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frequency. In the low frequency range at T 1, expression (31) becomes:

LA		T 1	20lg K A.	(32)

Therefore, in the low frequency range the Bode diagram can be plotted by horizontal straight line.

In the high frequency range, T 1, expression (31) is rewritten as:

LA

K

A

20lgT ;

(33)

20lg

20lg KA

T 1

T

changing the frequency 10 times yields:

LA 10

T 1

20lg K 20lg10T

20lg K 20lgT 20 LA

T 1

20.

(34)

Therefore in the range of high frequency is represented by straight line with scope of 20 dB/dec . So in the low and high frequency ranges asymptotic Bode diagram is constructed from straight lines. They intersect at a corner frequency S which is calculated

comparing (32) and (33) expressions:

20lg K 20lg K 20lgT s ;				20lgT s 0;
T s 1;	s	1	.	(35)
		T
The actual	LA and		the asymptotic LAa Bode dia-

grams, plotted correspondingly according to (31) and (32) - (33) expressions, are shown in Fig. 6.

51

Fig. 6. Bode plots of proportional (P), integrating (I) and first order system LA, LAa

Fig. 6 shows that Bode plot of the first order system is horizontal line at low frequencies with magnitude of 20lg KA dB. In

the middle range of frequencies at frequency s T1 , the difference

between magnitudes of actual and asymptotic diagrams reaches about 3 dB. Finally in the high frequency range actual and asymptotic diagrams coincide and slope of the diagram is 20 dB/dec . The phase angle dependence versus frequency also is plotted in semi log coordinates. The phase angle is calculated from expression:

	A arctan	ImGA j	.	(36)
		ReGA j

Dependence of phase angle versus frequency of the first order system is shown in Fig. 7.

52

Fig. 7. Dependence of phase angle versus frequency of proportional (P), integrating (I) and first order system LA, LAa

Figure 7 shows that the phase angle of the first order systemin the low frequency range is close to zero i. e., is similar to

that proportional system. In the middle frequency range phase reach-

es –45º at frequency T 1 . Finally, the phase angle approaches to –90º in the high frequency range. The asymptotic phase angle characteristic Aa is also presented in Fig. 7. According to asymptotic

characteristic it is possible to separate three ranges of frequency: low frequency range, while s /10 ; s /10 10 is middle fre-

quency range and 10 s is high frequency range.

The examples of the first order system are given in Fig. 8. System, shown in Fig. 8a differs from system shown in Fig. 3a

just with assumption non-zero friction force FT . For sliding movement:

FT kT v;

(37)

where kT is friction coefficient. The second Newton Law can be written in this way:

53

m	dv			m	dv			(38)
	dt	F	FT ;		dt	F	kT v.

Fig. 8. Examples of the first order systems

If the vector projections in the „x” axis are considered, then expression (38) is rewritten in scalar form as:

m dv k v F;

m

dv v

1

F.

(39)

dt

T

kT

dt

kT

If

the

input

and

output

signals

correspondingly

are:

u F

, y v

x

,

and

K

A

1

and

T

m

,

then equation (39)

be-

x

kT

kT

comes the same as the differential equation of first order system (Eq. (20)).

The electrical circuit, shown in Fig. 8 b is described by differential equation:

L di Ri u ;		(40)
dt	1

where i is current.

Output voltage according to Ohm‘s low is calculated as:

54

u2 iR;		i	u2	.	(41)
			R

Substituting (41) to (40) yields:

L	du2 u	2	u .	(42)
R dt			1

From Eq. (41) it is evident, the gain of the system KA 1 and time constant T L R .

Ideal and real differentiating systems also depend to first order systems. Ideal differentiating system is described as:

	a y b du		; y	b1	du	K	du	;	(43)
				a	dt		D dt
	0	1 dt
			0

where KD is gain of the system.

Step response of ideal differentiating system is Dirac impulse

t :

t	,	t 0;	(44)
	0,	t 0.

Dirac function is impulse function with limited signal power:

	t dt 1.	(45)

Step response of ideal differentiating system is shown in Fig. 9, where it corresponds to Dirac impulse t .

Transfer function of ideal differentiating system is:

a Y s b U s s		G		s	Y s	K		s. (46)
0	1		DI	U s			DI
		55

From expression (46) can be found frequency response and Bode plot. Frequency response is calculated as:

GDI j KDI j .

