreases toward zero at the ends. A Hertzian dipole, by contrast, is modeled with a uniform current distribution. This difference affects the radiation resistance formula.

The Hertzian dipole is therefore best understood as a theoretical antenna model rather than a direct representation of every practical short antenna.

It is useful for understanding several important antenna concepts:

• Radiation resistance
• Radiation pattern
• Electrical length
• Radiated power
• Directivity
• Antenna gain
• Relationship between antenna length and wavelength

The far-field radiation from a Hertzian dipole has the familiar dipole angular dependence, with maximum radiation broadside to the dipole and nulls along its axis.

Hertzian Dipole vs Practical Short Dipole

These terms should not automatically be treated as identical.

The distinction is important because the calculator specifically uses the Hertzian dipole radiation-resistance model.

What Is a Hertz Antenna Calculator?

A Hertz antenna calculator is a calculation tool that estimates key parameters of an electrically short Hertzian dipole from its frequency and physical length.

The calculator accepts:

1. Frequency in MHz
2. Physical length in meters

It then calculates:

• Wavelength
• Physical length
• Length-to-wavelength ratio
• Electrical length
• Radiation resistance
• Directivity
• Estimated gain
• Electrically short status

The wavelength is calculated first because antenna dimensions are normally evaluated relative to wavelength.

For a frequency entered in MHz, the calculator uses:

where:

• λ is wavelength in meters
• f is frequency in MHz

The calculator then compares the physical antenna length with the calculated wavelength.

This gives:

where l is the physical antenna length.

That ratio is one of the most important values in the calculator because it tells you how electrically large or small the antenna is.

Hertz Antenna Calculator Inputs

Frequency

The first input is Frequency, measured in MHz.

Frequency determines wavelength. As frequency increases, wavelength decreases. As frequency decreases, wavelength increases.

The calculator uses:

For example, at 100 MHz:

So a 100 MHz signal has an approximate free-space wavelength of 3 meters using the calculator's 300/f approximation.

A higher frequency produces a shorter wavelength.

For example:

This relationship is important when determining whether a fixed physical antenna is electrically short.

Physical Length

The second input is Physical Length, measured in meters.

For example:

• 0.01 m = 1 cm
• 0.03 m = 3 cm
• 0.10 m = 10 cm
• 1 m = 100 cm

The calculator compares this physical length with the wavelength.

Suppose the wavelength is 3 meters and the antenna is 0.03 meters long:

The antenna is therefore:

or 1% of a wavelength.

This ratio is more useful for antenna analysis than physical length alone because the same antenna can have different electrical sizes at different frequencies.

Hertz Antenna Calculator Formulas

The calculator uses several formulas to transform the two input values into useful antenna parameters.

Wavelength Formula

The calculator uses:

where:

• λ = wavelength in meters
• f = frequency in MHz

This is the first calculation performed.

For example, for 150 MHz:

Therefore, the wavelength is approximately 2 meters.

The exact speed of electromagnetic propagation depends on the medium, but this simplified relationship is commonly used for free-space calculations.

Length-to-Wavelength Ratio

The next calculation is:

This compares the physical antenna length with one complete wavelength.

For example:

• Physical length = 0.05 m
• Wavelength = 5 m

Then:

The antenna length is therefore 0.01 wavelength.

A smaller value means the antenna is electrically smaller relative to the signal wavelength.

Electrical Length

The calculator expresses the length-to-wavelength ratio in degrees:

For example, if:

then:

So the calculated electrical length is 3.6 degrees.

Electrical length is useful because antenna behavior is closely related to wavelength and phase rather than physical dimensions alone.

A full wavelength corresponds to 360 degrees. Therefore, an antenna that is 0.01 wavelength long has an electrical length of 3.6 degrees.

Radiation Resistance

The calculator uses the Hertzian dipole radiation-resistance equation:

where:

• R_r = radiation resistance in ohms
• l = physical length
• λ = wavelength

This formula is specifically associated with the ideal Hertzian dipole model. Engineering references distinguish it from the approximately 20π²(L/λ)² expression commonly used for a practical electrically short dipole with a triangular current distribution.

This distinction is important when interpreting the calculator's output.

The formula shows that radiation resistance depends on the square of the length-to-wavelength ratio:

Therefore, making an antenna electrically shorter can cause radiation resistance to decrease very quickly.

Directivity

The calculator returns:

Directivity describes how concentrated an antenna's radiation is in a particular direction compared with an isotropic radiator.

