To determine Find the time domain expression of i o for time t > 0 . Explanation Given data: Refer to given circuit in the textbook. The switch is in position ‘a’ for long time and moved to position ‘b’ at time t = 0 . Formula used: Write a general expression to calculate the impedance of a resistor in s -domain. Z R = R (1) Here, R is the value of resistance. Write a general expression to calculate the impedance of an inductor in s -domain. Z L = s L (2) Here, L is the value of inductance. Write a general expression to calculate the impedance of a capacitor in s -domain. Z C = 1 s C (3) Here, C is the value of capacitance. Calculation: The given circuit is redrawn as shown in Figure 1. For a DC circuit, at steady state condition for time t < 0 the switch is in position ‘a’ and the capacitor acts like open circuit and the inductor acts like short circuit. Now, the Figure 1 is reduced as shown in Figure 2. Refer to Figure 2, the capacitor is connected in parallel with the resistor R 3 . For the parallel connection, the voltage is same. Apply Kirchhoff’s current law in Figure 2. v o ( 0 − ) − 500 V 5 Ω + v o ( 0 − ) 25 Ω + v o ( 0 − ) 100 Ω = 0 v o ( 0 − ) 5 Ω − 500 V 5 Ω + v o ( 0 − ) 25 Ω + v o ( 0 − ) 100 Ω = 0 v o ( 0 − ) [ 1 5 Ω + 1 25 Ω + 1 100 Ω ] = 500 V 5 Ω v o ( 0 − ) [ 20 + 4 + 1 100 Ω ] = 500 V 5 Ω v o ( 0 − ) [ 25 100 Ω ] = 500 V 5 Ω Simplify the above equation to find v o ( 0 − ) . v o ( 0 − ) = ( 500 V 5 Ω ) ( 100 Ω 25 ) = 400 V Refer to Figure 2, the series connected inductor and resistor is connected in parallel with the resistor R 3 . For the parallel connection, the voltage is same and for the series connection current is same. The current through inductor is calculated as follows: i L ( 0 − ) = v o ( 0 − ) 25 Ω Substitute 400 V for v o ( 0 − ) in above equation to find i L ( 0 − ) . i L ( 0 − ) = 400 V 25 Ω = 16 A The capacitor voltage and inductor current is always continuous so that, v ( 0 ) = v o ( 0 − ) = v o ( 0 + ) = 400 V i ( 0 ) = i L ( 0 − ) = i L ( 0 + ) = 16 A For time t > 0 , the switch is moved to position ‘b’ and the Figure 1 is reduced as shown in Figure 3. Use equation (1) to find Z R 1 . Z R 1 = 5 Use equation (1) to find Z R 2 . Z R 2 = 25 Use equation (1) to find Z R 3 . Z R 3 = 100 Substitute 25 H for L in equation (2) to find Z L . Z L = s ( 25 H ) = 25 s Substitute 10 mF for C in equation (3) to find Z C . Z C = 1 s ( 10 mF ) = 1 s ( 10 × 10 − 3 F ) { ∵ 1 m = 10 − 3 } = 100 s Convert the Figure 3 into s -domain with initial conditions as shown in Figure 4. Redraw the Figure 4 as shown in Figure 5 with the appropriate values. Apply Kirchhoff’s current law in Figure 5. V o + 400 25 + 25 s + V o 100 + V o − 400 s ( 100 s ) = 0 V o 25 + 25 s + 400 25 + 25 s + V o 100 + V o ( 100 s ) − ( 400 s ) ( 100 s ) = 0 V o 25 + 25 s + 400 25 + 25 s + V o 100 + s V o 100 − 400 100 = 0 V o 25 + 25 s + V o 100 + s V o 100 = 400 100 − 400 25 + 25 s Simplify the above equation as follows: V o [ 1 25 ( s + 1 ) + 1 100 + s 100 ] = 4 − 400 25 + 25 s V o [ 4 + s + 1 + s ( s + 1 ) 100 ( s + 1 ) ] = 4 ( 25 + 25 s ) − 400 25 + 25 s V o [ 4 + s + 1 + s 2 + s 100 ( s + 1 ) ] = 100 + 100 s − 400 25 ( s + 1 ) V o [ s 2 + 2 s + 5 100 ( s + 1 ) ] = 100 s − 300 25 ( s + 1 ) Simplify the above equation to find V o . V o = ( 100 s − 300 25 ( s + 1 ) ) ( 100 ( s + 1 ) s 2 + 2 s + 5 ) = ( 100 ( s − 3 ) 25 ( s + 1 ) ) ( 100 ( s + 1 ) s 2 + 2 s + 5 ) = 400 ( s − 3 ) s 2 + 2 s + 5 Refer to Figure 5, the current I o is calculated as follows: I o = V o − 400 s ( 100 s ) Substitute 400 ( s − 3 ) s 2 + 2 s + 5 for V o in above equation to find I o

ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

11th Edition

ISBN: 9780134894300

Author: NILSSON

Publisher: PEARSON

See similar textbooks

Concept explainers

Current Division Method

The generalized current divider formula is given by the following relation below,

Videos

Question

Chapter 13, Problem 17P

To determine

Find the time domain expression of io for time t>0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 16P

Chapter 13, Problem 18P

Students have asked these similar questions

Do NOT WANT AI. need diagram fully labeled please

Calculate the current magnitude in the coils e1, e2 of theMagnetic circuit, if:ɸa = 3.00 x 10^-3 Wb, φb = 0.80 x 10^-3 Wb, ɸc = 2.20 x 10^-3 Wb L ab = 0.10 m,A ab = 5.0 cm^2L afeb = L acdb = 0.40 mA afeb = A acdb = 20 cm^2 MATERIAL CHARACTERISTICSH (At/m) 240 350 530 1300 5000 9000B (T) 0.7 0.9 1.1 1.3 1.5 1.6

A toroid magnetic circuit is composed of three sections A, B and C, thesection C has an air gap, section A has an 850 round coil thatconsumes a current of 1.2 A. the physical and magnetic properties of each sectionare: Section A: Length = 80 mm, Cross section = 120 mm^2, μr = 400 Section B: Length = 60 mm, Cross section = 40 mm^2, μr = 250 Section C: Length = 50 mm, Cross section = 200 mm^2, μr = 600 Gap: Length = 1 mm, Cross section = 40 mm^2, μr = 1 Calculate:The magnetic field density in each of the sections

Chapter 13 Solutions

ELECTRIC CIRCUITS-W/MASTERINGENGINEERING

Ch. 13.2 - The parallel circuit in Example 13.1 is placed in...Ch. 13.3 - Prob. 2AP Ch. 13.3 - The energy stored in the circuit shown is zero at...Ch. 13.3 - The dc current and dc voltage sources are applied...Ch. 13.3 - Prob. 6AP Ch. 13.3 - Using the results from Example 13.7 for the...Ch. 13.3 - The energy stored in the circuit shown is zero at...Ch. 13.4 - Derive the numerical expression for the transfer...Ch. 13.5 - Find (a) the unit step and (b) the unit impulse...Ch. 13.5 - The unit impulse response of a circuit is υo(t) =...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.

Find the time domain expression of i o for time t &gt; 0 .

Concept explainers

Videos

Find the time domain expression of io for time t>0.

Want to see the full answer?

Chapter 13 Solutions

Find the time domain expression of i o for time t > 0 .