Question 6.3.2: (a) Determine the transfer function Vo(s)/Vs(s) of the circu......
(a) Determine the transfer function V_{o}(s)/V_{s}(s) of the circuit shown in Figure 6.3.1. (b) Use a block-diagram representation of the circuit to show how the output voltage v_{o} affects the internal voltage v_{1}.
a. The energy in this circuit is stored in the two capacitors. Because the energy stored in a capacitor is expressed by Cv²/2, appropriate choices for the state variables are the voltages v_{1} and v_{o}. The capacitance relations are
v_{o}={\frac{1}{C}}\int i_{3}\qquad v_{1}={\frac{1}{C}}\int i_{2}which give the state equations
{\frac{d v_{o}}{d t}}={\frac{i_{3}}{C}} (1)
{\frac{d v_{1}}{d t}}={\frac{i_{2}}{C}} (2)
For the right-hand loop,
v_{1}-R i_{3}-v_{o}=0Thus
i_{3}={\frac{v_{1}-v_{o}}{R}} (3)
and equation (1) becomes
{\frac{d v_{o}}{d t}}={\frac{1}{R C}}(v_{1}-v_{o}) (4)
For the left-hand loop,
v_{s}-R i_{1}-v_{1}=0which gives
i_{1}={\frac{v_{s}-v_{1}}{R}} (5)
From conservation of charge and equations (3) and (5),
i_{2}=i_{1}-i_{3}={\frac{v_{s}-v_{1}}{R}}-{\frac{v_{1}-v_{o}}{R}}={\frac{1}{R}}(v_{s}-2v_{1}+v_{o}) (6)
Substituting this into equation (2) gives
{\frac{d v_{1}}{d t}}={\frac{1}{R C}}(v_{s}-2v_{1}+v_{o}) (7)
Equations (4) and (7) are the state equations. To obtain the transfer function V_{o}(s)/V_{s}(s),, transform these equations for zero initial conditions to obtain
s V_{o}(s)={\frac{1}{R C}}[V_{1}(s)-V_{o}(s)]s V_{1}(s)=\frac{1}{R C}[V_{s}(s)-2V_{1}(s)+V_{o}(s)]
Eliminating V_{{1}}(s) from these two equations gives
V_{o}(s)={\frac{1}{R^{2}C^{2}s^{2}+3R C s+1}}V_{s}(s)So the transfer function is
\frac{V_{o}(s)}{V_{s}(s)}=\frac{1}{R^{2}C^{2}s^{2}+3R C s+1}b. In the block diagram in Figure 6.3.2a, the left-hand inner loop is based on equations (2) and (6). The right-hand inner loop is based on equations (1) and (3). The diagram can be simplified by reducing each inner loop to a single block, as shown in part (b) of the figure.
This simpler diagram shows how the output voltage v_{o} affects the inner voltage v_{1} through the outer feedback loop.
