sibt.nsw.edu.au navitas.com Chapter 8 Basic RL and RC Circuits (Part 1) ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com KCL and KVL can be applied to circuits containing capacitors and inductors. Consider the following circuit, where is an independent but time varying voltage source. Assuming mesh currents 1 and 2 in the left and right mesh, respectively: Left mesh : 201 + = or 201 + 1 1 2 = Right Mesh 102 + = or 102 + 1 = 1 (2 1) Introduction ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Similarly, applying KCL on the node above the capacitor: 1 20 + 1 + 1 2 10 = 0 And then, the top right node: 2 1 10 + 1 = 0 Introduction ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com The solution of these integro-differential equations will give us detailed insight about this circuit. Until we know how to solve them, we can not analyze circuits involving capacitors, inductors and time varying sources. However, when independent sources are active in the circuit, capacitors and inductors store energy. In this situation, circuit behaviour is called Forced Response or Steady State Response. When independent sources are switched off, capacitors and inductors can provide energy over a limited period of time. The behaviour of source free circuit is called Natural Response or Transient Response. Introduction ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Consider a simple series RL circuit which was connected to a source for a long time. The source is removed from the circuit at = 0. In the source free circuit, magnetic field in the inductor will continue pushing the current for a while. But, it will gradually slow down and eventually stops after some time. Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Assuming the current through the inductor at = 0 is 0. Applying KVL: + = 0 + = 0 + = 0 Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Here, we are interested in finding a solution for () such that it satisfies KVL equation and 0 = 0. Separating () in KVL equation: = 0 () = 0 ln ln 0 = Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis ln 0 = 0 = = 0 The above solution satisfies KVL equation as well yields 0 = 0. Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis = 0 Current in source free RL circuit starts from 0 and decays exponentially with time. 0 mainly depends upon the source connected to the circuit at t < 0 and is not really a function of R and L. However, how quickly or slowly current decays from its initial value depends upon R and L values. Hence, our transient analysis will focus on R and L values. Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Plotting 0 versus . 0 = / Let = = time constant 0 = When t = 0, 0 = 1 Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis At t = , 0 = 1 = 0.3679 At t = 2, 0 = 2 = 0.135 At t = 3, 0 = 3 = 0.049 At t = 4, 0 = 4 = 0.018 At t = 5, 0 = 5 = 0.006 Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis of RL Circuit Hence, theoretically, current will continue to flow until = ∞ . But, practically, current will become zero at = 5. For small values of = , current will decay quickly. For large values of = , current will decay slowly. RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Power and Energy In a source free RL circuit, power delivered by inductor is dissipated by resistor. = 2 = 0 2 2 Total energy supplied by the inductor can be calculated by integrating power: = 0 ∞ = 0 2 0 ∞ 2 = 0 2 2 0 1 = 1 2 0 2 Source Free RL Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Consider a simple series RC circuit which was connected to a source for a long time. The source is removed from the circuit at = 0. In the source free circuit, charge stored in the capacitor will provide the current for a while. But, it will gradually slow down and eventually stops after some time. Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Assuming the voltage across capacitor at = 0 is 0. Applying KCL: + = 0 + = 0 + 1 = 0 Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Here, we are interested in finding a solution for () such that it satisfies KCL equation and 0 = 0. Separating () in KCL equation: = 1 0 () = 1 0 ln ln 0 = 1 Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis ln 0 = 1 0 = 1 = 0 1 The above solution satisfies KCL equation as well yields 0 = 0. Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis = 0 1 Voltage in source free RC circuit starts from 0 and decays exponentially with time. 0 mainly depends upon the source connected to the circuit at t < 0 and is not really a function of R and C. However, how quickly or slowly voltage decays from its initial value depends upon R and C values. Hence, our transient analysis will focus on R and C values. Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Plotting 0 versus . 0 = Let = = time constant 0 = When t = 0, 0 = 1 Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis At t = , 0 = 1 = 0.3679 At t = 2, 0 = 2 = 0.135 At t = 3, 0 = 3 = 0.049 At t = 4, 0 = 4 = 0.018 At t = 5, 0 = 5 = 0.006 Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Transient Analysis Hence, theoretically, voltage will continue to stay until = ∞ . But, practically, volatge will become zero at = 5. For small values of = , voltage will decay quickly. For large values of = , voltage will decay slowly. Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com Power and Energy In a source free RC circuit, power delivered by inductor is dissipated by resistor. = 2 = 0 2 2 Total energy supplied by the inductor can be calculated by integrating power: = 0 ∞ = 0 2 0 ∞ 2 = 0 2 2 0 1 = 1 2 0 2 Source Free RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin sibt.nsw.edu.au navitas.com If an RL circuit has multiple resistors and inductors which can be solved using series and parallel combinations, the time constant becomes: = Similarly, if an RC circuit has multiple resistors and capacitors which can be solved using series and parallel combinations, the time constant becomes: = General RL and RC Circuits ENGN 110 Principles of Electric Circuits Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
