Impedance in AC Circuits - Online Tutor, Practice Problems & Exam Prep

Impedance in AC circuits combines resistance and reactance, similar to how voltage and current interact in series and parallel configurations. In a series RC circuit, the total voltage is calculated using the Pythagorean theorem, leading to the equation for impedance: R^2+(1ωC)^2. Understanding impedance is crucial for analyzing AC circuits effectively.

Impedance in AC Circuits

Hey guys. In this video, we want to talk about this quantity of a circuit called impedance. Okay? It's going to be very similar to reactants. So, let's get to it. We know how to find the current in any AC circuit with a single element. Okay? So we saw an AC source connected to a resistor, an AC source connected to a capacitor, an AC source connected to an inductor. The maximum of a resistor to just be its resistance. Okay, because reactance and resistance are the same for resistors. Now there are 2 types of circuits for combining multiple elements we know. There are series circuits and parallel circuits. There are also types of circuits that have neither series nor parallel connections but those you are not going to encounter when discussing AC circuits. They will either be purely series or purely parallel. Okay? Whenever an AC circuit has multiple elements in series, the current phasors all line up. They are all in phase. Okay? And that just comes from the fact that the current is the same for all elements in series. That's the definition of a series connection. Whenever an AC circuit has multiple elements in parallel, the voltage phasors line up. The voltage phasors are in phase. Right? That's because the voltage for all elements in parallel is the same. That's just the definition of a parallel connection. Okay?

Now let's consider one particular circuit, which happens to be an AC source connected in series to a resistor and a capacitor. Okay. In this case, so let's draw that circuit. Here we have our AC source. Here we have our resistor and our capacitor connected to our AC source. I have defined in this case the voltage across the resistor and the capacitor to be V_RC. Notice that that voltage, the voltage across both of those elements is the same as the voltage across the source, V_max. Okay? So those have to be the same, the same maximum at least. Okay. In this case, the maximum voltage across the resistor and capacitor V_RC will not simply be the sum of V_R and V_C. Okay? Remember that we have equations for both of these. This is simply I_R and this is I_XC. It's not just going to be the sum of those 2 because the maximum voltages don't appear at the same time. Okay? Instead, of voltage phasors. This is one of the particular reasons why we use phasors because you can add them like vectors. Okay? So this is a series circuit. We are going to have both elements current phasors to be in parallel. Now a resistor always has its voltage phasor in series with its current phasor.

So right here we have the voltage phasor of the resistor. Now a capacitor always has its voltage phasor lagging by 90 degrees to its current phasor. So right here, we have the voltage phasor of the capacitor. Okay? So what is the total voltage going to be? Well, DZ_RC is going to be the Pythagorean theorem, DZ_R squared plus DZ_C squared. Right? This is the vector sum of those 2 phasors. So this is going to be I_max²R², which is just the maximum voltage across the resistor plus I_max²XC², which is just the maximum voltage across the capacitor. Okay? We want to rewrite this like Ohm's law, like having a reactance, like having a resistance, and we rewrite it with this variable Z. And Z we call the impedance of this AC circuit, which acts like the effective reactance of the entire circuit with all the elements taken into account. Okay? And the maximum current output by a source is always going to be defined in terms of the reactants. It's always going to be defined as V _max/z. Okay? In this particular case, the case of a series RC circuit, we see that the reactance is just R² plus and I'm going to substitute in the capacitive reactance. This is 1 over omega² c². Okay. This is the impedance of a series RC circuit. But that's only for a series RC circuit. The impedance can be found for multiple different types of circuits and it's all found the same way. You draw the phasor diagram and you do the vector sum of something that you're looking for. That will lead you to the impedance. Okay? Let's do an example to illustrate that. What's the impedance of an AC circuit with a resistor and an inductor in series? Okay? In this case, once again, since they are in series, the current is going to be the same for both of them. Right, so this is the current.

Now the voltage across a resistor is always in phase with its current. So this is the voltage across a resistor. And the voltage across an inductor always leads its current by 90. So this is the voltage across the inductor. It's leading by 90. So what is the maximum voltage in this circuit? It's just going to be the vector sum of those 2 voltage phasors. So I'm going to use the Pythagorean theorem. Right? Now I'm going to substitute in I_max²R² for the maximum voltage across the resistor, and I_max²XL² for the maximum voltage across an inductor. This factor of I_max they both share so I can factor that out. And the inductive reactance, I can substitute in terms of the angular frequency. And don't forget, this term right here is the reactance, sorry, is the impedance. So the impedance of this circuit is the square root of R² plus omega² L². Okay and this is absolutely different from the impedance of a series RC circuit that we saw before this. Alright guys, that wraps up our discussion on impedance. Thanks for watching.

