4/14/16 - Inductor and Capacitor First Order Cicruits

We are told that inductors follow the same rules to combine as resistors.

Parallel : Add in inverse

Series: Add normally

He is the equivalent inductance from the above circuit.

Next we look at the equations for the voltage and current across inductors and capaticors, as well as their time constants.

Here we can say that the voltage across the capacitor has to be equal to the negative voltage across the resistor, and simplifying yields a first order differential equation.

Here we solve for the voltage across an capacitor as a function of time. We find that the voltage depends an exponential function with a time constant of 1/RC.

We are told that a general rule of thumb for a capacitor to be completely depleted in about five time constants. Solving for 1% of the potential left on a capacitor, we find that it is roughly 4.6 time constants. Rounding up to 5, we can say that after 5 time constants, there will be less than 1% of the original potential left on the capacitor.

Since we define power in terms of Watts, which is volts times amperes, we can find the rate of energy leaving the capacitor as a function of time.

Given a simple circuit, we calculate the voltage and current as a function of time based on the resistance and capacitance values.

Given a range of values for resistance and capacitance values, we calculate the range of time constants that can be produced from it.

Here we hook up the Analog Discovery to a simple circuit with two resistors and a capacitor, we hook up the oscilloscope probes across the capacitor to measure the potential.

We set up Waveforms' oscilloscope function to have a trigger of about 1V and record the climb in potential across the capacitor leads. We can see that it is an exponential function.

Same deal with letting it discharge, an exponential function.

Next, we set up a square wave as an input to the circuit, we find the time constant to be roughly 15ms.

There is also a Kirby Eric in the corner.

Here is the output of the square function as an input.

An attempted close up of the graph, didn't turn out too well from a different phone.

Here we expanded the curve to measure approximately how long it took for the capacitor to fully charge.

We measured that it took approximately 75ms, which is just about 5 of the time constants we calculated.

The zoomed in discharage graph.

Next, we tackle the current across an inductor. We can use our methods to solve a first order differential equation and find the current as a function of time across the inductor. As all general solutions to first order differential solutions, the solution follows a exponential format similar to the capacitor.

Next, we given a basic circuit with a resistor in series with a rather large inductor of 5H, we calculate the time constant and then right the current as a function of time. We then integrate the current function, and multiply by some constants to find the voltage as a function of time.

Now a more complicated circuit, we see the DC voltage source is disconnected at t > 0. We find the equivalent resistance for the resistors and then calculate the time constant. We use a basic current divider equation to find the initial current and solve for our current function.

So we pluck a inductor in the same spot as the capacitor and now measure the voltage across the inductor.

We input a square wave with a frequency of 100kHz and observe the outcome.

We can see the the voltage drops fast across the inductor, which follows our exponential equation.

The charge and dischrage of the inductor due to the wavegen.

Here we calculated our time constant for the circuit and measured the actual time it took. We found it to be roughly 5 time constants like predicted.