Tuned Circuits

Andy Collinson and Ron J

Used extensively in radio and electronics, tuned circuits are the major components in filters, oscillators and frequency selective networks.

Reactance is the opposition to change of electric voltage or current due to the inductance or capacitance in a circuit. If the circuit contains inductors, then the reactance is inductive. If the circuit contains capacitors, then the reactance is capacitive. Reactance is not constant and varies according to the frequency of the input signal.

Inductive Reactance
The reactance for an inductor is given by the following equation: XL = 2 π F L

Where XL is the inductive reactance in ohms, F is the frequency in hertz and L the inductance of the coil in Henrys.

An inductance calculator is shown below, enter your inductance and frequency to calculate inductive reactance.

Inductive Reactance Calculator
Enter Inductance:
 XL = ?
Enter Frequency:

To reset the calculator hit refresh or press "F5" on your keyboard.

Q Factor

q-factor diagram As an inductor is made from a coil of wire it will always contain resistance. The Quality factor, or Q factor is the ratio of energy stored in a coil to the energy lost during one cycle. The energy losses are due to internal resistance of the coil and absorption of the magnetic field. Coils made with thick wire will have less resistance and higher Q than a coil made with a smaller diameter wire. A high Q parallel tuned circuit will have a narrow bandwidth at resonance and low Q circuits will have wider bandwidths. Q factor is calculated by:
Q = 2π FL

where r is coil internal resistance in Ohms, F is frequency in Hz and L inductance of coil in Henries.

Capacitive Reactance
The reactance of a capacitor may be found using the following formula:
Xc = 1
2 π FC

This is similar to the inductor, with C in farads and f in hertz. The capacitive reactance will be in ohms. A capacitive reactance calculator is shown below, enter your capacitance and frequency to calculate reactance.

Capacitive Reactance Calculator
Enter Capacitance:
 XC = ?
Enter Frequency:

To reset the calculator hit refresh or press "F5" on your keyboard.

Parallel Tuned Circuit
A tuned circuit arranged as a parallel combination of an inductor and capacitor, will have a resonant frequency, Fr that is calculated with the following equation.
where Fr is given in hertz when L is in Henrys and C in Farads. You may find the following Resonant circuit calculator useful.

Resonant Frequency Calculator
Resonant Frequency:

Calculators to Determine L or C Value from Resonant Frequency
Sometimes you may want to make a radio or tuned circuit to operate at a particular frequency, e.g. 14MHz. The resonant frequency formula can be used in reverse, enter your wanted frequency and then your inductor or capacitor value from your parts box to find the appropriate value.
Inductor Value

Capacitor Value

Hartley Oscillator
The Hartley oscillator uses two coils (or a single centre tapped coil) and a parallel tuning capacitor. The frequency of oscillation is given by :
fosc = 1
2π √ C (L1 + L2)

Fosc is in ohms when L is in Henrys and C in Farads.

Hartley Oscillator Calculator
Capacitance :
Inductance 1:
Inductance 2:
   Frequency :

Ron has also kindly made a Hartley oscillator calculator. Enter values for inductance and capacitance, press calculate to find the oscillator frequency. Refresh the page (F5) to clear data.

Colpitts Oscillator
The Colpitts oscillator uses two capacitors and a single coil, the oscillator frequency is determined by: The formula is the same as the resonant circuit for both the Hartley and Colpitts oscillators, expect that the total inductance is summed in the Hartley case and the capacitors are added in series in the Colpitts oscillator. An example is shown in the analysis section.

Colpitts Oscillator Calculator
Capacitor 1 :  
Capacitor 2 :  
Inductance : 
 Frequency : 

Ron has also kindly made a Colpitts oscillator calculator. Enter values for the two capacitors and inductor, press calculate to find the oscillator frequency. Refresh the page (F5) to clear data.

Air Spaced Coils
An air spaced coil is usually wound over a former (like a plastic tube or drill bit) and the former removed. Sometimes for rigidity the former is left in place. The images below from left to right show a coil being wound on a drill bit, the completed coil, and finally two commercially made formers. The red core is air spaced coil, the grey core has a ferrite slug. The slug increases inductance and final value "tuned" by adjusting the slug.

The equation to calculate inductance for air spaced coils is below:

L = r2 n2
9r + 10l

Where L is the inductance in uH, n = number of turns on the coil, r is the radius of the coil in inches, and l (small L ) is the length of the coil in inches.
If you want to work with coil diameters, the as radius = diameter /2 the formula is re-arranged as below:

L = d2 n2
18d + 40l

As before, all dimensions in inches and inductance in uH. If you know the coil radius and length want to know how many turns to make a particular inductance, then the equation is re-arranged as shown below :

n = L (9r + 10l)

Alternatively if you want to work with coil diamter and length then use this equation :

n = L (18d + 40l)

Ron J has kindly made a calculator, select your dimensions, select your dimensions in inches or centimeters and your desired inductance. As much work is done at RF and VHF the inductance is set to microhenries, uH. The calculate button computes the turns required.

Turns Calculator
† Diameter :
Length    :
Inductance µH :  Turns
† External Diameter

Suppose you have a plastic former with outside diamter 7.2mm and 7mm long and you want to make a 580nH inductor. The values entered into the turns calculator are 0.72, 0.7, 0.58 (as 0.58uH = 580nH). The calulator predicts 10.79 turns. In practise you would either wind 10 or 11 turns and compress or expand the length of the coil for the exact value. These calculations will produce a coil very close to measured values.

Skin Effect
skin effect At dc and low frequencies current flow is distributed evenly throughout a conductor. As frequency increases there is a tendency for the current to flow on the outer surface of the conductor, this is known as the skin effect. The skin effect is frequency sensitive and as frequency is increased, the depth of current flow in the outer layer is reduced. On the left diagram, Dw represents overall wire diameter and Ds the skin depth. The gradient represents curremt flow, small in the centre and greatest at the extremeties. Skin effect lowers Q factor and decreases inductance by around 2%. The skin depth, Ds in metres is calculated by:
Ds = √(107ρ/(μr F )) / (2π)

where ρ is the conductor resistivity (ohm metres), μr is the relative permeability and F is the frequency (Hz).

1 Skin Effect in Conductors
2 Air Coil Calculations by Harry Lythall
3 Introduction to the Air Coil, University Surrey, EE Department.
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