In the circuit of Problem 88, at what time after switch S is closed is the voltage across the co

In the circuit of Problem 88, at what time after switch S is...

Key Concepts

Time Constant

The time constant, denoted by ? (tau), is the product of the resistance (R) and capacitance (C) in the circuit. It is a critical measure that defines the rate at which the circuit transitions from its initial value to its final steady-state value, with the capacitor typically reaching about 63.2% of its final voltage after one time constant.

Exponential Charging Behavior

In RC circuits, when a step input is applied, the voltage across the capacitor increases according to the equation V(t) = V_final * (1 - e^(-t/?)). To determine the time at which the capacitor's voltage reaches a specific fraction (for example, 50% of its final value), one sets the equation equal to that fraction and solves for t, thereby highlighting the significance of the natural logarithm in the analysis.

RC Circuit Transient Analysis

This concept involves studying how voltages and currents in an RC circuit change after a sudden disturbance, like the closing of a switch. The transient response is characterized by gradual changes that follow exponential behavior, distinguishing it from the steady state, which is reached after a sufficient lapse of time.

Transcript

00:01 To calculate time after which voltage drop across the combination of three capacitors will be 50 % of its final state.

00:09 Let's do this.

00:10 So as we know, as we ask to calculate for the combination of capacitors, we first going to calculate equivalent capacitance and equivalent resistance.

00:24 So let's start with the resistance because it's straightforward.

00:27 It's just a sum of r1 and r2 because the x2.

00:32 Connected in theory so it equals to 25 plus 33 oms which is 58 oms now let's calculate equivalent capacitance so equivalent capacity 1 and c2 are connected to imperial therefore the equivalent capacitance is sum of c1 and c2 which is 35 micro ferrets and c1 and c2 and c2 and c 2 is c1 is connected in c3, therefore 1 over c1 to c3, which is 1 over c1 is a sum of 1 over c1 2 and 1 over c3 or c12 plus c3 over c12 multiplied by c3.

01:46 So therefore equivalent capacitance which is labeled that c equals to c12 multiplied by c3, divided by c12 plus c3.

01:59 So let's calculate this number.

02:02 It equals to 35 microferrets multiplied by c3, which is 46 microferrets.

02:15 So then it's divided by 35 microferrets plus 46 microferrets.

02:21 So let's calculate this number.

02:32 So it equals to 19.

02:53 0 .8765 microferrets.

02:56 Yeah, we're going to round and calculate the number of decimal places later.

03:05 So now let's proceed to the second step of our solution...