5. Frequency Response

1

B

J

Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

5. Frequency Response

Transfer FunctionIn a circuit we will distinguish between input and output. The inputs are the independent current and voltage sources, also said excitations. The output are the branch currents and the tensions (branch voltages, node voltages or any potential difference between two nodes).

In a linear, time independent circuit for an input-output pair a transfer function (or network function) is defined. This is also the case of sinusoidal isofrequencylinear circuit analysis in the frequency space (Steinmetz transform). The transfer function is the ratio between an output and an input when the other sources, except the one considered, are switched off.

In Out

=Out F In

The transfer function can be defined in the frequency domain [voltages and currents: V and I ].

Linear, time-independent

network


As it had been stated by the superposition principle, in a linear circuit any voltage vr and any current is can be expressed as a linear combination of the p independent tension sources and the q independent current sources:

Vr = αr1V01 + αr2V02 + …. + αrpV0p + Zr1I01 + Zr2I02 + …. + ZrqI0q

Is = Ys1V01 + Ys2V02 + …. + YspV0p + βs1I01 + βs2I02 + …. + βsqI0q

The coefficients, Zrj, Ysj, αri, βsj are the transfer functions of the r voltages and the s tensions when coupled two by two to the p tension sources and the q current sources. The transfer functions αri and βsj are complex dimensionless constants. Zrj has the dimension of an impedance (Ω). The Ysj have the dimension of a an admittance (S = 1/Ω).

Transfer Function


Frequency Response

The frequency response of a circuit is the variation in its behavior with change in signal frequency.

As the impedances of a circuit are functions of the frequency of the specific AC signal, the response of a circuit to signals of same amplitude and different frequencies are different.

As the impedances of a circuit are dependant on the frequency at which they operate, the transfer function (or network function), defined for linear and time independent circuits in the frequency domain, is a complex function of the frequency ω : H = H(w).When X(w) and Y(w) are the input the output considered, the transfer function H(w) is:

H(ω ) = Y(ω )X(ω )


Frequency Response


Since the inputs and the outputs can be either voltage or current at any place in the circuit, there are four possible tranfer functions:

H(ω) = Vout(ω)Vin(ω)

, Voltage Gain

H(ω) = Iout (ω)Iin(ω)

, Current Gain

H(ω) = Vout(ω)Iin(ω)

, Tranfer Impedance

H(ω) = Iout (ω)Vin(ω)

, Tranfer Admittance

where the subscrips "in" refer to the inputs and the subsripts "out" refer to the outputs. H(ω) = H(ω)e jϕ , is the transfer function where H(ω) is the magnitude and ϕ is the phase.

In the RC circuit the input signal vS(t) = Vcosωt is the input signal. The output is given by the voltage across the capacitor.

Frequency ResponseLowpass filter

VS

R

C-+ Vout

. .

Vin = VS = R+ 1jωC

⎛⎝⎜

⎞⎠⎟I, Vout =

1jωCI

H(ω) = H(ω)e jϕ=VoutVS= 1 jωCR +1 jωC

=1-jω ω0( )1+ ω ω0( )2

H(ω) = 1

1+ ω ω0( )2 and ϕ = - tan-1 ω

ω0

H(ω) is the voltage gain, ω0 = 1 RC is the cutoff frequency. At ω = 0, H = 1 and ϕ = 0.At ω = ω0 , H = 1 2 = 0,707 and ϕ = - 45°.

ω

H

ω0=1/RC0

0,7071

ω

ϕ

ω0=1/RC0

- 45°

- 90°

ω/ω0 0 1 2 3 10 20 100

H 1 0.71 0.45 0.32 0.1 0.05 0.01

ϕ 0 -45° -63° -72° -84° -87° -90°

I.

In the RL circuit the input signal vS(t) = Vcosωt is the input signal. The output is given by the voltage across the inductor.

Frequency ResponseHighpass filter

I

VS

R

-+ Vout

. ..

VS = Vin = R+jωL( )I, Vout = jωL I

H(ω) = VoutVS

= jωLR + jωL

= jω ω0

1+jω ω0

where ω0 = R L is the catoff frequency.At ω = 0, H = 0 and φ = 90°.At the cutoff frequency ω0 , H = 1 2 and φ = 45°.

ω

φ

ω0=R/L0

- 45°

- 90°

L

ω

H

ω0=R/L0

0,7071


In the RLC series circuit the input signal vS(t) = Vcosωt is the input signal. The output is given by the voltage across the resistor.

Frequency ResponseBandpass filter

I

VS

C

-+ Vout

. ..

Vin = VS = R+j ωL −1 ωC( )⎡⎣ ⎤⎦I, Vout = R I

H(ω) = VoutVS

= RR+j ωL −1 ωC( )

Here H(0) = 0 and H(∞) = 0. The bandpass filter passes a band of frequencies ω1 <ω <ω2centered on ω0, the center frequency, given by

ω 0=1LC

For ω = ω1 and ω = ω2 the amplitude of the voltage gain is H = 1 2 = 0,707.

ω

H

ω10

0,7071

R

L

ω2


In the RLC series circuit the input signal vS(t) = Vcosωt is the input signal. The output is given by the voltage across the series inductor and capacitor.

Frequency ResponseBandstop filter

VS = R+j ωL −1 ωC( )⎡⎣ ⎤⎦I, Vout = j ωL −1 ωC( )I

H(ω) = VoutVS

= j ωL −1 ωC( )R+j ωL −1 ωC( )

Here H(0) =1 and H(∞) = 1. The bandstop filter stop or eliminates all frequencies withina band of frequencies, ω1 <ω <ω2, centered on

ω 0=1LC

For ω = ω1 and ω = ω2 the amplitude of the voltage gain is H = 1 2 = 0,707.

ω

H

ω10

0,7071

I

VS

C-+

Vout

.

.

. R

L

ω2


Frequency ResponseIdeal Frequency Response Filters

ω

H

0

1

ω0 ω

H

0

1

ω0

ω

H

ω10

1

ω2 ω

H

ω10

1

ω2

Lowpass filter

Bandstop filter

Highpass filter

Bandpass filter

Passive filters consist only of passive elements (R, L and C). Their gain cannot be greater than 1 as passive elements cannot deliver energy to the network. They can require bulky and expensive inductors; they perform poorly at frequencies below the audio range (300 Hz < f < 3000 Hz). Active filters contain active elements as transistors and operational amplifiers.

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activefilter filtroattivo

adjustableloadresistance

resistenzadicaricovariabile

bandpass filter filtropassa-banda

bandstop filter filtroarresta-banda

bandwidth larghezzadibanda

currentgain guadagnodicorrente

cutofffrequency frequenzaditaglio

frequencyresponse risposta infrequenza

halfpowerfrequency frequenzadimetàpotenza

highpass filter filtropassa-alto

lowpass filter filtropassa-basso

passivefilter filtropassivo

qualityfactor fattorediqualità

resonancefrequency frequenzadirisonanza

voltagegain guadagnoditensione

Terminology English – Italian


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5. Frequency Response