Transfer functions and block diagrams
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| block diagram | sơ đồ khối |
| characteristic equation/ˌkærɪktəˈrɪstɪk ɪˈkweɪʒn/ | phương trình đặc trưng |
Three rules of block arithmetic
- A block diagram 方框图 multiplies along series paths, adds on parallel paths, and closes loops with $T = \dfrac{G}{1+GH}$.
- One line derives the feedback formula: $e = r - Hy$ and $y = Ge$ give $y(1+GH) = Gr$. The equation $1 + GH = 0$ is the characteristic equation 特征方程. The characteristic equation is the closed-loop denominator set to zero; its roots are the closed-loop poles.
$G_1 = \dfrac{1}{s+1}$, $G_2 = \dfrac{1}{s+2}$, $K = 6$, unity feedback: $T = \dfrac{6}{(s+1)(s+2)+6} = \dfrac{6}{s^{2}+3s+8}$, so $\omega_n = 2.83$ rad/s and $\zeta = 0.530$.
{exercise:uni-ee1042-series}
G = 10/[s(s+2)] with sensor feedback H = 0.2. The closed-loop denominator is —
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1 + GH = 0 gives s(s+2) + 10×0.2 = s² + 2s + 2, i.e. ζ = 0.707. The sensor gain scales the loop, not the plant.
Reduce inner loops first
- Collapse the innermost loop, multiply the series blocks, then close the next loop out: the discipline keeps the denominator visible at every step.
- Inner loop $\dfrac{2}{s}$ with feedback 0.5 becomes $\dfrac{2/s}{1+2/s\cdot0.5} = \dfrac{2}{s+1}$; in series with $\dfrac{1}{s+1}$ it gives $G = \dfrac{2}{(s+1)^{2}}$.
Close the outer unity loop: $T = \dfrac{2}{s^{2}+2s+3}$. Read it off: $\omega_n = \sqrt{3} = 1.73$ rad/s, $\zeta = 0.577$. A tangled diagram became two numbers.
{exercise:uni-ee1042-nested}
An inner loop 2/s with feedback 0.5, in series with 1/(s+1), inside a unity-feedback outer loop closes to —
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Inner: 2/(s+1). Series: 2/(s+1)². Outer unity loop: divide by 1 + 2/(s+1)², giving 2/(s² + 2s + 3) with ω_n = 1.73 and ζ = 0.577.
Where reductions go wrong
- The return path carries $GH$, not $G$: closing as $\dfrac{G}{1+G}$ when $H \ne 1$ quietly changes the loop.
- The feedback summer subtracts: $e = r - Hy$. Adding instead flips the characteristic equation to $1 - GH = 0$ and every stability claim with it.
- Mason's gain rule does all of this in one formula; reach for it only when block algebra genuinely tangles.
{exercise:uni-ee1042-sign}
A summer that ADDS Hy instead of subtracting it makes the characteristic equation —
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With e = r + Hy the closed loop is G/(1 − GH): positive feedback. The poles move to where 1 − GH = 0, which usually means instability.