Download presentation
Presentation is loading. Please wait.
Published byAldous Hood Modified over 9 years ago
1
Capacitors in a circuit 1
2
Example Let’s find the current through the following capacitor with v(t)=5 cos(2000 t)V. C = 30 F What if v(t) = 5A? = 0 (A capacitor “looks like” an open circuit to DC) 2
3
Inductors Inductor is a passive circuit element which stores energy in its magnetic field. It consists of a coil of wire wrapped around a core. 3
4
Inductance Inductors are defined by their inductance (L) measured in henrys (H), where is the magnetic flux through the core of the inductor. 4
5
Inductors in a circuit = L i 5
6
Example Let’s find the inductor voltage assuming 1)i(t) = 5 A, then 2)i(t) = 5 cos(2000 t) A. L = 3 mH 1) = 0 (An inductor “looks like” a short circuit to DC) 2) 6
7
20 mH 10 50 cos (100 t) V Let’s find v in this circuit. Apply KVL: You can either solve the differential equation, or transform the problem from the time domain to the frequency domain. Then the differential equation becomes a linear algebraic equation. 7
8
Time Domain Representation Frequency Domain (Phasor Representation) Circuit Component Sources M cos ( t + ) V or AM/ V or A Resistors R R/0 InductorsL H j L = L/90 CapacitorsC F 1/(j C) = 1/( C)/-90 8 Time Domain: Frequency Domain: Re Im Argand Diagram M Transforming to the frequency domain is done using phasors. Start by writing each circuit quantity as a phasor.
9
In the frequency domain, R, L, and C values are expressed in ohms. These ohmic values are called “impedances.” Impedance (Z) = Resistance (R) + j Reactance (X) Z = R + jX Ohm’s Law becomes: V = I Z 9
10
20 mH 10 50 cos (100 t) V Let’s find i. V max Apply KVL: 50/0 V 10/0 j L = j (100)(20 10 -3 ) j 2 = 2/90 50/0 = (10/0 ) i + (2/90 ) i Solve: 50/0 = (10/0 + 2/90 ) i 50/0 = (10.2/11.3 ) i i = = 4.9/-11.3 Transform back to time domain: i (t) = 4.9 cos (100 t – 11.3 ) A 10 Remember: j = 1/90
11
20 mH 10 50 cos (100 t) V Now let’s find v. Apply Ohm’s Law: 50/0 V 10/0 j L = j (100)(20 10 -3 ) j 2 = 2/90 i (t) = 4.9 cos (100 t – 11.3 ) A i = 4.9/-11.3 v = i Z = (4.9/-11.3 ) (2/90 ) = 9.8/78.7 V Transform back to time domain: v (t) = 9.8 cos (100 t + 78.7 ) V 11
12
Let’s find v and i. 10 k 10 F 50 cos (10t + 30 ) V 50/30 V 10 4 /0 1/(j C) = 1/[(10)(10 10 -6 )/-90 ] = 10 4 /-90 Apply KVL: 50/30 = 10 4 /0 i + 10 4 /-90 i Solve: 50/30 = (10 4 /0 + 10 4 /-90 ) i 50/30 = (1.414 10 4 /-45 ) i i = 3.54/75 mA Ohm’s Law: v = (3.54 10 -3 /75 ) (10 4 /-90 ) = 35.4/-15 V i (t) = 3.54 cos (10 t + 75 ) mA v (t) = 35.4 cos (10 t – 15 ) V 12
13
13 v s (t) = 50 cos (10 t + 30 ) V i (t) = 3.54 cos (10 t + 75 ) mA v c (t) = 35.4 cos (10 t – 15 ) V v r (t) = 35.4 cos (10 t + 75 ) V Time Domain v s = 50/30 V i = 3.54/75 mA v c = 35.4/– 15 V v r = 35.4/75 V Frequency Domain Re Im
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.