Mysterious Circuit Q

So, you found an old piece of RF test equipment at a rummage sale. A friend who is a Ham radio enthusiast tells you it is a 'Q meter', and it appears to be in working order - great!

But, what is a Q meter?

A Q meter was used to measure the resonant rise of voltage across either of the reactive elements in a tuned circuit. This measurement resulted from injecting a small known radio frequency voltage across a very small series resistor which connected to the resonant circuit. The magnified or resonant voltage rise was then measured by a vacuum tube voltmeter (translation: high input impedance voltmeter). Since the amount of injected voltage was accurately known, the resonant voltage rise or circuit magnification factor could be directly calibrated in terms of the Q of the coil being measured. Although still popular with radio hobbyists, they have been largely replaced by network analyzers.

Why do I want to measure Q?

Q is a very common measure of performance for resonant circuits. It expresses a ratio of the total energy stored versus the energy dissipated in a circuit during one cycle of operation.

Using Q we can determine:

1. The damping effect when current is decaying in a resonant circuit.

2. Phase angle and power factor of tuned circuits.

3. Antenna characteristics.

4. Transmission-line parameters.

5. Selectivity of a tuned circuit.

6. The RF impedance of a coil.

7. The RF loss angle of a capacitor.

8. Dielectric constants.

Q and Series Resonant Circuits

In series resonant circuits, Q is calculated using the effective series resistance (Rs), a theoretical value representing all losses including those of the resonant coil and capacitor. Using Rs, Q is expressed as:

Q = ωL/Rs =1/(C x Rs); where ω = 2π x f.

Rearranging we find: Rs = ωL/Q = 1/Q x ωC - usually a very small quantity.

Q and Parallel Resonant Circuits

In a parallel resonant circuit, Q is calculated using the effective parallel resistance (Rp) a reflected value often represented by a parallel resistor connected across the tuned circuit. Using Rp, Q is expressed as:

Q = Rp/ωL = ωC x Rp

Again, by rearranging, we see Rp = Q x ωL = Q/ωC - usually a very large quantity.

Here we see that the impedance of a parallel resonant circuit is Q times the impedance of the reactive elements. In the series case, the current flowing at resonance is Q times the normal current flow.

Q and Damped Oscillation

In a damped oscillating circuit, Q expresses the logarithmic decrement ς of the circuit.

Therefore:

ς = π/Q = Rs/2f = 2 x (π^2) x f x C x Rs

As we know, R/2L is the damping coefficient -. since the equation for current decay is:

I2 / I0 = e ^(-RT/2L).

ς, which multiplies the damping coefficient by f, accounts for frequency.

Q and Phase Angle.

The vector relationship between the current and driving voltage in a resonant circuit is the familiar equation:

tanθ = ωL/Rs = Q; where θ is the phase angle

Q and Power Factor.

The relationship between Q and the power factor (cosθ) of an inductor or the ratio of the total effective resistance to the total circuit impedance can be shown to be:

Q = 1/cosθ; where θ is the phase angle.

Glen Taylor is the owner of eWerks Inc. (http://www.ewerksinc.com) for over 25 years he has been involved in the design and manufacture of industrial electronics. Glen holds a Master of Science degree (with distinction) in Electronic Product Development from the University of Bolton in the UK, a CID certification in circuit board design from IPC Designers Council, and is an authorized Microchip Design Partner. He is also a senior member of IEEE - Institute of Electrical and Electronic Engineers and ISA - International Society for Automation. For more tutorials visit his blog at http://www.Thingsthatgoblink.com

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