Products
Rewards 
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Please sign up to our reward program to support us in return and take advantage of the incredible listed offers.

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY 
BUSINESS MANAGER

Advertise your business, and reach millions of students around the world.

HOLOOLY 
TABLES

All the data tables that you may search for.

HOLOOLY 
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY 
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY 
HELP DESK

Need Help? We got you covered.

Chapter 7

Q. 7.EX.3

Bridged Tee Circuit in State-Variable Form
Determine the state-space equations for the circuit shown in Fig. 2.25.

Step-by-Step

Verified Solution

In order to write the equations in the state-variable form (i.e., a set of simultaneous first-order differential equations), we select the capacitor voltages v_1 \text{ and } v_2 as the state elements (i.e., x = [v_1 v_2 ]^T ) and v_i as the input (i.e., u = v_i ). Here v_1 = v_2, v_2 = v_1 − v_3, and still v_1 = v_i. Thus v_1 = v_i, v_2 = v_1 , and v_3 = v_i − v_2 . In terms of v_1 and v_2 , Eq. (2.34) is

\frac{v_1 − v_i}{R_1} + \frac{v_1 − (v_i − v_2 )}{R_2} + C_1 \frac{dv_1}{dt} = 0.

-\frac{v_{1}-v_{2}}{R_{1}}+\frac{v_{2}-v_{3}}{R_{2}}+C_{1} \frac{d v_{2}}{d t}=0 ,           (2.34)

Rearranging this equation into standard form, we get

\frac{dv_1}{dt} = − \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) v_1 − \frac{1}{C_1} \left( \frac{1}{R_2} \right) v_2 + \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) v_i.                   (7.6)

In terms of v_1 and v_2 , Eq. (2.35) is

\frac{v_i − v_2 − v_1}{R_2} + C_2 \frac{d}{dt}(v_i − v_2 − v_i) = 0.

\frac{v_{3}-v_{2}}{R_{2}}+C_{2} \frac{d\left(v_{3}-v_{1}\right)}{d t}=0               (2.35)

In standard form, the equation is

\frac{dv_2}{dt} = − \frac{v_1}{C_2R_2} − \frac{v_2}{C_2R_2} + \frac{v_i}{C_2R_2}.             (7.7)

Equations (2.34)–(2.35) are entirely equivalent to the state-variable form, Eqs. (7.6) and (7.7), in describing the circuit. The standard matrix definitions are

\pmb F = \left[\begin{matrix} -\frac{1}{C_1}\left(\frac{1}{R_1}+ \frac{1}{R_2} \right) & -\frac{1}{C_1} \left( \frac{1}{R_2}\right)\\ -\frac{1}{C_2R_2} & -\frac{1}{C_2R_2} \end{matrix} \right] , \\ \pmb G = \left[\begin{matrix} \frac{1}{C_1}\left( \frac{1}{R_1} + \frac{1}{R_2}\right) \\ \frac{1}{C_2R_2} \end{matrix} \right] , \\ \pmb H = \left[\begin{matrix} 0 & -1 \end{matrix} \right] , \ \ \ J =1 .