AC Circuits & Transformers, Lecture notes of Electronics

Download AC Circuits & Transformers and more Lecture notes Electronics in PDF only on Docsity!

Series Resonance

Series Resonance

• Resonance is a condition in a RLC circuit in which the reactive and capacitive reactance are equal in

magnitude, thereby resulting in a purely resistive Impedance.

• (^) Resonance circuits are used for constructing filters and used in many other applications.

Consider the series RLC circuit shown in figure:

Net reactance of the circuit : X= (XL – XC) ohm

Series Resonance

Effects at resonant frequency:

1. Net reactance XL – Xc = 0

2. Net impedance is minimum Z=R ohm

3. The current in the circuit is maximum I= V⁄Z= V⁄R

4. Since current is maximum , power absorbed by the circuit will also be maximum

5. VL =I.XL and VC = I.Xc and the two voltage drops are equal in magnitude but opposite in phase. Hence,

they cancel out each other. The two reactance’s taken together act as short circuit since no voltage

develops across them. The applied voltage V drops entirely across R so that V = VR as shown in figure

below.

6. Power factor at resonance =1.

Series Resonance Resonant frequency : The frequency at which net reactance is zero given from the relation XL – XC = 0 or XL = XC or ωL = 1/ωC 𝜔

= 1 𝐿𝐶 ( 2 𝜋𝑓𝑜)

= 1 𝐿𝐶 𝑓𝑜 = 1 2 𝜋ξ𝐿𝐶 Variation of impedance z with frequency: a. At resonant frequency : Supply frequency f = fo , the circuit is at resonance and resistive, Z = R. b. At low frequencies : Supply frequency f > fo , XL > XC, the circuit is inductive and Z = R + j X. c. At high frequencies : Supply frequency f < fo , XL < XC, the circuit is 1 Capacitive and Z = R – j X

Series Resonance Bandwidth of a circuit is given by the range of frequencies which lie between two points on either side of the resonant frequencies fo where current falls to 1/ 2 of its maximum value at resonance. Narrower the bandwidth, higher the selectivity of the circuit. A As shown in figure the bandwidth is given by Δf = (f 2 – f 1 ) Hz or Δω = (ω 2 – ω 1 ) rad/sec. This range of frequencies (bandwidth), current is equal to or greater than I 0 / 2 where I 0 = V/R --- maximum current at resonance. For series resonant circuit bandwidth is given as 𝐵𝑊=

ሺ𝐻𝑧ሻ=

𝑟𝑎𝑑/𝑠𝑒𝑐 Bandwidth = R / L (rad/sec) f 1 and f 2 are the frequencies at which the current is exactly = I 0 / 2. These frequencies f 1 and f 2 are called as the upper and the lower cutoff frequencies respectively. 𝑓 2 = 𝑓𝑟 + 𝐵𝑊 2 𝑓 1 = 𝑓𝑟 − 𝐵𝑊 2

Series Resonance Q-Factor of series circuit : In the case of series R-L-C circuit it is defined as equal to the voltage magnification in the circuit at resonance. At resonance I 0 = R V = Imax, Since XL = Xc VL = Vc = I 0 XL = I 0 XC Supply voltage V = IR So Voltage magnification =

V

VL

= I R

I XL

0 (^0) = R CR

X

R

L

R

X L C

0

So Q – factor    tan 0 2 0    R f L R L Where Q is power factor of the coil Since, LC f

0 ^ Or^ LC f 1 2  0  so Q = C

L

R

In fact, Q of series circuit may be written as : Q = ω 0 Band width = C L R LC^ R L R L L R 0 0 0 1     

𝑄= 1 𝑅 ඨ 𝐿 𝐶

Numericals on Series Resonance

Numericals on Series Resonance

Q.2)

Numericals on Series Resonance

Q.3)

Numericals on Series Resonance

Numericals on Series Resonance