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We know that impedance, Z, is complex.

I'm told that $$V = IZ$$ But that rearranges to $$Z = \frac{V}I$$

But both voltage and current are real numbers, so how can a real number divided by another real number give a complex number? What have I done wrong?

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    \$\begingroup\$ V and I in such a case ARE complex numbers are well. Usually their magnitudes are given along with an angle, if no angle is given, you can (usually) assume an angle of zero degrees. \$\endgroup\$ – Jarrod Christman Dec 23 '14 at 18:16
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Complex \$Z\$ has a meaning only for alternating currents/voltages, so your assumption that \$V\$ and \$I\$ are real is wrong. They can be represented as complex as well. Or you can work with effective values, which are real, but then you will have to take an absolute value of \$Z\$ as well.
UPD: More about representing electrical quantities in a complex for can be found in this nice lecture

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  • \$\begingroup\$ I see, thanks, that makes more sense. So how do you represent V and I as complex? Also, by your last sentence, do you mean at a specific point in time Z is real? (Making it just resistance, and impedance just varying resistance?) \$\endgroup\$ – ACarter Dec 23 '14 at 18:40
  • \$\begingroup\$ My last sentence means, that you can take effective values of voltage and current (which is the amplitude divided by square root of 2), and their ratio will give you the absolute value of Z, which is a real number (think of it as a "length" of the Z vector in the complex plane). \$\endgroup\$ – Eugene Sh. Dec 23 '14 at 18:44
  • \$\begingroup\$ Ah I see, so using the RMS values. Thanks. Also, how do you represent voltage or current as a complex number? Thankyou very much \$\endgroup\$ – ACarter Dec 23 '14 at 19:44
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    \$\begingroup\$ Using Phasors: cas.umkc.edu/physics/wrobel/phy250/lecture4.pdf \$\endgroup\$ – Eugene Sh. Dec 23 '14 at 19:47
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    \$\begingroup\$ You didn't paste the information into the answer. Only the link. Stack Exchange sites generally prefer to have the actual content in the answer, in case the link breaks in the future. We don't need the whole lecture - just the bit that explains what complex values for V and I actually mean, or a summary thereof. \$\endgroup\$ – Dawood says reinstate Monica Dec 23 '14 at 21:53
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Complex doesn't mean unearthly.

When you say Z is complex. That means there will be a phase shift between the V and I components. Complex notations helps represent that phase relationship between the V and I (simply called phase shift) in a understandable/manageable way.

For a purely ohmic impedance( represented with a real number). The V and I remain in-phase.

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When one learns about circuits, one encounters two versions of Ohm's law. The first is \$V = IR\$, which only applies to resistors considered in isolation. If you use basic circuit analysis techniques on a purely resistive circuit, the equation you get is going to be basically of the form \$\mathcal{E} - IR = 0\$. However, when you introduce capacitors, inductors, and AC voltage sources, circuit analysis starts giving you differential equations. These equations can be expressed as second-order nonhomogeneous ODEs for current, in the general form

$$L\frac{\mathrm{d}^2 I}{\mathrm{d}t^2} + R\frac{\mathrm{d} I}{\mathrm{d}t} + \frac{I}{C} = \mathcal{E}\cos(\omega t)\tag{1}$$

In this equation, all the quantities are real. But a common technique used to solve these differential equations is to consider a related equation with complex variables,

$$L\frac{\mathrm{d}^2 \tilde I}{\mathrm{d}t^2} + R\frac{\mathrm{d} \tilde I}{\mathrm{d}t} + \frac{\tilde I}{C} = \mathcal{E}e^{i\omega t}\tag{2}$$

If you define \$\tilde I(t)\$ such that \$I(t) = \operatorname{Re}\tilde{I}(t)\$, then you'll notice that equation (1) is just the real part of equation (2). But equation (2) is easier to solve, because exponential functions are easier to work with than sines and cosines. So we solve equation (1) by solving equation (2) and then taking the real part.

So anyway, you can solve equation (2) for \$\tilde{I}(t)\$, and you can plug that into

$$\begin{align} \tilde{V}_C &= \frac{1}{C}\int\tilde{I}\,\mathrm{d}t & \tilde{V}_R &= \tilde{I}R & \tilde{V}_L &= L\frac{\mathrm{d}\tilde{I}}{\mathrm{d}t} \tag{3} \end{align}$$

to calculate the quantities \$\tilde{V}_{C,R,L}\$ for any capacitor, resistor, or inductor in the circuit. Note that the real parts of equations (3) are the usual voltage laws for circuit elements, which is why you can justify thinking of the \$\tilde{V}\$'s as "complex voltages". Again, you get the real solution by solving the complex equations and then taking the real part at the end.

Now, an interesting thing about exponential functions is that

$$\begin{align} \int e^{i\omega t}\,\mathrm{d}t &= \frac{1}{i\omega}e^{i\omega t} & \frac{\mathrm{d}}{\mathrm{d}t} e^{i\omega t} &= i\omega e^{i\omega t} \end{align}$$

so, assuming \$\tilde{I}\$ is an exponential of this form, you can express equations (3) as

$$\begin{align} \tilde{V}_C &= \underbrace{\frac{1}{i\omega C}}_{Z_C}\tilde{I} & \tilde{V}_R &= \underbrace{R}_{Z_R}\tilde{I} & \tilde{V}_L &= \underbrace{i\omega L}_{Z_L} \tilde{I} \end{align}$$

This is the origin of complex impedance: assuming that \$\tilde{I}\$ is a complex exponential whose real part represents the actual current, then \$\tilde{V}\$ is a complex exponential whose real part represents the actual voltage, and you can calculate it as \$\tilde{V} = Z\tilde{I}\$, where \$Z\$ is a complex quantity that depends on the circuit element in question. This is handy because, not only is it easier to solve the relevant differential equations by using complex exponentials, but the exponential also incorporates the time dependence of a sinusoidal current or voltage.

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This answer is decidedly terse to provoke a thought reaction. Please don't be offended.

Ohm's law is complex. And at the same time it is real for any real values. That's the very reason we have effects like reflections in a wire for high frequency signals. Because U=ZI must hold true for a wire and U=RI must hold true for each point in a wire (because instantaneous voltage and current at a singular point in a wire are real values), the net effect is that for any non-DC signal, there also must be a wave and a returning wave in a wire. It's just because of Ohm's Law that we must study high frequency design and why we must treat certain wires as "transmission lines".

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So a complex number has real and and imaginary part. So when a number is real view it as the imaginary part is zero and the other way round when you have an imaginary number.

The Z impedance is expressed as a complex quantity because it helps you study the effects of changing frequency in the frequency domain. The Z impedance becomes zero only when the current or voltages are not in same phase. Example say I= 5 cos(wt) and V - 10 cos(wt+ 50) here the current lags the voltage by phase of 50 and hence the impedance turns out to be complex value with both real and imaginary parts being non zero. Say if the voltage didn't have the +50 phase shift then Z would become just 1/2 ohms just like the case in dc circuits when the current and voltage don't vary with time and hence no time phase shift is there to cause the impedance to have a non zero imaginary component to it.

So finally imaginary components arise in the case only when there is phase shifts.

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The complex quantity Z can be expressed as a magnitude and an angle (or a modulus and argument).

  • The magnitude of Z equals the magnitude of V divided by the magnitude of I.
  • The angle (or argument) of Z equals the phase shift between the voltage and the current.
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