# RC and RL Circuits

#### Inductors, Capacitors, Transformers

In this section, we'll be looking at RC and RL circuits. RC and RL circuits are nothing more than applications for resistors, inductors, and capacitors. We will primarily look at RC circuits since they are more commonly used and especially they're widely used with computers circuits and digital circuits which will be our focus. RL will be presented, but it will be in an abbreviated form.

### Characteristics With Sine Wave Inputs

Here, we have the diagram here. We see here is an RC circuit. Here, we look at the sign waves that would be resultant across current, across the resistor, across the capacitor, and then the applied voltage, and then this is phasor diagram.

In a series circuit, current is identical through all components. Since this is a series circuit, the current that is applied to this component will be the same as applied as to this one. Here, we see the sine wave for current and it will be identical through all components and that is simply the characteristic of a series circuits.

The voltage. Here, we have the voltage across the resistor, across the capacitor, and the applied voltage can all be still found using ohms law. The voltage across the resistor is simply going to be current times the resistance of a component and the voltage across the cap will be found using the current times the capacitive reactance.

There is a nine-degree phase shift between VR and VC. Notice here with the sine waves here we have the voltage that would we see in an O-scope across the resistor, and you'll notice that it is in phase with the current through the circuit. You'll notice that the capacitor voltage, notice it is 90 degrees out of phase with the resistor. Here, we have times zero on the resistor times zero on the cap is lagging by 90 degrees.

The phasor shows a diagram of an RC circuit. Since current is the same throughout the circuit it is the reference. Here, we have the reference point here is going to be current through the resistor. The phasor for the cap is drawn 90 degrees away from current because the voltage across the cap lags current by 90 degrees. In our phasor diagram, if we go back 90 degrees, there's where we'll find the voltage across the capacitor.

You'll notice that the dashed line here is indicating the point where we will see our applied voltage because our applied voltage is going to be the sum of what is across VR and what is across VC. Even on our diagram, we can see that. Notice here we have VR, VC lags by 90 degrees, and you'll notice that the applied voltage is somewhere in between these two because this value is going to be a sum of VR and VC. Our calculation of that's going to be on the next page.

### Voltage Drops and Phase Relationships

Solving voltage drops in a series RC circuit is accomplished by using Kirchhoff's voltage law to sum the component voltages using phasor addition. This is our process here to calculate for applied voltage. We would take the square root of the voltage across the resistor squared plus the voltage across the capacitor squared. The impedance or the resistance in a series RC circuit is calculated by and here we have impedance. It's going to be the square root of R squared plus the reactance squared.

We've taken a brief look at introducing this subject of RC circuits, and we're looking specifically at this point with the characteristics of an RC circuit with sine wave inputs.