Logarithmic frequency characteristics of elementary links

Lecture 10

LOGARITHMIC FREQUENCY CHARACTERISTICS OF ELEMENTARY LINKS

Lecture plan:

1.  Logarithmic frequency characteristics of position links

2.  Logarithmic frequency characteristics of integrating links

3.  Logarithmic frequency characteristics of differentiating links

4.  Construction of system’s logarithmic frequency characteristic

As we said before, in certain cases for amplitude and phase frequency characteristics more comfortable is logarithmic scale.

In logarithmic coordinates frequency axis marked proportional to lg w.  This axis divided in decades: each decade accords to frequency rising in 10 times, for example, if decade beginning accords to value  then this decade ends by value

Logarithmic scale for amplitude marked in decibels: One bell accords to power multiplication in 10 times, two Bells – in 100 timed, etc. Formula for translation of linear amplitude characteristics in logarithmic scale:

L(w)=20lg|W(i w)|=20lgA(w). [dB].

Decibel is tenth part of Bell.

Coefficient 20 derives from A(w) isn't a power ratio, but amplitude ratio. Increase of amplitude in 10 times accords to power increase in 100 times. Hence, one decibel accords to amplitude increase in  times.

Negative values of L(w) accords to amplitude decrease. For phase frequency characteristics linear scale used.

The most important feature of logarithmic frequency characteristic, that in logarithmic coordinates amplitude functions represents by asymptotic straight lines even they represents by curves in linear scale. For phase characteristics: each curve in logarithmic scale keep their shape for all variety of certain type links. This curve shifted along frequency axis only.

Now we show logarithmic frequency characteristics for main elementary dynamic link set.

1. Non-inertia link :                            W(p)=K

L(w)=20lgK=const

j(w)=0=const.

2. Aperiodic link:                                

L_l(w)20lgK – straight line parallel to frequency axis.

L_h(w)20lgK-20lgK-20lgw - straight line with constant slope.

To find frequency of break combine both of cases:

L_l(w)=L_h(w)

20lgK=20lgK-20lgT-20lgw₀ Þ

Logarithmic frequency characteristic of aperiodic link represented in fig.1.

For phase frequency characteristics as in linear scale:

3. Oscillation link:      

Analog to previous:

 

Asymptote slope: -40 dB/dec.

Break frequency:

- it is resonance frequency.

For phase frequency characteristic:

Value of peak DL according to frequency w₀ depends on damping coefficient x. The more x the less value of DL. For x®0 (conservative link) we have infinity big value of peak of amplitude characteristic resonance frequency.

4. Integrating link     

It’s easy to find, that

  For w=1: L(w)=20lgK;

For any value of frequency; logarithmic amplitude frequency characteristic of this link can be exactly represented through straight line with slope  –20 dB/dec:

L(10)=20lgK-20lg10=20lgK-20.

As we find for linear scale:

5. Differentiating link    W(p)=Kp.

For this link L(w) is straight line slope +20 dB/dec. And for w=1 L(w)=20lgK. .

6. Differentiating link with slowing

W(p)=K(Tp+1)

Logarithmic amplitude for low frequency: for high frequency:

Hence for frequency this characteristic represents through straight line with slope +20 dB/dec.

On studied facts base we can derive next rule for logarithmic frequency amplitude-phase characteristics building.

It transfer function of system can be represented through fraction:

then we can build general logarithmic characteristic as geometry sum of elementary links characteristics.

For w=1     L(w)=20lgK.

Slope of low-frequency amplitude characteristic equal to 20(M-L) dB/dec. Value (M-L)=N we shall call as astatism order. This number shows how many zero roots has denominator of open-loop system’s transfer function. Frequency axis divided by break frequencies in some segments. In each segment slope of amplitude characteristic will accord to combination of break frequencies: each value of , where T_j-time constant of aperiodic link  - slope is (-20 dB/dec); for , where T_k – time constant of oscillation link – slope is (-40 dB/dec).

Phase frequency characteristic received as geometry sum of according phase characteristics of elementary links too.

Example:

logarithmic frequency characteristic represented in fig.3