(47)

Polar plot of expression (47) coincides with imaginary axis and begins at the origin at 0 as well as approaches to infinity at

. Polar plot of GD1 j is shown in Fig. 10. Magnitude of Bode diagram is calculated as:

LDI 20lg KDI 20lg KDI 20lg . (48)

Expression (48) is straight line with 20 dB/dec slope. Frequency, at which amplitude of ideal differentiating system is equal to zero, is obtained by setting to zero the left hand side of expression (48):

20lg K	DI	20lg	,		K	1.	(49)
		0			0	DI

LDI is shown in Fig. 11.

Phase dependence versus frequency is calculated as:

	DI arctan	ImGDI j	90 .	(50)
		ReGDI j

Thus DI is horizontal straight with ordinate, equal to 90º.

It is shown in Fig. 12. Ideal differentiating system has no prototypes in real techniques. Instead of that is used real differentiating system comprised from ideal differentiating system and first order system. It is described as:

a dy	a	y b du .	(51)
1 dt	0	1 dt
			56

Using notations KDR b1 / a0 andT a1 / a0 , differential equation can be rewritten in this way:

T dy	y KDR du	;	(52)
dt	dt

where: KDR is gain of real differentiating system and T is time constant.

Left sides of Eqs. (51) and (20) are the same, thus the solution of homogenous equation will have the same form as (21). Analytical expression of general solution at u 1 t is:

y	K	DR			t		(53)
			exp			.
	T
					T

Step response of real differentiating system is shown in Fig. 9. As the solution of homogenous system is the same as that of the first order system, thus step response is characterized by the same settling time tpp 3 4T .

57

Fig. 9. Step responses of ideal (DI) and real (DR) differentating systems

According to (52) the transfer function of real differentiating system

	is:	KDR s
	G		.	(54)
	DR	Ts 1
	Transfer function of real differentiating			system in frequency
	domain is expressed as:

	GDR j	KDR j	KDR	T 2 j .
		Tj 1
				1 T 2

Real and imaginary parts of expression (55) has the form:

x Re G j			KDRT 2	; y Im G j			KDR	.
							1 T 2
		1 T 2

The real part of GDR j can be be presented in this way:

2		x
			.
		T KDR Tx

Substituting ω to expression (56) yields:

(55)

(56)

(57)

	KDR			x
y			T KD Tx				KDR Tx	x	.	(58)
	1 T	2		x				T KDR Tx
			T KD Tx
Dividing		both		sides of			Eq. (58) by	KD Tx and		taking

square of those, also considering 0 : y 0 gives:

58

y

2

x

2

2

KDR

;

y

x

x

0;

T KDR

Tx

T

(59)

KDR Tx

0.

y 0;

y

Parametric equation (59) is rewritten in this way:

K

DR

K

DR

2

K

DR

2

y2 x2 2

x

;

2T

2T

2T

y 0;

(60)

K

DR

2

K

DR

2

y2

x

.

2T

2T

y 0.

The first equation of the set (60) describes circle with centre

coordinates

x0 0, KDR

2T

and

radius

R KDR

2T . The se-

cond equation states the polar plot being over the real axis as half cycle. Polar plot of real differentiating system is shown in Fig. 10.

Fig. 10. Polar plot if ideal (DI) and real (DR) differentiating systems Magnitude of real differentiating system is calculated from expres-

sion (55) as:

59

	GDR j			KDR		T j			KDR T 2 1	.
				1 T 2					1 T 2

(61)

Bode plots uses magnitudes, expressed by decibels, thus:

LDR 20lg GDR j 20lg KDR T 22 1 . 1 T

(62)

Expression (62) can be directly used to plot Bode diagram by computer, but in many cases the asymptotic Bode plots are constructed. For this at first two frequency ranges are analysed: low frequency range at T 1 and high frequency range at T 1. At low frequency range expression (62) becomes:

LDR		T 1	20lg KDR .	(63)

Expression (63) is the same as (47) for ideal differentiating system. It describes the straight line with slope 20 dB/dec. At the high frequency zone expression (62) is rewritten in this way:

L					20lg	KDR	.	(64)
DR				T 1		T
Expression (64) corresponds to horizontal line with ordinate,
equal to KDR	T decibels. Finally, the corner frequency s							can be

found, where both straights intersect. Set equal expression (63) to (64) and get:

20lg K	DR		20lg	KDR	;			1	.	(65)
		s		T			s	T
			60