AC Circuits and Impedance
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Written by Bryce Ringwood   
Saturday, 27 April 2013 18:15
 

Introduction

In this article we look at AC circuits and electrical impedance. If you can't follow the maths - don't worry too much, rather try to get an overall understanding from the text. At the end of the article, I'm going to demonstrate how to calculate the impedance of a circuit using a calculator. 

We haven't really discussed "Circuits" in any organised way, except for an article in "Experimenter's Corner" where Ohm's law is made the subject of a simple 'C' programming excercise. There is also a circuit calculator for Ohm's law in the "Calcs" section of this web site.

 

These two articles summarize DC circuits and if we were to just use "pure" resistors and RMS values of AC voltage, we would not have to worry any further. Indeed, in the world of light-bulbs (prior to compact fluorescent) and electric kettles, we don't worry. We just use 230 volts (AC, RMS), talk of 2kW elements and carry on. We can, because to all intents and purposes these things are pure resistances and Ohm's law applies. 

Radio receivers contain a number of AC circuits. For example, the heater supplies to the valves, the audio circuitry and the RF and IF circuits  all contain examples of AC (Alternating Current) . When it comes to DC circuitry, we are clear as to what to do. With AC, a mental fog sets in.

What we would like is a kind of Ohm's Law for AC circuits in which there are combinations of resistors, capacitors and Inductors. Fortunately, we can provide exactly that, but it requires complex number arithmetic. This is why I provided a complex number calculator and (possibly) is why a number of pocket calculators (e.g. HP 35s) have a complex number feature.( Please take a look at the maths review articles to get you back up to speed with complex numbers.)

Voltages and currents in AC circuits have a magnitude  and a phase  as illustrated below:

Illustrating Phase Relationship

The amplitude of the voltages is 2.5 Volts, and the phase difference is $\pi \over 4$ or 45 degrees.

For convenience, we rather represent voltage and current as complex quantities, for example voltage:

\displaystyle{{\bf V} = |{\bf V}|e^{j(\omega t + \phi_{voltage})}=a + jb}

and current, similarly:
\displaystyle{{\bf I}=|{\bf I}|e^{j(\omega t +\phi_{current}) }}

where:
\displaystyle{\omega = 2 \pi f}

$\displaystyle{f}$ is the AC frequency, and $\displaystyle{\phi_{voltage}}$ and $\displaystyle{\phi_{current}}$ are the phase angles for voltage and current. Complex quantities are represented in bold face.

Finally, Ohm's Law for AC circuits uses impedance. Impedance is the AC circuit equivalent of resistance in DC circuits. As may be expected, it is best represented complex quantity:

\displaystyle{\bf V = I \times Z}
where $\bf Z$ is the impedance. A simple resistor  has an impedance of $\displaystyle{R + j0}$ - the value marked on the body. You will hear people talk of impedance when thay are referring to resistance. It can be confusing, but don't worry.

Why did I put the word "represented"  in italics? The answer is that in the real world we are dealing with phase angles and magnitudes. We could equally well have represented these as mathematical quantities called vectors, but they require unfamiliar arithmetic operations that are not available on a calculator.

Inductive Reactance

An inductor consists of a number of turns of wire around a core of soft iron, ferrite, or even – air. The core has a property of magnetic permeability. When we apply a DC voltage to the inductor, the core becomes magnetized by virtue of the current flowing through the wire. The wire has some resistance, otherwise the current through the coil would be infinite and everything would get hot and melt. When we apply an AC voltage at a low frequency, the magnetic field generated in the core simply rises and falls in sympathy with the AC voltage.

As the frequency increases, the collapsing magnetic field in the core induces an opposing current, so that the current in the core can't keep pace with the applied voltage. The rise and fall of the current in the inductor becomes out of step with the applied voltage, and lags behind it by a certain amount. The voltage and current are out of phase, with the current lagging behind the voltage.

In a “pure” inductor that has zero resistance and no capacitance, the current would lag behind the voltage by 90 degrees. BUT we don't work in degrees, so we rather say $\displaystyle {\pi} \over 2$ radians.

A Brief Analysis

In the article on inductances, we saw that $\displaystyle{V=L{dI \over dt}}$.
Now we can write $\displaystyle{{\bf V} = |{\bf V}| e^{(j \omega  t)} = L{d{\bf I} \over dt}}$ and solve, giving us:
$\displaystyle{{{{|\bf V}|e^{(j \omega t)}} \over{j  \omega}}={ \bf I} L}$ So $\displaystyle{{\bf V} = j \omega L {\bf I}}$
and $\displaystyle{{\bf Z} = 0 + j \omega L}$ The magnitude of $\bf Z$ is the reactance $\displaystyle{{X_L}=\omega L}$

The axes represent the complex plane, so that inductive reactance is a purely complex quantity. To put it another way, the impedance of an inductor is a pure complex quantity.

