UPDATE: 1/31/20
I’ve just updated this section for the 2020 version of the No Nonsense Extra Class License Study Guide. Please read that post instead of this one….73. Dan
It’s hard to believe that it’s been four years since I wrote the first edition of my “No Nonsense” Extra Class study guide. It has been, though, and now it’s time to update the study guide to cover the 2016 – 2020 question pool. Here’s the first updated section.
There have been several question changes in this section. They removed questions that asked you to calculate the resonant frequency or half-power bandwidth and added questions talking about Q and the consequences of having lower or higher Q.
Here’s the updated section:
E5A – Resonance and Q: characteristics of resonant circuits; series and parallel resonance; Q; half-power bandwidth; phase relationships in reactive circuits
Resonance is one of the coolest things in electronics. Resonant circuits are what makes radio, as we know it, possible.
What is resonance? Well, a circuit is said to be resonant when the inductive reactance and capacitive reactance are equal to one another. That is to say, when
2πfL = 1/2πfC
where L is the inductance in henries and C is the capacitance in farads.
For a given L and a given C, this happens at only one frequency:
f = 1/2π√(LC)
This frequency is called the resonant frequency. Resonance in an electrical circuit is the frequency at which the capacitive reactance equals the inductive reactance.(E5A02)
Let’s calculate a few resonant frequencies, using questions from the Extra question pool as examples:
The resonant frequency of a series RLC circuit if R is 22 ohms, L is 50 microhenrys and C is 40 picofarads is 3.56 MHz. (E5A14)
f = 1/2π√(LC) = 1/(6.28 x √(50×10-6 x 40×10-12)) = 1/(2.8 x 10-7) = 3.56 MHz
Notice that it really doesn’t matter what the value of the resistance is. The resonant frequency would be the same is R = 220 ohms or 2.2 Mohms.
The resonant frequency of a parallel RLC circuit if R is 33 ohms, L is 50 microhenrys and C is 10 picofarads is 7.12 MHz. (E5A16)
f = 1/2π√(LC) = 1/(6.28x√(50×10-6 x 10×10-12)) = 1/(1.4×10-7) = 7.12 MHz
When an inductor and a capacitor are connected in series, the impedance of the series circuit at the resonant frequency is zero because the reactances are equal and opposite at that frequency. If there is a resistor in the circuit, that resistor alone contributes to the impedance. Therefore, the magnitude of the impedance of a series RLC circuit at resonance is approximately equal to circuit resistance. (E5A03)
The magnitude of the current at the input of a series RLC circuit is at maximum as the frequency goes through resonance. (E5A05) The reason for this is that neither the capacitor or inductor adds to the overall circuit impedance at the resonant frequency.
When the inductor and capacitor are connected in parallel, the impedances are again equal and opposite to one another at the resonant frequency, but because they are in parallel, the circuit is effectively an open circuit. Consequently, the magnitude of the impedance of a circuit with a resistor, an inductor and a capacitor all in parallel, at resonance, is approximately equal to circuit resistance. (E5A04)
Because a parallel LC circuit is effectively an open circuit at resonance, the magnitude of the current at the input of a parallel RLC circuit at resonance is at minimum. (E5A07) The magnitude of the circulating current within the components of a parallel LC circuit at resonance is at a maximum. (E5A06) Resonance can cause the voltage across reactances in series to be larger than the voltage applied to them. (E5A01)
Another consequence of the inductive and capacitive reactances canceling each other is that there is no phase shift at the resonant frequency. The phase relationship between the current through and the voltage across a series resonant circuit at resonance is that the voltage and current are in phase. (E5A08)
Ideally, a series LC circuit would have zero impedance at the resonant frequency, while a parallel LC circuit would have an infinite impedance at the resonant frequency. In the real world, of course, resonant circuits don’t act this way. To describe how closely a circuit behaves like an ideal resonant circuit, we use the quality factor, or Q. Because the inductive reactance equals the capacitive reactance at the resonant frequency, the Q of an RLC parallel circuit is the resistance divided by the reactance of either the inductance or capacitance (E5A09):
Q = R/XL or R/XC
The Q of an RLC series resonant circuit is the reactance of either the inductance or capacitance divided by the resistance (E5A10):
Q = XL/R or XC/R
Basically, the higher the Q, the more a resonant circuit behaves like an ideal resonant circuit,and the higher the Q, the lower the resistive losses in a circuit. Lower losses can increase Q for inductors and capacitors. (E5A15) An effect of increasing Q in a resonant circuit is that internal voltages and circulating currents increase. (E5A13)
Q is an important parameter when designing impedance-matching circuits. The result of increasing the Q of an impedance-matching circuit is that matching bandwidth is decreased. (E5A17) A circuit with a lower Q will yield a wider bandwidth, but at the cost of increased losses.
A parameter of a resonant circuit that is related to Q is the half-power bandwidth. The half-power bandwidth is the bandwidth over which a series resonant circuit will pass half the power of the input signal and over which a parallel resonant circuit will reject half the power of an input signal.
We can use the Q of a circuit to calculate the half-power bandwidth:
BW = f/Q
Let’s look at some examples:
The half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 7.1 MHz and a Q of 150 is 47.3 kHz. (E5A11)
BW = f/Q = 7.1 x 106/150 = 47.3 x 103 = 47.3 kHz
What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 3.7 MHz and a Q of 118 is 31.4 kHz. (E5A12)
BW = f/Q = 3.5 x 106/118 = 31.4 x 103 = 31.4 kHz
Dan KB6NU says
17 questions down, 700 to go!
David Ryeburn VE7EZM and AF7BZ says
You wrote “Ideally, a series resonant circuit would have zero impedance at the resonant frequency and an infinite impedance at all others. A parallel resonant circuit would have an infinite impedance at the resonant frequency and be zero at all others.” The off resonance part of this is incorrect. With lossless components, off resonance the impedance of a series circuit would be the net reactance after cancellation between inductive and capacitive reactance. For a parallel circuit, the net impedance would be the reactance remaining after cancellation between the susceptance in one branch and the susceptance in the other.
Dan KB6NU says
D’oh. That was kind of stupid wasn’t it? I’ve sort of combined those two sentences and attached it to the following paragraph.
David Ryeburn VE7EZM and AF7BZ says
You wrote “A circuit with a lower Q will yield a wider bandwidth, but at the cost of increased losses.” This confuses loaded Q and unloaded Q. Loss depends upon the ratio between them. A matching network with low loaded Q but made from components with high unloaded Q will be very efficient; indeed, the lower the loaded Q (for the same unloaded Q), the wider the bandwidth and the higher (not lower) the efficiency. This is why L networks can be more efficient than Pi networks or T networks (which are in effect two L networks back-to-back, with at least one of the two having higher loaded Q than an L network accomplishing the same match would have had).
Dan KB6NU says
Hmmmmm. I may have to research this more. Do you know any book or article that explains this?