For the electrically short dipole pattern, maximum directivity is 1.5, which corresponds to approximately 1.76 dB relative to isotropic radiation.

The calculator displays 1.50 as the directivity value.

Estimated Gain

The calculator displays:

This corresponds to the 1.5 directivity value expressed logarithmically:

However, directivity and gain are not technically identical. Gain accounts for antenna losses, with gain related to directivity through radiation efficiency.

Therefore, the 1.76 dBi value in this calculator should be treated as an idealized estimate based on the calculator's model, not as a measured gain value for a real antenna.

What Does "Electrically Short" Mean?

An antenna can be physically small but its electrical size depends on its wavelength.

For example, a 10 cm antenna might be:

• Very small at a low frequency
• Moderately sized at a higher frequency
• A significant fraction of a wavelength at an even higher frequency

The Hertz antenna calculator classifies the antenna as electrically short when:

That means the physical length is no more than 2% of one wavelength according to this calculator's implementation.

For example:

is classified as:

Electrically Short: Yes

But:

is classified as:

Electrically Short: No

The 0.02 threshold should be understood as the classification rule used by this calculator. Different antenna texts can use different definitions or approximations for "electrically short," so the threshold should not be interpreted as a universal boundary for every antenna analysis.

How to Use the Hertz Antenna Calculator

Using the calculator is straightforward.

Step 1: Enter the frequency

Enter the operating frequency in MHz.

For example:

100 MHz

Step 2: Enter the physical length

Enter the antenna length in meters.

For example:

0.03 m

Step 3: Calculate the wavelength

The calculator determines:

Step 4: Calculate the length-to-wavelength ratio

The result is:

0.010000 λ

Step 5: Calculate electrical length

Step 6: Calculate radiation resistance

The calculator uses:

which gives approximately:

Step 7: Review the remaining results

The calculator also provides:

• Directivity: 1.50
• Estimated gain: 1.76 dBi
• Electrically short: Yes

This gives a quick theoretical picture of the antenna's electrical size and radiation resistance.

Real-Life Example: 100 MHz Hertzian Dipole

Consider an RF engineering student studying an extremely short antenna for a 100 MHz signal.

The student has an antenna with a physical length of:

0.03 meters

The operating frequency is:

100 MHz

The goal is to determine whether the antenna is electrically short and understand its approximate radiation resistance.

Step 1: Find the wavelength

The wavelength is 3 meters.

Step 2: Find the length-to-wavelength ratio

Therefore:

The antenna is only 1% of a wavelength long.

Step 3: Find the electrical length

The electrical length is therefore:

3.6°

Step 4: Find radiation resistance

Using the calculator's Hertzian dipole equation:

The result is approximately:

This is a very small radiation resistance.

Step 5: Determine whether it is electrically short

The calculator uses:

Since:

the calculator returns:

Electrically Short: Yes

What does this example tell us?

The 3 cm antenna is physically small compared with the 3 m wavelength at 100 MHz.

Its electrical size is only 0.01 wavelength, or 3.6 degrees. The Hertzian dipole model therefore predicts very low radiation resistance.

In a practical antenna system, such low radiation resistance can make efficient power transfer more challenging because losses in conductors, connections, matching networks, and other components can become significant compared with the radiation resistance.

This is one reason why very short antennas often require careful impedance matching and efficiency analysis.

Hertz Antenna Calculator Use Cases

RF and Antenna Education

The calculator can help students understand the relationship between:

• Frequency
• Wavelength
• Antenna length
• Electrical length
• Radiation resistance

Instead of working through every equation manually, students can change the inputs and immediately see how the results respond.

Short Antenna Research

Researchers can use the calculator for quick theoretical estimates when investigating electrically small antenna dimensions.

It is useful for initial calculations before moving to more advanced electromagnetic analysis.

Preliminary Antenna Design

An engineer can use the tool to determine whether a proposed antenna length is extremely small compared with the operating wavelength.

It can also provide a first estimate of radiation resistance using the Hertzian dipole model.

Comparing Antenna Sizes

Suppose the operating frequency remains fixed at 100 MHz.

The wavelength remains:

Now compare different physical lengths:

This makes it easy to see how physical length changes electrical size.

Laboratory and Classroom Work

The calculator can also be used in:

• Electromagnetics courses
• RF engineering classes
• Antenna laboratories
• Engineering assignments
• Antenna theory demonstrations

It can help students connect mathematical formulas with physical antenna dimensions.

