Understanding Frequency Effects on Impedance: Resistors, Capacitors & Inductors

[viola.geek]

While working in a lab this summer, I've been reading up on impedance and AC circuits as much as I can, considering I have to present at the symposium here in August. I'm really confused about a few...parts of impedance, for lack of a better term.

How does frequency affect impedance, in general?
I know: Z = R + jX, Z = R + Z_L + Z_C = R + j(X_L - X_C) and X_C = 1/\omega*C, X_L = \omega*L
But a more general statement would help when it comes to explaining/summing everything up on how frequency affects impedance, especially when explaining it to people who have never studied impedance in depth but have only heard the term before.

How does frequency affect resistors, capacitors, and inductors? Is it just the equations for impedance for each of these?

And when I see |Z| and \phi, am I to think of polar plots of impedance? Or something to that effect?

Sorry if I repeated any questions already answered here. I did a search and found nothing. :)

 

[Bob S]

These are good questions, and anyone who plans to major in physics or EE should understand them.
First, resistors are easy. V = I*R; voltage = amps times resistance; power = I²R = V²/R etc. Learn to read the stripes on resistors; black, brown, red orange, yellow, green, blue, violet, grey, white. I would give you a memorable little ditty but the Forum would object.
Second, capacitors. Capacitors store charge. Charge is the time integral of current. Capacitors do not dissipate energy, however. When you put current into a capacitor, the voltage increases. Q = C*V; dQ/dt = I = C*dV/dt. If the voltage is sinusoidal, dV/dt = jwV, where j = a 90 degree phase shift and w = 2 pi*f where f = frequency (e.g., 60 hertz). So I = jwC*V, or V= I/jwC.
Similar for inductances, except they store magnetic energy (current), not charge. V= L*dI/dt = jw*L*I where L = inductance.
So a series RLC circuit has an impedance Z = V/I = R + jwL + 1/jwC = R + jwL - j/wC. Note that j/wC can equal jwL and cancel it out at a certain frequency where w² = 1/LC. This called a series resonance.
Most homes have refrigerators and other appliances that have electric motors. Electric motors have an impedance that is partly inductive, so the household voltage and current are slightly out of phase. V = I(R + jwL). The power is then I*V*cos(θ), where cos(θ) is called the power factor.
Complex impedance is often plotted on polar plots; reactive (inductive) = +y, resistive = + x. and capacitive = -y.
Many years ago, I worked part time at an electronics lab while I was in college. very valuable experience.

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω

 

[viola.geek]

Thank you for your response.

But what do we mean when we say "phase shift?" Is that just for the polar plot, with a vector |Z| and an angle \phi? I know you can show impedance data on a Bode plot, which has the log |Z| and log ω, but I don't understand why it's the log of each value. Why not just plot |Z| vs. ω? (I asked around in my lab, but no one could explain it.)

I have yet to take an electronic measurements course (or E&M...both classes are this next school year), so maybe this is why I'm confused about this, but what do you mean by "the voltage is sinusoidal" and "j = 90 degree phase shift?"

So a series RLC circuit has an impedance Z = V/I = R + jwL + 1/jwC = R + jwL - j/wC. Note that j/wC can equal jwL and cancel it out at a certain frequency where w2 = 1/LC. This called a series resonance.

Is this true for parallel circuits too?

 

[Bob S]

But what do we mean when we say "phase shift?" Is that just for the polar plot, with a vector |Z| and an angle \phi? I know you can show impedance data on a Bode plot, which has the log |Z| and log ω, but I don't understand why it's the log of each value. Why not just plot |Z| vs. ω? (I asked around in my lab, but no one could explain it.)
but what do you mean by "the voltage is sinusoidal" and "j = 90 degree phase shift?"?

All our ac power is sinusoidal, meaning that V(t) = V₀ sin(wt) = V₀ sin (2 pi f t)
If Both the voltage and current are in phase, V = V₀ sin(wt) and I = I₀ sin (wt).
If the current leads the voltage by 90 degrees, then V = V₀ sin(wt) and I = I₀ cos (wt). This a 90 degree phase shift.

Z = V/I = R + jwL + 1/jwC = R + jwL - j/wC.
Is this true for parallel circuits too?

For parallel resonance, 1/Z = 1/jwL + jwC = (1 - w²LC)/jwL

 

[Naty1]

Resistance is the opposition to current flow...big resistances do not carry current easily and tend to get hot. Impedance is a more general form of opposition to current flow. which indudes resistance, reflecting the opposition of capacitors and inductors to current flow.

In general frequency alters impedance, and hence current flow varies,but in different ways for different circuit components (capacitors and inductors) and does not generally effect resistance at moderate frequencies. At extremely high frequencies even the wiriing in a circuit, that is the wiring sitting on a lab table, can exhibit capacitance and inductance..

In the US the standard alternating voltage and frequency is 120 volts and 60 hertz, that is, 60 cycles per second. In Europe, I'm not sure what the stand voltage is, but standard frequency is 50 hertz.

Try wikipedia at http://en.wikipedia.org/wiki/Electrical_impedance or read any introductory tex on ac circuits.

I also worked in electronics labs at at NYU and Manhattan College; a great experience. Be sure you make the effort to learn as much as you can, you'll be gald you did. good luck.

 

[viola.geek]

So, then, impedance's relation to frequency is the equation(s)? (Just to clarify.)
Edit: 1/Z = 1/jωL + jωC --> always the equation for impedance in a parallel circuit? or just that resonant frequency?

Sorry for asking so many questions! I just want to make sure I understand this.

In a Bode plot, log |Z| is plotted against log ω, and phi against log ω. Why is it the logarithm of each value, except for phi? (My project involves lithium ion batteries, and since I need to cover the background on EIS in my presentation, I'm also trying to anticipate questions...)