第四章
4-1 (1) $ \frac{\alpha}{s(s+\alpha)} $
(2) $ \frac{2s+1}{s^{2}+1} $
(3) $ \frac{1}{(s+2)^{2}} $
(4) $ \frac{2}{(s+1)^{2}+4} $
(5) $ \frac{s+3}{(s+1)^{2}} $
(6) $ \frac{1}{s+\beta}-\frac{s+\beta}{(s+\beta)^{2}+\alpha^{2}} $
(7) $ \frac{2}{s^{3}}+\frac{2}{s^{2}} $
(8) $ 2 - \frac{3}{s + 7} $
(9) $ \frac{\beta}{(s+\alpha)^{2}-\beta^{2}} $
(10) $ \frac{1}{2}\left(\frac{1}{s}+\frac{s}{s^{2}+4\Omega^{2}}\right) $
(11) $ \frac{1}{(s+a)(s+\beta)} $
(12) $ \frac{(s+1)e^{-a}}{(s+1)^2+\omega^2} $
(13) $ \frac{(s+2)e^{-(s-1)}}{(s+1)^{2}} $
(14) $ aF(as + 1) $
(15) $ aF(as + a^{2}) $
(16) $ \frac{1}{4}\left[\frac{3s^{2}-27}{(s^{2}+9)^{2}}+\frac{s^{2}-81}{(s^{2}+81)^{2}}\right] $
(17) $ \frac{2s^{3}-24s}{(s^{2}+4)^{3}} $
(18) $ -\ln\left(\frac{s}{s+\alpha}\right) $
(19) $ \ln\left(\frac{s+5}{s+3}\right) $
(20) $ \frac{\pi}{2} - \arctan\left(\frac{s}{\alpha}\right) $
4-2 (1) $ \frac{\omega}{s^{2}+\omega^{2}}\left(1+\mathrm{e}^{-\frac{T}{2}s}\right) $
(2) $ \frac{\omega\cos\varphi+s\sin\varphi}{s^{2}+\omega^{2}} $
4-3 (1) $ \frac{1}{s+1}e^{-2(s+1)} $
(2) $ \frac{1}{s+1}e^{-2s} $
(3) $ \frac{e^{2}}{s+1} $
(4) $ \frac{2\cos 2+s\sin 2}{s^{2}+4}e^{-s} $
(5) $ \frac{1}{s^2}[1-(1+s)e^{-s}]e^{-s} $
4-4 (1) $ e^{-t} $
(2) $ 2e^{-\frac{s}{2}t} $
(3) $ \frac{4}{3}(1-\mathrm{e}^{-\frac{3}{2}t}) $
(4) $ \frac{1}{5}\left[1 - \cos(\sqrt{5}t)\right] $
(5) $ \frac{3}{2}(e^{-2t}-e^{-4t}) $
(6) $ 6e^{-4t} - 3e^{-2t} $
(7) $ \sin t + \delta(t) $
(8) $ e^{2t} - e^{t} $
(9) $ 1 - e^{-\frac{t}{RC}} $
(10) $ 1 - 2e^{-\frac{t}{RC}} $
(11) $ \frac{RC\omega}{1+(RC\omega)^2}\left[\mathrm{e}^{-\frac{t}{RC}}-\cos(\omega t)+\frac{1}{RC\omega}\sin(\omega t)\right] $
(12) $ 7e^{-3t} - 3e^{-2t} $
(13) $ \frac{100}{199}(49e^{-t}+150e^{-200t}) $
(14) $ \mathrm{e}^{-t}(t^{2}-t+1)-\mathrm{e}^{-2t} $
(15) $ \frac{A}{K}\sin(Kt) $
(16) $ \frac{1}{6}\left[\frac{\sqrt{3}}{3}\sin(\sqrt{3}t) - t\cos(\sqrt{3}t)\right] $
(17) $ \frac{-a}{(\alpha-a)^2+\beta^2}\left\{\mathrm{e}^{-at}-\left[\cos(\beta t)+\frac{\alpha^2+\beta^2-a\alpha}{a\beta}\sin(\beta t)\right]\mathrm{e}^{-at}\right\} $