Impedance of a Parallel LR AC Circuit

Hey, guys. Let's do an example using impedance. Okay? What's the impedance of a parallel LR circuit? Okay, LRAC circuit. Okay. So it has an inductor and a resistor in it and let's draw the phasor diagram for this when they're parallel. When they're parallel don't forget their voltage phasors are in phase because all elements in parallel have the same voltage. So I'm gonna draw some voltage phasor right here. Now for the resistor, the current and voltage phasors are always going to be in phase. So here is the current phasor for the resistor. Now the voltage phasor for the inductor is always going to lead its current phasor. So I need to draw the current phasor for the inductor as lagging by 90 degrees. Current in this circuit is going to be given by the vector sum of these two current phasors. So using the Pythagorean theorem I get this.

Okay now I want to rewrite everything in terms of the maximum voltage. So this is V_max over the impedance. Remember this this is the definition of the impedance that the maximum current produced by the source is just the maximum voltage of the source divided by the impedance. This is going to be the square root of V_max² over R² plus V_max² over XL². Why do both of these terms also have V_max? Because this is a parallel circuit so everything has the same voltage as the source, right? So I can cancel all of these terms of V_max. So this tells me that one over Z, the impedance, is the square root of 1 over R² and I can substitute in the equation for the inductive reactants and this becomes 1 over omega² L², and this is our impedance for a parallel LR circuit. Okay. For parallel circuits, you're never quite going to define the impedance. You're always going to define 1 over impedance like this. But this is a perfectly fine way of representing the answer. Alright, guys? Thanks for watching.

V max R R 2 V max XL XL 2

What's the impedance of a parallel RC AC circuit?

• A. Z = sqrt(R² - 1/ω²C²)
• B. Z = sqrt(1/R² + ω²C²)
• C. 1/Z = sqrt(1/R² + ω²C²)
• D. 1/Z = sqrt(1/R² + 1/ω²C²)

An AC source operates at a maximum voltage of 120 V and an angular frequency of 377 s^-1 . If this source is connected in parallel to a 15 Ω resistor and in parallel to a 0.20 mF capacitor, answer the following questions: 

a) What is the maximum current produced by the source? 

b) What is the maximum current through the resistor? 

c) What is the maximum current through the capacitor?

• A. a) 12 V b) 8A c) 4A
• B. a) 12 V b) 8A c) 9A
• C. a) 7.8 V b) 7.6A c) 0.2A
• D. a) 8.9 V b) 8A c) 4A

Here’s what students ask on this topic:

What is impedance in an AC circuit?

Impedance in an AC circuit is a measure of the opposition that a circuit presents to the flow of alternating current (AC). It combines both resistance (R) and reactance (X), which includes inductive reactance (X_L) and capacitive reactance (X_C). Impedance is represented by the symbol Z and is measured in ohms (Ω). The formula for impedance in a series RC circuit is given by:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Understanding impedance is crucial for analyzing how AC circuits behave under different conditions.

How do you calculate the impedance of a series RC circuit?

To calculate the impedance (Z) of a series RC circuit, you need to consider both the resistance (R) and the capacitive reactance (X_C). The formula for impedance in a series RC circuit is:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Here, ω (omega) is the angular frequency of the AC source, and C is the capacitance. This formula is derived using the Pythagorean theorem to combine the resistance and reactance vectors.

What is the difference between impedance and reactance?

Impedance (Z) and reactance (X) are related but distinct concepts in AC circuits. Impedance is the total opposition to AC and includes both resistance (R) and reactance (X). Reactance, on the other hand, is the opposition to AC due to inductors and capacitors only. Reactance can be inductive (X_L) or capacitive (X_C). The formula for impedance in a series RC circuit is:

$\sqrt{R^2+{\frac{1}{\omega C}}^2}$

Impedance is a more comprehensive measure as it accounts for both resistance and reactance.

How does impedance affect the current in an AC circuit?

Impedance (Z) affects the current (I) in an AC circuit by determining the total opposition to the flow of current. According to Ohm's Law for AC circuits, the current is given by:

$\frac{V}{Z}$

where V is the voltage. Higher impedance results in lower current for a given voltage, and vice versa. Impedance combines both resistance (R) and reactance (X), so it provides a complete picture of how the circuit will behave under AC conditions.

What is the impedance of a series RL circuit?

The impedance (Z) of a series RL circuit, which includes a resistor (R) and an inductor (L), is calculated using the following formula:

$\sqrt{R^2+{\omega }^L}$

Here, ω (omega) is the angular frequency of the AC source, and L is the inductance. This formula is derived by combining the resistance and inductive reactance (X_L) vectors using the Pythagorean theorem.