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The first thing I recommend is to rewrite the equation in a low-entropy format in which the numerator and the denominator are unitless. In your case, as underlined by a concerned citizen, the presence of the zero at the origin does not change the exercise. A second-order polynomial can be put under the following normalized form: \$D(s)=1+b_1s+b_2s^2\$. Then, ...


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It is incorrect to say that the loop-gain must be stable in order for the complete, closed loop system, to be stable. Note: in the system \$ F = \frac{G}{1+GH} \$, loop gain is \$GH\$, not \$1+GH\$, as you stated. It may be true that instability in the path \$GH\$ could cause closed-loop instability (for example, if \$H\$ itself is an ill-behaved entity), ...


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As mentioned in the book, We have to perform polynomial division. When we divide a polynomial of order \$n\$ with another of order \$n\$, the quotient is a constant and the remainder is a poly nominal of order \$n-1\$. So, $$ \color{green}{\text{quotient}} \times \color{blue}{\text{divisor}} + \color{red}{\text{remainder}} = \color{green}{b_n} \cdot \color{...


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The "aggregate", "net", "combined", or "compound" transfer function of cascaded amplifier stages is sometimes quoted to be the simple product of the transfer functions of the individual stages, but this is only true if every stage implements a simple direct proportionality, like \$ V_{OUT} = -3 \times V_{IN} \$. Your ...


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It's a bit of a tricky case. You actually have two inputs: \$V_{in} \$ and \$V_p \$, the voltage source connected to + in your second op-amp. Both these sources affect \$V_o \$ so to find the effect from each source you need to use superposition. simulate this circuit – Schematic created using CircuitLab Case 1: \$V_p =0\$ The node equations become $$\...


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The answer of your question can be easily found in any authentic book of Control Systems. Nevertheless, there is a direct expression to determine the angle of breakaway/in which is: theta = (pi)/n where n is the total number of root locus branches approaching/leaving the point(breakaway/in point). If there are 2 branches, let's say leaving from the breakaway ...


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Quick Overview I diagrammed things out about like the following (which includes some sympy script needed to perform a full solution): The automated solution is horribly long and not at all useful. So this implies a different approach is needed. I don't have the time to go through all of how I'd approach this. Not right now. But I may be able to outline what ...


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Log time ago I try to analyze this circuit. For low frequency (bass) we have a "simple" case. simulate this circuit – Schematic created using CircuitLab And raw (high entropy) transfer function will look like this: $$H(s) = -\frac{R_2 + P_{3b} + C_2 R_2 (P_{3a} + P_{3b})s }{R_1 + P_{3a} + C_2 R_1 (P_{3a} + P_{3b}) s}$$ And for the max bass ...


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