(18) $ \frac{1}{(\beta^2 + \alpha^2 - \omega^2)^2 + (2\alpha\omega)^2}\left\{(\beta^2 + \alpha^2 - \omega^2)\cos(\omega t) + 2\alpha\omega\sin(\omega t) + \mathrm{e}^{-\alpha t}\left[(\omega^2 - \alpha^2 - \beta^2)\cos(\beta t) - \frac{\alpha}{\beta}(\omega^2 + \alpha^2 + \beta^2)\sin(\beta t)\right]\right\} $
(19) $ \frac{1}{4}[1 - \cos(t - 1)]u(t - 1) $
(20) $ \frac{1}{t}(e^{-9t}-1) $
(1) $ f(0_{+})=1 $, $ f(\infty)=0 $
(2) $ f(0_{+})=0, f(\infty)=0 $
4-6 $ E\left(1+\frac{R}{r}\mathrm{e}^{-\frac{R}{L^{\prime}}}\right)u(t) $
$$ 4-7\quad\frac{R_{2}E}{R_{1}+R_{2}}\left(1-\mathrm{e}^{-\frac{R_{1}+R_{2}}{R_{1}R_{2}C}t}\right)u\left(t\right) $$
$$ 4-8\quad E\Bigg[\frac{R_{2}}{R_{1}+R_{2}}+\bigg(\frac{C_{1}}{C_{1}+C_{2}}-\frac{R_{2}}{R_{1}+R_{2}}\bigg)\mathrm{e}^{-\frac{R_{1}+R_{2}}{R_{1}R_{2}(C_{1}+C_{2})}t}\Bigg]u\left(t\right) $$
4-9 设符号 $ \alpha=\frac{1}{2RC}\omega_{0}=\frac{1}{\sqrt{LC}}\omega_{d}^{2}=\omega_{0}^{2}-\alpha^{2} $
$$ i\left(t\right)=\frac{E}{R}\left[1-\frac{2\alpha}{\omega_{\mathrm{d}}}\mathrm{e}^{-a t}\sin\left(\omega_{\mathrm{d}}t\right)\right]u\left(t\right) $$
4-10 （1）设符号 $ \alpha=\frac{R+R_{0}}{2RR_{0}C} $ $ \omega_{0}=\frac{1}{\sqrt{LC}} $
$ \omega_{d}^{2}=\omega_{0}^{2}-\alpha^{2} $且假设 $ \alpha<\omega_{0} $
$$ h\left(t\right)=\frac{1}{R C}\mathrm{e}^{-\alpha t}\left[\cos\left(\omega_{\mathrm{d}}t\right)-\frac{\alpha}{\omega_{\mathrm{d}}}\sin\left(\omega_{\mathrm{d}}t\right)\right]u\left(t\right) $$
（2）设符号 $ \alpha=\frac{1}{R_{1}R_{2}C_{1}C_{2}} $ $ \beta=R_{1}C_{1}+R_{1}C_{2}+R_{2}C_{2} $
$$ p_{1}=\frac{\alpha}{2}\left(-\beta+\sqrt{\beta^{2}-\frac{4}{\alpha}}\right)\quad p_{2}=\frac{\alpha}{2}\left(-\beta-\sqrt{\beta^{2}-\frac{4}{\alpha}}\right) $$
$$ h\left(t\right)=\delta\left(t\right)+\frac{1}{p_{2}-p_{1}}\left[\left(p_{1}\alpha\beta+\alpha\right)\mathrm{e}^{p_{1}t}-\left(p_{2}\alpha\beta+\alpha\right)\mathrm{e}^{p_{2}t}\right]u\left(t\right) $$
