Chapter 4 · 2 days ago · Chapter 4 b. Selectivity. Resonance • Resonance in AC circuit is a special frequency determined by the values of the: Resistance, capacitance and inductance

Chapter 4b. Selectivity

Resonance

• Resonance in AC circuit is a special frequency

determined by the values of the: Resistance,

capacitance and inductance

• For series resonance the condition of resonance

is characterized by minimum impedance and

zero phase

• Parallel resonance, which is more common in

electronic practice, requires a more careful

definition

Series Resonance

• The resonance of a series RLC circuit occurs

when the inductive and capacitive reactance are

equal in magnitude but cancel each other

because the are 1800 degrees apart in phase

• The sharp minimum in impedance which occurs

is useful in tuning applications

• The sharpness of the minimum depends on the

values of R and is characterized by the “Q” of the

circuit.

Series RLC Circuit Variation reactance with frequency

Series Resonance

𝑋𝐿 = 𝑋𝐶

𝑋𝐿 = 2𝜋𝑓𝑟𝐿

𝑋𝐶 =1

2𝜋𝑓𝑟𝐶

2𝜋𝑓𝑟𝐿 =1

2𝜋𝑓𝑟𝐶

𝑓𝑟 =1

2𝜋 𝐿𝐶

𝐿 =1

4𝜋2𝑓2𝐶𝐶 =

1

4𝜋2𝑓2𝐿

Example: The capacitance that will resonate at a frequency of

18 MHz with a 12-μH inductor is determined as follows:

𝐶 =1

4𝜋2𝑓2𝐿=

1

39,478 18 × 106 2(12 × 10−6)

𝐶 =1

39,478 3,24 × 1014 2(12 × 10−6)= 6,5 × 10−12𝐹 𝑜𝑟 6,5 𝑝𝐹

Series Resonant

• Basic definition of resonance in a series tuned

circuit is the point at which XL equals XC

• With this condition, only the resistance of the

circuit impedes the current

• The total circuit impedance at resonance is Z = R

• Resonance in a series tuned circuit can also be

defined as th epoint at which the circuit

impedance is lowest and the circuit current is

highest

Series Resonant

• Since the circuit is resistive at resonance, the

current is in phase with the applied voltage

• Above the resonance frequency, the inductive

reactance is higher than the capacitive

reactance, and the inductor votage drop is

greater than the capacitor voltage drop

• Therefore the circuit is inductive, and the

current will lag the applied voltage

Series Resonant

• Below resonance, the capacitive reactance is

higher than the inductive reactance

• The net reactance is capacitive, producing a

leading current in the circuit

• The capacitor voltage drop is higher than the

inductor voltage drop

Series Resonant

Series Resonant

• Figure above is plot of frequency and phase shift

of the current in the circuit with respect to

frequency

• At very low frequencies, the capacitive reactance

is much greater than the inductive reactance,

therefore the current in the circuit is very low

because of the high impedance

Series Resonant

• If the circuit is predominantly capacitive, the

current leads the voltage by nearly 900

• As the frequency increases, XC goes down and XL

goes up

• The amount of leading phase shift decreases

• As the value of reactances approach one another,

the current begin to rise

Series Resonant

• When XL equals XC their effects cancel and the

impedance in the circuit is just that of the

resistance

• This produces a current peak, where the current

is in phase with the voltage (00)

Series Resonant

• As the frequency continues to rise, XL becomes

greater than XC

• The impedance of th ecircuit increases and the

current decreases

• The circuit will predominantly inductive, the

current lags the applied voltage

Series Resonant

• If the output voltage were being taken from across

the resistor in figure (a), the response curve and

phase angle of the voltage would correspond to

those in figure (b)

(a) (b)

Series Resonant

• The current is

highest in a region

centered on the

resonant frequency

• The narrow

frequency range

over which the

current is highest is

called the

bandwidth

• The current is highest in a region centered on the

resonant frequency

• The narrow frequency range over which the

current is highest is called the bandwidth

Series Resonant

• The upper and lower boundaries of the

bandwidth are defined by two cutoff frequencies

designated f1 and f2• These cutoff frequencies occur where the current

amplitude is 70.7 percent of the peak current

• In the figure, the peak current is 2 mA, and the

current at both the lower (f1) and upper (f2) cutoff

frequency is 0.707 of 2 mA, or 1.414 mA

Series Resonant

• Current levels at which the response is down

70.7 percent are called the half-power point

because the power at the cutoff frequencies is

one-half the power peak of the curve

𝑃 = 𝐼2𝑅 = 0.707 𝐼𝑝𝑒𝑎𝑘2𝑅 = 0.5 𝐼𝑝𝑒𝑎𝑘

2𝑅

Series Resonant

• The bandwidth BW of the tuned circuit is defined

as the difference between the upper and lower

cutoff frequencies:

𝐵𝑊 = 𝑓2 − 𝑓1

Series Resonant

• For example, assuming a resonant frequency of

75 kHz and upper and lower cutoff frequencies of

76.5 and 73.5 kHz, respectively, the bandwidth

is BW = 76.5 – 73.5 = 3 kHz

Selectivity and Q of a Circuit

• Resonant circuit are used to respond selectivity

to signal of a given frequency while

discriminating against signals of different

frequencies

• If the response of the circuit is more narrowly

peaked around the chosen frequency, we say

that the circuit has higher “selectivity”.


• A “quality factor” Q, as described below, is a

measure of that selectivity, and we speak of a

circuit having a “high Q” if it is more narrowly

selective

• The selectivity of a circuit is dependent upon the

amount of resistance in the circuit

• The smaller the resistance, the higher the “Q” for

given values of L and C


• The bandwidth of a resonant circuit is

determined by the Q of the circuit

• Recall the Q of an inductor is ration of the

inductive reactance to the circuit resistance,

which includes the resistance of the inductor

plus any additional series resistance:

𝑄 =𝑋𝐿

𝑋𝑇and 𝐵𝑊 =

𝑓𝑟

𝑄


• If the Q of a circuit resonant at 18 MHz is 50,

then the bandwidth is BW = 18/50 = 0.36 MHz =

360 kHz

• The higher of resistance value, then lower the Q

value, the lower the Q, caused the higher of BW



• Question: What is the Q if Assuming a resonant

frequency of 75 kHz and upper and lower cutoff

frequencies of 76.5 and 73.5 kHz?

• Answer : 𝑄 =𝑓𝑟

𝐵𝑊=

75 𝑘𝐻𝑧

76.5 −73.5=

75 𝑘𝐻𝑧

3= 25


• Since the bandwidth is approximately centered

on the resonant frequency, f1 is the sae distance

from fr as f2 is from fr• This fact allows you to calculate the resonant

frequency by knowing only the cutoff

frequencies:

𝑓𝑟 = 𝑓1 × 𝑓2


• For example: if f1 = 175 kHz and f2 = 178 kHz,

the resonan frequency is

𝑓𝑟 = 175 × 103 × 178 × 103 = 176.5 𝑘𝐻𝑧


• For a linear frequency scale, you can calculate

the center or resonant frequency by using an

average of the cutoff frequencies

𝑓𝑟 =𝑓1 + 𝑓22






Chapter 4 · 2 days ago · Chapter 4 b. Selectivity. Resonance • Resonance in AC circuit is a special frequency determined by the values of the: Resistance, capacitance and inductance

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