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Problem 4.06PP
Consider the DC motor control system with rate (tachometer) feedback shown in Fig.(a).
(a) Find values for K and so that the system of Fig.(b) has the same transfer function as the 
system of Fig.(a).
(b) Determine the system type with respect to tracking dr and compute the system Kv in terms of 
parameter fC and .
(c) Does the addition of tachometer feedback with positive kt increase or decrease Kv? 
Figure Control system
Step-by-step solution
step 1 of 7
(a)
Refer to Figure 4.25 (a) in the textbook.
The simplified circuit of Figure 4.25 (a) is shown in Figure 1.
Figure 1
Step 2 of 7
Calculate the closed loop transfer function.
K K „K ^
KKJC,
~ * [ s ( i+ r ^ ) + /a : j f c ^ ]+ A M : . j r ,
KKJC.
(')
■ +KKJcJa + KKJC,
K K,K ,
(1)
step 3 of 7
Refer to Figure 4.25 (b) in the textbook.
Calculate the closed loop transfer function from the Figure 4.25 (b).
K'
i ( l + r ^ ) + A T '( l + * » 
9 K'
(2)
Step 4 of 7
Compare the equations (1) and (2).
K K.K .trt _
*
And
K% = KK.k,
KKJC.Substitute i f o r jc 'in K% = KKJc,
' K .
Therefore, the values of j f ' and are
k ' K,
Step 5 of 7
(b)
The inner loop in Figure 4.25 (a) is. 
A A .
! ( * ) =
j ( i+ r , i ) + A a r , t , i
KK.
s(1 + t, s + K K J .) 
KK,
s '0 + T .^ i-K K .k , )
^e denominator of the tr 
jre, the type of the syst
Hence, the type of the system is jT y p e l
From the denominator of the transfer function, it is clear that one pole exists at the origin. 
Therefore, the type of the system is 1.
Step 6 of 7
The value of K , is. 
JCy s lim4Z.(4)
s lim ^i-**
s lim
KK.
— 5 ( i+ r . i + i a : . t , )
KK.
■ - - i+ r , s + /a r , * ,
KK.
1 + X K jt,
K.=
KV,
( since fiom equation (3) w e have^ 
[K 'k ; = K K J , )
k,{l+K'k;)
Therefore, the value of K , in terms of j f ' and is
Step 7 of 7
(C)
The velocity error constant is.
Kk;
■(3)
From equation (3). it is clear that, is inversely proportional to k,- 
That is.
The addition of tachometer feedback with positive k, Idecrwisfsl AT, ■