Circuit Theory

Basics

PASSIVE SIGN CONVENTION

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In an electrical circuit, some elements supply power and some elements consume it.

In the above circuit, the battery supplies power and the lightbulb consumes it.

In terms of V and I:

As we know, $$p (t)=\frac {dW}{dt} $$Rewritting this as $$p(t)=\frac {dW}{dq}×\frac{dq}{dt}$$Since $v (t)=dW/dq $ and $i (t)=dq/dt $$$p (t)=v (t)\;i (t) $$ 

Horsepower:

This term was used by James Watt for marketing his improvised steam engine, as it was the easy way to explain people at that time.
“This engine can do the work of $10$ horses = $10$ HP engine”$$1\;HP=746 \;Watts$$

Nowadays we are using this term to measure the output power of engines, motors, etc.

Power vs Energy:

If someone says, “Hey man… I can solve this math problem within $10$ minutes” is nothing but the rate of work-done or power. 

The above statement doesn’t mean… he can do $144$ problems within $24$ hours or $4320$ problems within a month or $1,576,800$ problems within a year… Right??? 

Let us say, on an average he can work for $6$ hours per day, $5$ days per week… so $6×6×5×52$.. less than or equal to $9360$ problems per year. This is what his capability or energy. 

Hence,

Power = How fast he can do work

Energy = How much amount of work he can actually do 

From the definition, we can say “How much work one can do = How fast one can do that work × How much time he/she can work” 

  • In other words, “Energy = Power × time period”
  • If power is constant over the time period, $$E\;=\;P×T$$
  • If power varies with time, then$$E= \int_0^T p(t) dt $$

Energy:

Energy is nothing but the capability to do work.

“Energy = Power × time period”

  • If power is constant over the time period, $$E\;=\;P×T$$
  • If power varies with time, then

$$E= \int_0^T p(t) dt $$