How Frequency Affects a Hertzian Dipole

Frequency has a direct effect on wavelength.

The relationship is:

As frequency increases, wavelength decreases.

As frequency decreases, wavelength increases.

Suppose an antenna has a fixed physical length of 0.03 meters.

At 10 MHz:

Therefore:

At 100 MHz:

Therefore:

At 300 MHz:

Therefore:

The same 3 cm physical antenna therefore has three different electrical sizes at these frequencies.

This is a key antenna-design concept: physical size alone does not tell you whether an antenna is electrically small.

The operating frequency must also be considered.

How Physical Length Affects Radiation Resistance

The calculator uses:

The square is important.

If the length-to-wavelength ratio doubles, the calculated Hertzian radiation resistance becomes four times larger, assuming all other factors remain unchanged.

For example, if the ratio changes from:

to:

the ratio has doubled.

Because radiation resistance is proportional to the square:

So the calculated radiation resistance becomes four times larger.

This explains why extremely short antennas can have very low radiation resistance.

A practical antenna can also have additional losses. The radiation resistance represents the equivalent resistance associated with radiated power; it does not represent every resistance or loss present in a real antenna system.

Hertzian Dipole vs Half-Wave Dipole

A Hertzian dipole and a half-wave dipole represent very different antenna models.

A Hertzian dipole is an idealized, electrically very small antenna. Its current distribution is treated as uniform, making it particularly useful for mathematical analysis.

A half-wave dipole is much larger electrically and has a different current distribution and radiation behavior.

The distinction can be summarized as follows:

You should therefore not use the Hertz antenna calculator as a substitute for a half-wave dipole calculator.

Understanding the Calculator Results

The calculator produces several results.

Wavelength

Wavelength is the distance represented by one complete cycle of the electromagnetic wave.

The calculator reports it in meters.

Physical Length

This is the antenna length entered by the user.

Length / Wavelength Ratio

This shows the antenna length as a fraction of one wavelength.

For example:

means the antenna is 1% of one wavelength long.

Electrical Length

This converts the length-to-wavelength ratio into degrees.

For example:

Radiation Resistance

This estimates the radiation resistance using the Hertzian dipole equation:

Directivity

The calculator returns:

1.50

For an ideal short-dipole radiation pattern, maximum directivity is 1.5.

Estimated Gain

The calculator returns:

1.76 dBi

This represents the idealized 1.5 directivity value expressed relative to an isotropic radiator.

For an actual antenna, gain can be lower because gain incorporates losses through radiation efficiency.

Electrically Short

This returns either:

Yes

or:

No

The calculator uses:

as its classification rule.

Example: Changing Antenna Length at 100 MHz

At 100 MHz:

Now consider four antenna lengths.

0.01 m

This is below 0.02, so the calculator classifies it as electrically short.

0.03 m

This is also electrically short.

0.06 m

This is exactly at the calculator's threshold and is therefore classified as electrically short.

0.15 m

This exceeds the 0.02 threshold, so the calculator returns:

Electrically Short: No

This example shows why the length-to-wavelength ratio is more useful than looking at antenna length alone.

Limitations of the Hertz Antenna Calculator

The Hertz antenna calculator is a simplified analytical tool. It should not be treated as a complete electromagnetic simulation.

The calculator does not model every factor that can affect a practical antenna.

For example, it does not account for:

• Conductor losses
• Ground effects
• Ground conductivity
• Nearby buildings
• Human-body interaction
• Feed-line losses
• Matching-network losses
• Mutual coupling
• Detailed installation environment
• Antenna loading
• Manufacturing variations
• Full electromagnetic field simulation

The radiation-resistance calculation is also based on the ideal Hertzian dipole model. A practical electrically short dipole can have a different radiation resistance because its current distribution is not uniform. Engineering references specifically note the difference between the 80π² Hertzian-dipole expression and the approximately 20π² practical electrically-short-dipole expression.

The calculator's directivity and gain outputs are also simplified. It returns 1.50 for directivity and 1.76 dBi for estimated gain based on the ideal dipole model.

For a real antenna project, these values should not replace electromagnetic simulation or measurement.

A practical design may require tools such as an electromagnetic solver, impedance measurement, network analysis, or antenna-range testing.

Frequently Asked Questions

What is a Hertz antenna calculator?

