142

FIRST-ORDER CIRCUITS

[CHAP. 7

Table 7-1(b) Step and Impulse Responses in RL Circuits RL circuit

Unit Step Response

Unit Impulse Response

vs ¼ uðtÞ ( v ¼ eRt=L uðtÞ i ¼ ð1=RÞð1  eRt=L ÞuðtÞ

vs ¼ ðtÞ ( hv ¼ ðR=LÞeRt=L uðtÞ þ ðtÞ hi ¼ ð1=LÞeRt=L uðtÞ

is ¼ uðtÞ ( v ¼ ReRt=L uðtÞ

is ¼ ðtÞ ( hv ¼ ðR2 =LÞeRt=L uðtÞ þ RðtÞ

i ¼ ð1  eRt=L ÞuðtÞ

hi ¼ ðR=LÞeRt=L uðtÞ

Fig. 7-18

The natural response ih ðtÞ is the solution of Ri þ Lðdi=dtÞ ¼ 0; i.e., the case with a zero forcing function. Following an argument similar to that of Section 7.4 we obtain ih ðtÞ ¼ AeRt=L

ð17bÞ

The forced response ip ðtÞ is a function which satisﬁes (16) for t > 0.

The only such function is

ip ðtÞ ¼ I0 est After substituting ip in (16), I0 is found to be I0 ¼ V0 =ðR þ LsÞ. boundary condition ið0þ Þ ¼ 0 is also satisﬁed. Therefore, iðtÞ ¼

ð17cÞ By choosing A ¼ V0 = ðR þ LsÞ, the

V0 ðest  eRt=L ÞuðtÞ R þ Ls

ð17dÞ

Special Case. If the forcing function has the same exponent as that of the natural response ðs ¼ R=LÞ, the forced response needs to be ip ðtÞ ¼ I0 teRt=L . This can be veriﬁed by substitution in (16), which also yields I0 ¼ V0 =L The natural response is the same as (17b). The total response is then iðtÞ ¼ ip ðtÞ þ ih ðtÞ ¼ ðI0 t þ AÞeRt=L From ið0 Þ ¼ ið0þ Þ ¼ 0 we ﬁnd A ¼ 0, and so iðtÞ ¼ I0 teLt=R uðtÞ, where I0 ¼ V0 =L.

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An