4-11 设 $ \omega_{0}=\frac{1}{\sqrt{LC}} $ $ i(t)=\frac{E}{2L\omega_{0}}\sin(\omega_{0}t)u(t) $
4-12 -0.1te^{-t}u(t)
4-13 (a)
$$ \frac{s}{RC\left(s^{2}+\frac{3}{RC}s+\frac{1}{R^{2}C^{2}}\right)} $$
$$ -\frac{s-\frac{1}{RC}}{s+\frac{1}{RC}} $$
$$ \frac{1}{6} $$
4-14
$$ \frac{R}{2}\left(\frac{1}{L-M}\mathrm{e}^{-\frac{R}{L-M}t}-\frac{1}{L+M}\mathrm{e}^{-\frac{R}{L+M}t}\right)u\left(t\right) $$
(2)
$$ \frac{1}{2}\left(\mathrm{e}^{-\frac{R}{L+M}t}-\mathrm{e}^{-\frac{R}{L-M}t}\right)u(t) $$
$$ 4-\mathbf{15}\quad\frac{E}{2}\mathrm{e}^{-20t}u\left(t\right)-\frac{E}{40T}\left\{\left(1-\mathrm{e}^{-20t}\right)u\left(t\right)-\left[1-\mathrm{e}^{-20\left(t-T\right)}\right]u\left(t-T\right)\right\} $$
4-16 (1) $ H(s)=\frac{K}{s^{2}+(3-K)s+1} $
(2) 当 K=2 时， $ h(t)=\frac{4}{\sqrt{3}}\mathrm{e}^{-\frac{1}{2}t}\sin\left(\frac{\sqrt{3}}{2}t\right)u(t) $
$$ 4-17\quad\frac{2E}{3}\bigg[\delta\big(t\big)+\frac{1}{12}\mathrm{e}^{-\frac{t}{6}}u\big(t\big)\bigg] $$
4-18 $ H(s)=\frac{s^{2}+2s+1-g^{3}}{3s^{3}+10s^{2}+11s+4+2g^{3}} $
4-19 $ \frac{F_{1}(s)}{1-e^{-sT}} $
4-20 (1)
$$ \frac{1}{s(1+\mathrm{e}^{-\frac{sT}{2}})} $$
(2)
$$ \frac{\omega}{s^{2}+\omega^{2}}\frac{1+\mathrm{e}^{-\frac{sT}{2}}}{1-\mathrm{e}^{-\frac{sT}{2}}} $$
4-21
(1)
$$ \sum_{n=0}^{\infty}f(nT)\mathrm{e}^{-nsT} $$
(2)
$$ \frac{1}{1-\mathrm{e}^{-\left(a+s\right)T}} $$
4-23
(a)
$$ H(s)=1+\frac{1}{s+1}\quad i(t)=\delta(t)-\mathrm{e}^{-2t}u(t) $$
(b)
$$ H(s)=2-\frac{1}{s+1}\quad i(t)=\frac{1}{2}\delta(t)+\frac{1}{4}\mathrm{e}^{-\frac{t}{2}}u(t) $$
(c)
$$ H(s)=1+\frac{2s}{4s^{2}+1} $$
$$ i\left(t\right)=\delta(t)+\left[-\frac{1}{2}\mathrm{c o s}\left(\frac{\sqrt{3}}{4}t\right)+\frac{1}{2\sqrt{3}}\mathrm{s i n}\left(\frac{\sqrt{3}}{4}t\right)\right]\mathrm{e}^{-\frac{t}{4}}u(t) $$
(d)
$$ H(s)=\frac{10\left(s^{2}+\frac{s}{20}+\frac{1}{4}\right)}{s(s+5)} $$
$$ i(t)=\frac{1}{10}\left\{\delta(t)+\left[\frac{99}{20}\cos\left(\frac{\sqrt{399}}{40}t\right)-\frac{299}{20\sqrt{399}}\sin\left(\frac{\sqrt{399}}{40}t\right)\right]\mathrm{e}^{-\frac{t}{40}}u(t)\right\} $$