A Hertz antenna calculator is a tool for estimating important parameters of a Hertzian dipole. It uses frequency and physical antenna length to calculate wavelength, electrical length, length-to-wavelength ratio, radiation resistance, directivity, estimated gain, and electrically short status.

What is a Hertzian dipole?

A Hertzian dipole is an idealized electrically small dipole antenna model with a uniform current distribution. It is primarily used for electromagnetic and antenna theory.

How do you calculate antenna wavelength?

For frequency expressed in MHz, the calculator uses:

where wavelength is in meters.

For example, at 100 MHz:

What is the Hertzian dipole radiation resistance formula?

The Hertzian dipole radiation-resistance formula used by this calculator is:

This expression applies to the ideal Hertzian dipole model.

What does 0.02 lambda mean?

0.02 λ means that the antenna's physical length is 2% of one wavelength.

For example, if the wavelength is 3 meters:

So 0.06 meters is 0.02 wavelength at 100 MHz.

When is an antenna electrically short?

For this calculator, an antenna is classified as electrically short when:

This is the calculator's defined threshold and should not be treated as a universal definition for every antenna analysis.

Why does a short dipole have low radiation resistance?

For the Hertzian dipole model, radiation resistance is proportional to the square of the length-to-wavelength ratio:

Therefore, as the antenna becomes much shorter than the wavelength, its radiation resistance becomes very small.

Can this calculator design a practical antenna?

It can provide useful theoretical estimates, but it does not perform complete practical antenna design. Real antennas are affected by conductor losses, feed systems, ground conditions, nearby objects, matching networks, and installation environments.

Is a Hertzian dipole the same as a half-wave dipole?

No. A Hertzian dipole is an idealized electrically very short dipole with a uniform current model. A half-wave dipole is much larger electrically and has a different current distribution and radiation behavior.

What is the difference between directivity and gain?

Directivity describes directional concentration of radiation without accounting for antenna losses. Gain incorporates antenna efficiency. In general:

where e_rad is radiation efficiency.

Hertz Antenna Calculator vs Other Antenna Calculators

Different antenna calculators solve different problems.

A Hertz antenna calculator is appropriate when you want to analyze an idealized electrically short Hertzian dipole.

Other tools may be more appropriate for different antenna types:

• Dipole antenna calculator: General dipole dimensions and characteristics
• Half-wave dipole calculator: Half-wave dipole dimensions
• Quarter-wave antenna calculator: Quarter-wave monopole dimensions
• Monopole calculator: Monopole antenna analysis
• Wavelength calculator: Frequency-to-wavelength conversion
• Radiation resistance calculator: Radiation-resistance calculations
• Antenna gain calculator: Gain and directivity relationships
• Electrical length calculator: Electrical dimensions relative to wavelength

For an antenna-design website, these tools work well as part of a broader antenna-calculator cluster.

Practical Design Tips

When using the Hertz antenna calculator, keep the following points in mind.

Use the correct units

Enter frequency in MHz and physical length in meters.

Entering centimeters when the calculator expects meters can produce a result that is 100 times different from the intended physical length.

Check the wavelength first

The wavelength provides context for every other result.

Once you know wavelength, the physical antenna length becomes much easier to interpret.

Look at the length-to-wavelength ratio

Do not evaluate antenna size using physical length alone.

A 10 cm antenna has very different electrical behavior at different frequencies because its wavelength changes with frequency.

Pay attention to radiation resistance

Very small radiation resistance can be important in practical antenna design because losses may become significant compared with the radiation resistance.

Do not treat estimated gain as measured gain

The calculator's 1.76 dBi output is an idealized model-based value. A real antenna can have lower gain because of losses and other practical factors.

Use simulation for advanced designs

If you are developing a real antenna, the calculator can be a useful first step, but detailed electromagnetic simulation and measurement may be necessary for final validation.

Key Takeaways

The Hertz antenna calculator provides a quick way to analyze an idealized Hertzian dipole using frequency and physical antenna length.

The main calculations are:

for wavelength,

for electrical size, and

for Hertzian dipole radiation resistance.

The calculator also converts the length-to-wavelength ratio into electrical degrees and determines whether the antenna meets its defined electrically short criterion of l/λ ≤ 0.02.

For the ideal short-dipole model, a directivity of 1.5 corresponds to approximately 1.76 dBi.

The most important point is that the calculator is an analytical estimation tool. It is excellent for understanding the relationship between frequency, wavelength, antenna size, and radiation resistance, but practical antenna performance depends on many additional factors.