The Laws of Circuit - you can learn and practice by just reading

copyright. Charles Kim 2006

 

Before and After t=0 Analyses

 

t<0 analysis and t>0 analysis: When switching is involved in a circuit with R, C, and/or L, with DC source(s), we analyze the circuit behavior before the switching and after the switching event. Since the switching event is usually assumed to be occurred at time t=0, the analysis of the circuit before the switching is normally called t<0 analysis, and the one after switching as t>0 analysis. In these analyses, one thing we have to know (and assume) is that (if you read your problem statement very carefully, it usually says that the circuit has been in such such state for long time before the switching occurs at t=0, then it asks you to find either v(t) or i(t) for t>0 ). A t<0 analysis is 'steady state' analysis and t>0 analysis is 'transient analysis'. The point is that before t<0, the circuit has been there for long time. How long is a long time? Well, you give me a time and my 'long time' is slightly longer than your time. That means really really long time. Actually, in practical sense, "long time" means that capacitors in the circuit are fully charged (this means there is no current flowing through the capacitors. Why? When capacitors are charged, they are charged until the capacitor voltage is the same as the source voltage. Then, current cannot flow since there is no potential difference between the capacitor and the source. Or you can say this way: the current through a capacitor is determined by Capacitance (C) times the derivative of voltage over time., i.e., i = C dv/dt. Since the source is DC voltage/current, in steady state, dv/dt becomes 0, because whatever V is it is a constant. ) If there is no current through an element, that element can be modeled as an open circuit. This means in "steady state" or t<0 analysis, capacitor can be modeled as open circuit. How'bout an inductor? In the inductor, t<0 period, it is fully charged until it cannot be charged. In inductor current is charged in magnetic energy form. When it's fully charged, the voltage across the inductor is zero (that's why there is no charge action). This can be similarly argued using the voltage equation across the inductor: V=L di/dt. In DC source circuit, the current i is a constant value, so di/dt becomes zero. Therefore voltage across the inductor is zero. Similarly again, an element whose voltage across is zero can be modeled as a short circuit. Why? No voltage drop means no resistance by Ohm's Law. No resistance means R=0, which means two nodes are just connected by a wire. Summary of t<0 analysis: (1) steady-state is assumed by a 'long time' phrase; (2) C is replaced by an open circuit (remember, voltage can be developed across an open circuit, though. A 12V battery sitting in your garage, with two terminals are open, as you know, the voltage between two terminals is 12V if the battery not very old. So in the t<0 analysis, do not overlook the voltage across the two terminals of the capacitor. This voltage across the capacitor is the voltage maintained until the switching event at t=0 and it is called as "initial voltage" Vc(0) ).  Also, (3) L is replaced by a short circuit (Remember, current flows through the short circuit. This current amount is the current flowing through the inductor until just before the switching event at t=0. This current at t=0 is called "initial current" or iL(0)).

Now t>0 analysis. As I said above, after switching event there is no "long time" phrase. Instead, it asks just v(t) for t>0. This actually means "v(t) for just a few milliseconds after t=0". In other words, the question for t>0 is just very short transition or transient period right after switching. If the question were "find v(t) after a long time", the question is another steady state analysis after the switching event. Then you'd apply the same approach for t<0 analysis. But that's not the case here. I am saying again, t>0 analysis is NOT steady state analysis, but is transient analysis which focuses only the first a few millisecondw after the switching event. Think about your life. Before you move to this University, your schedule had been quite a routine. Wake-up, school, dinner, homework, TV, chatting, sleep, then wake-up again. I would say your life was quite a 'steady' and in a 'steady state'. Now after a few years at the university in the new town, you are doing the same thing only without your parents' incessant check-ups. Well, you'd say you're in a steady-state on campus. But I guess you still remember your first orientation in a huge setting and the first night at a hotel in a strange town, folloed by a good-bye to your parents, new roommates, address change, big load of class material, homeworks and exams, no excuses, etc. And I can safely guess you'd say the experience as passing through hell fire. But just after a semester, you're snuggled and your phone bill is much lowered and now you don't have to grab a flight in the middle of the first semester in order to see if your bed is still in tact. So the first semester can be called 'transient' period. To end the long story, t>0 analysis is myopic, short-sighted, short-lived analysis. In the t>0 analysis, a target voltage or current is a time varying entity, whether you have a DC source or not. That's why voltage across even a resistor is labeled as v(t), not just v. Remember in the transient period, everything is time varying. Therefore in a new circuit (altered by the switching action), you express the target variable (v(t) or i(t)) using KCL or KVL. The resultant equation is either first order or second order differential equation. And they need initial conditions: initial voltage or current. They are the ones already found from t<0 analysis.

 

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