Class Activity (Resonant Circuits and Filters)

Series and parallel resonant circuit have an important role in electronic communication systems. In a

series resonant circuit magnitude of Xl is equal with magnitude of Xc. Therefore, impedance

decreasing and current increasing.





ωL = 1/ωC , ω₀^2 = 1 / (LC) ,   2πfo = ω<sub>0 , fo is resonant frequency of a series circuit. Under resonant 

condition circuit is purely resistive. So Z = R and current increasing because I = V/R .</sub>

Average power at resonant frequency is P(avg) = (Vm)^2 / (2R) .

 At a certain frequencies ω = ω ₁ , ω ₂ , called the half-power

frequencies the dissipated power is half the maximum value.

P(ω₁) = P(ω₂) = (Vm/radical(2))^2  / 2R = (Vm)^2 / (4R) .   ω₀² = ω₁* ω₂

B is a half-power bandwidth     B = ω ₂ − ω ₁

_{Q = 2(pi) (peak energy stored in the circuit)/(energy dissipated by the circuit in one period at resonance)}

The quality factor of a circuit in resonant:   Q = (ωoL)/R , B = R/L = ωo / Q

Changes of Q Relative to Resistance

High-Q circuits are used often in communications networks.

                                                       ω₁ ≈ ωo - (B/2)  ,   ω₂ ≈ ωo + (B/2) 

Parallel Circuit resonant

In contrast with a series circuit, in parallel circuit impedance is maximum and current is minimum.

 A filter is a circuit that design to select a specific frequency and reject others. 

1. A low-pass filter (LPF) passes low frequencies and stops high frequencies.

A Low pass RC Circuit

Transfer function for a RC series low pass filter.

When f is low, XC is high and Vc = Vout is maximum. When f is high, XC is minimum 

and Vc = Vout is minimum.

When f is low, XL is low and VR = Vout is maximum. When f is high, XL is maximum 

and VR = Vout is minimum.

2. A high-pass filter (HPF) passes high frequencies and rejects low frequencies.

When f is low, XC is high and VR = Vout is minimum. When f is 

high, XC is minimum and VR= Vout is maximum.

When f is low, XL is low and VL = Vout is minimum. When f is 

high, XL is maximum and VL = Vout is maximum.

3. A band-pass filter (BPF) passes frequencies within a frequency band and blocks or attenuates

frequencies outside the band.

This is a series RLC circuit. In resonant frequency Z = R and current is 

maximum so VR = Vout is also maximum.

4. A band-stop filter (BSF) passes frequencies outside a frequency band and blocks or attenuates

frequencies within the band

.

This is a series RLC circuit. In resonant frequency Z = R and current is 

maximum so VR is maximum, and Vout is minimum.