Capacitive Reactance

As we saw in the section on capacitors, they behave a little bit like batteries. As you charge them, the current diminishes and the voltage rises. They do the opposite of an inductor, in that the current leads the voltage across the capacitor plates.

Brief Analysis

In a circuit containing capacitance:
$\displaystyle{{\bf I}=C{d{\bf V}\over dt}}$, where $\displaystyle {{\bf I}={|\bf I|}e^{j \omega t}}$  solving

$\displaystyle{{|\bf I|}{e^{j \omega t}\over {j \omega }}=C{\bf V}}$, rearranging and substituting

$\displaystyle{{\bf V}={ ({1\over{j \omega C}}){\bf I}}}$ giving an impedance $\displaystyle{{\bf Z}= {0 -{({j \over {\omega C}})}}}$ . The capacitative reactance, $\displaystyle{{X_C}={1\over{\omega C}}}$

Circuits Containing combinations of R,L and C

Arrmed with a complex number calculator, the analysis of AC circuits becomes as easy as it is for DC circuits.

Components in series:
$\displaystyle{\bf Z_{total}= Z_1+Z_2+Z_3+ .....}$


Components in parallel
$\displaystyle{\bf{1 \over Z_{total}}={1 \over Z_1}+{1 \over Z_2}+{1 \over Z_3} + ...}$
and $\displaystyle{\bf Z_{total}= {{{Z_1}{Z_2}}\over {Z_1 + Z_2}}}$ for two elements in parallel.

Let's Check it out !

Even if you haven't quite followed the maths up to now - try to master this little bit.

Example Ladder Network

Example Ladder Network - Passive Components

The example is from the HP65 EE Pak1 "Impedance of a Ladder Network" (EE-1-04A) Hewlett Packard, 1974.

The frequency is 4.0MHz

Element Formula Impedance New Impedance
50 Ohms $R + j0.0$ 50 + j0.0  
2400pF $0.0 -{ j \over \omega C}$ 0 - j16.578640 4.95252 - j14.936
2.56µH $0.0 + j \omega L  $ 0 + j64.33982 4.95252 + j49.4033
796pF $0.0 -{ j \over \omega C}$ 0 - j 49.9855 497.621 + j8.512

 

Having worked out the impedances for each element, here are the HP 35s keystrokes:

50 ENTER ENTER ENTER      // Fill the stack with the first value Z1
0i16.57864 +/-            // Enter Z2 The +/- key used to be the CHS key
×                         // Z1×Z2
x ⇔ y                    // Interchange the x and y stack values 
last x                    // Z2 (The complex no. calculator has no last x - you will have to
                          // store and recall.)
+                         // Z1 + Z2
÷                         // Z1×Z2 /(Z1 + Z2) or Z1||Z2
(result) 4.95252i14.936   // Combined impedance of R and 1st 
ENTER ENTER ENTER         // Fill stack with new value of Z1
0i64.33982                // Z2
+                         // Z1 + Z2
(result)4.95252i49.4033   // Combined impedance of 1st R L and C
ENTER ENTER ENTER         // Fill stack with new value of Z1 
0i49.9855 +/-             // New value of Z2
×                         // Z1×Z2
x ⇔ y
last x                    // Z2
+                         // Z1 + Z2 
÷                         // Z1×Z2 / (Z1 + Z2)
(result)497.612i8.512     // Done!
abs 497.69 (Magnitude of Z)

arg 0.97 (Phase angle in degrees)(Nearly a pure resistance of 500 Ohms at 4 MHz)

I could not find a Rectangular to Polar key on The HP, but you can use an rθ display mode.

Admittedly - its still a lot of PT.. Maybe I will port the "Impedance of a Ladder Network"  program by HP to the calcs section.

Summary

This feature is a brief introduction to AC Circuit theory. Here are the main points:

  • AC Voltage, Current and Impedance have magnitude and phase
  • Impedance is the ratio of Voltage to Current in an AC circuit. It is measured in Ohms
  • Voltage, Current and Impedance can conveniently be expressed as complex numbers
  • The operator j in electrical engineering is the same as the mathematical operator i
  • The article demonstrates the use of a calculator to evaluate the impedance of passive circuit elements in a ladder network
  • A complex number calculator reduces the complexity of AC calculations to the level of DC calculations. No more horrendous formulae.

References

Nelkon M and Parker.P, "Advanced Level Physics",Heinemann,1974
Horowitz P and Hill W,"The Art of Electronics",Cambridge University Press,1988 (You might find this a bit confusing. Its the only confusing section in the whole book.)

Also see Wikipedia's "Electrical Impedance" article. It takes fewer liberties than I have taken.

Last Updated on Tuesday, 11 June 2013 13:33
 
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