4-24 (a)
$$ H(s)=\frac{C_{1}}{C_{1}+C_{2}}\frac{s+\frac{1}{C_{1}R}}{s+\frac{1}{(C_{1}+C_{2})R}} $$
$$ v_{2}(t)=\frac{C_{1}}{C_{1}+C_{2}}\bigg[\delta(t)+\frac{C_{2}}{C_{1}(C_{1}+C_{2})R}\mathrm{e}^{-\frac{t}{R(C_{1}+C_{2})}}u(t)\bigg] $$
(b)
$$ H(s)=\frac{L_{2}}{L_{1}+L_{2}}\frac{s}{s+\frac{R}{L_{1}+L_{2}}} $$
$$ v_{2}(t)=\frac{L_{2}}{L_{1}+L_{2}}\left[\delta(t)-\frac{R}{L_{1}+L_{2}}\mathrm{e}^{-\frac{R}{L_{1}+L_{2}}t}u(t)\right] $$
(c)
$$ H(s)=\frac{s}{10s^{2}+s+10} $$
$$ v_{2}(t)=\frac{1}{10}\mathrm{e}^{-\frac{t}{20}}\left[\cos\left(\frac{\sqrt{399}}{20}t\right)-\frac{1}{\sqrt{399}}\sin\left(\frac{\sqrt{399}}{20}t\right)\right]u\left(t\right) $$
(d)
$$ H(s)=\frac{0.1s}{s+1} $$
$$ v_{2}(t)=0.1\left[\delta(t)-\mathrm{e}^{-t}u(t)\right] $$
4-25
$$ Z(s)=Z_{1}+\frac{1}{Y_{2}+\frac{1}{Z_{3}+\frac{1}{Y_{4}+\frac{1}{Z_{5}+\frac{1}{Y_{6}+\frac{1}{Z_{7}+\frac{1}{Y_{8}}}}}}}} $$
4-26 (a)
$$ \frac{s^{2}}{s^{2}+3s+1} $$
(b)
$$ \frac{s^{2}}{s^{2}+3s+1} $$
(c)
$$ \frac{1}{(4s^{2}+1)^{2}+(4s^{2}+1)-1} $$
(d)
$$ \frac{s^{3}}{(s^{2}+1)^{2}+(s^{2}+1)-1} $$
$$ \frac{3}{2}\delta(t)+\left(\mathrm{e}^{-2t}+8\mathrm{e}^{3t}\right)u(t) $$
$$ 4-28 $$
$$ \bigg(1-\frac{1}{2}\mathrm{e}^{-2t}\bigg)u\left(t\right) $$
4-29
(1)
$$ H(s)=\frac{5}{s^{2}+s+5} $$
(2) 极点 $ p_{1,2}=\frac{-1\pm j\sqrt{19}}{2} $
(3)
$$ h\left(t\right)=\frac{10}{\sqrt{19}}\mathrm{e}^{-\frac{t}{2}}\sin\left(\frac{\sqrt{19}}{2}t\right)u\left(t\right) $$
$$ g(t)=1-\mathrm{e}^{-\frac{t}{2}}\left[\cos\left(\frac{\sqrt{19}}{2}t\right)+\frac{1}{\sqrt{19}}\sin\left(\frac{\sqrt{19}}{2}t\right)\right]u(t) $$
4-30
$$ \begin{aligned}v_{2}(t)=\frac{5}{2}\Bigg\{&-\frac{48}{37}\mathrm{cos}t+\frac{8}{37}\mathrm{sin}t+\\ \mathrm{e}^{-\frac{t}{16}}&\Bigg[\frac{48}{37}\mathrm{cos}\bigg(\frac{\sqrt{63}}{16}t\bigg)-\frac{80}{37\sqrt{63}}\mathrm{sin}\bigg(\frac{\sqrt{63}}{16}t\bigg)\Bigg]\Bigg\}u(t)\end{aligned} $$
其中前两项为强迫响应，后两项为自由响应
4-31
(1)
$$ H(s)=\frac{s+1}{(s+1)^{2}}=\frac{1}{s+1} $$
(2)
