### Circuit Types

There are many ways to combine electronic parts to make circuits. Let’s focus on three components. First will be an inductor, indicated by the letter L. In a circuit it stores energy in a magnetic field. We’ll also use a capacitor, C, which stores energy as an electric charge. Finally, we’ll mix in a resistor, R, which manages current flow. All of these will be connected to a theoretical voltage source like a battery.

Combine the inductor, L, and a capacitor, C, and you get an LC circuit. An LC circuit is a resonant electrical circuit. It generates a resonant frequency. An LC circuit is also called a tank circuit or tuned circuit.

*LC circuit schematic*

The second type of circuit is an RLC circuit, which is also a resonant electrical circuit. We’ve simply added a resistor, R, to the other components. These components can be connected in series or parallel. It all depends on the purpose of the circuit.

So what is resonance? It’s when the capacitor and inductor, L and C oscillate, cycling with each other. The inductor discharges, which charges the capacitor. Then the capacitor discharges, charging the inductor.nLC and RLC circuits have a resonant frequency. That is a very specific frequency for the circuit where it oscillates at the greatest amplitude. Think of a guitar string resonating at a certain note.

*RLC series circuit schematic*

Let’s measure the voltage in a series RLC circuit. Due to **resonance **the voltage across reactances is higher than the voltage applied to the entire circuit. Put 12 volts in, and you’ll measure a higher voltage in the circuit.

*RLC parallel circuit schematic*

When those RLC components are connected in parallel, that impacts the current. The magnitude of the current at the input of a parallel *RLC* circuit at resonance is **minimum**. In other words, the circuit is drawing minimum current.

Take the resistor away, and you get a different effect. Circulating current within the components of a parallel *LC* circuit at resonance **is at a maximum****.** In other words, the current circulating within the circuit itself is increased when resonant.

Note that those answers look really similar! But, when you take away the resistor, the current changes.

So we’ve measured voltage and current, now we’ll measure impedance of an RLC circuit at resonance. The magnitude of the impedance of a **parallel** RLC circuit at resonance is **approximately equal to circuit resistance. **

Guess what, the magnitude of the impedance of a **series** RLC circuit at resonance is **approximately equal to circuit resistance. **

Did you catch that? The magnitude of impedance is the same in a parallel or series RLC circuit. So these two questions have the same answer.

If we have the values for L, and C, we can also calculate the resonant frequency of a circuit.

The formula for calculating resonant frequency is:

**f = 1 / (2π√(LC)) **

Let’s say we have a circuit that has R = 33 ohms, L = 50 microhenries, and C = 10 picofarads. Look at that equation. While you are given R, you don’t need it in the calculation. On this question, it’s

- L * C in this case is 50 * 10 so put 500 into the brackets,
- Derive the square root of 500, which is 22.36 rounded.
- 2π times 22.36 is 140.492
- 1 over 140.492 is 0.00712 rounded

That points you to your answer of **7.12 MHz.**

That’s a lot of math, and there are simpler formulas and calculations on the exam. So it might help to just remember this one and the next. The resonant frequency of an RLC circuit if R is 22 ohms, L is 50 microhenries and C is 40 picofarads is **3.56 MHz****.** However, here’s the math if you want to see it.

- L * C is 50 * 40 = 2000,
- √2000 = 44.72 rounded.
- 2π times 44.72 = 280.984 rounded
- 1 over 280.984 is 0.00356 rounded
- The answer is 3.56 MHz.

### Reactive Power

Reactive power is the current and voltage released by capacitors and inductors working together. This results in resonance.

The phase of the circuit looks at the flow of current through the components. For example, consider a series resonant circuit. When both the current through the circuit and the voltage across the circuit are at resonance, **the voltage and current are in phase. **

Imagine ideal inductors and capacitors in an RLC circuit. The self-resonance of each component is created by the **component’s nominal and parasitic reactance.** In this ideal circuit **energy is stored in magnetic or electric fields, but power is not dissipated. **

In most RLC circuits, current and voltage are **90 degrees out of phase.** That phase change is different depending on the component.

First, the relationship between the AC current through a capacitor and the voltage across a capacitor. In this case the **current leads voltage by 90 degrees****.** Think of this with an oscilloscope measuring current and voltage. The current will come through first to charge the capacitor, then voltage can flow.

It’s the other way around for an inductor. In an inductor **voltage leads current by 90 degrees. **

What about power consumption in an RLC circuit? Consider a 100-ohm resistor in series with a 100-ohm inductive reactance drawing 1 ampere of power. That circuit consumes **100 watts** of power.

### Phase Angles

A radio wave is in a constant cycle up and down. If you select a point in that cycle, you can express that point as an angle. This is called a phase angle and it’s a point where you can measure the voltage and current of the circuit.

It is possible to calculate the phase angle, and here’s the formula.

θ = arctan(X / R)

Here are the three series RLC circuit phase angle questions. We’ll plug these inputs into the formula and share the answers. All three ask for the voltage across and the current through a series RLC circuit:

- if XC is 500 ohms, R is 1 kilohm, and XL is 250 ohms, phase angle is
**14.0 degrees with the voltage lagging the current****.** - if XC is 300 ohms, R is 100 ohms, and XL is 100 ohms, phase angle is
**63 degrees with the voltage lagging the current.** - if XC is 25 ohms, R is 100 ohms, and XL is 75 ohms, phase angle is
**27 degrees with the voltage leading the current.**

Moving on, you know that the term impedance means the resistance of a circuit to AC. Instead of how resistant a circuit is, what if you looked at how conductive a circuit is? The ability of current to flow through a circuit is called admittance. Admittance is **the inverse of impedance****.** It has a component called susceptance, which is **the imaginary part of admittance****.** The letter **B** is commonly used to represent susceptance.

This applies to our polar angles. You can convert impedance in polar form to an equivalent admittance. To do that **take the reciprocal of the magnitude and change the sign of the angle.**

So what happens to the magnitude of a pure reactance when it is converted to a susceptance?**It is replaced by its reciprocal.**