$$ [\begin{array}{r}{v_{2}(0)-i_{1}(0)}\end{array}t\mathrm{e}^{-t}+i_{1}(0)\mathrm{e}^{-t} $$
4-32
(1)
$$ H(s)=\frac{s^{2}+\frac{1}{LC}}{s^{2}+\frac{1}{RC}s+\frac{1}{LC}} $$
(2)
$$ LC=\frac{1}{4} $$
$$ (1-2t)\mathrm{e}^{-2t}u(t) $$
$$ \begin{aligned}4-33\quad&v_{2}(t)=\underbrace{2\mathrm{e}^{-t}}_{ 自由 }+\underbrace{\frac{1}{2}\mathrm{e}^{-3t}}_{ 强迫 }\end{aligned} $$
完全响应即瞬态响应，稳态响应为零
4-35
$$ H(s)=\frac{5(s^{3}+4s^{2}+5s)}{s^{3}+5s^{2}+16s+30} $$
4-36
$$ K_{1}=-\frac{a-3}{3} $$
$$ 4-37 $$
(1)
$$ Z_{1}=-\frac{R}{L}\qquad p_{1,2}=-\frac{R}{2L}\pm\mathrm{j}\sqrt{\frac{1}{L C}-\frac{R^{2}}{4L^{2}}} $$
(2) $ R = 1 \, \Omega $ $ L = \frac{1}{3} \, H $ $ C = \frac{1}{10} \, F $
4-39（a）低通 （b）带通 （c）高通
(d) 带通 (e) 带通 (f) 带阻
(g) 高通 (h) 带通-带阻
4~40 (a) $ H(s)=\frac{L_{1}L_{2}Cs^{3}+L_{1}s}{L_{1}L_{2}Cs^{3}+RC(L_{1}+L_{2})s^{2}+L_{1}s+R} $
(b) $ H(s)=\frac{L_{1}L_{2}C_{1}s^{2}+L_{2}}{L_{1}L_{2}(C_{1}+C_{2})s^{2}+L_{1}+L_{2}} $
$$ H(s)=\frac{L_{2}C_{1}s^{2}}{L_{1}L_{2}C_{1}C_{2}s^{4}+\left(L_{1}C_{1}+L_{2}C_{2}+L_{2}C_{1}\right)s^{2}+1} $$
4-41 $ H(s)=\frac{s^{2}-s+1}{s^{2}+s+1} $，是全通
4-42 (a) 是最小相移，其他都是非最小相移
4-43 $ H(s)=\frac{s-\frac{1}{RC}}{s+\frac{1}{RC}} $ ，是全通
4 - 44 $ H(s) = -\frac{s^{2} - \frac{1}{R_{1}C_{1}R_{2}C_{2}}}{\left(s + \frac{1}{R_{1}C_{1}}\right)\left(s + \frac{1}{R_{2}C_{2}}\right)} $
当 $ R_{1}C_{1}=R_{2}C_{2} $ 时构成全通
4-45 (1) $ H(s)=\frac{ks}{s^{2}+(4-k)s+4} $
(2) $ k \leq 4 $
(3) $ h(t) = 4\cos(2t)u(t) $
4-46 (1) $ H(s)=\frac{k}{s^{2}+(3-k)s+1} $
(2) $ k \leqslant 3 $,稳定
4 - 47 $ H(s) = \frac{K}{1 - KF} = \frac{\beta}{CR_{i}} \left[ \frac{s}{s^{2} + \left( \frac{G}{C} - \frac{\beta F}{R_{i} C} \right) s + \frac{1}{LC} } \right] $
当 $ G = \frac{\beta F}{R_{i}} $ 时极点之实部等于零
4 - 48 $ H(s)=\frac{As^{2}}{s^{2}+\left(\frac{C_{1}+C_{2}}{RC_{1}C_{2}}+\frac{1-A}{R_{2}C_{1}}\right)s+\frac{1}{R_{1}R_{2}C_{1}C_{2}}} $
当满足 $ A \leqslant 1 + \frac{R_{2}}{R_{1}} + \frac{R_{2}C_{1}}{R_{1}C_{2}} $
4 - 49 $ H(s)=\frac{RM}{L^{2}-M^{2}}\frac{s}{\left(s+\frac{R}{L-M}\right)\left(s+\frac{R}{L+M}\right)} $