Fullscreen HTML5Med Res (??MB)HTML4

This is arguably the handiest tool anyone involved in electronic design could wish for! It avoids the need to make impedance or reactance calculations and there is no need to revise long-forgotten formulas. Reactance Chart for easy RC, RL or LC network design W ith this reactance chart, you can easily check the -3dB rolloff of a simple RC (resistor-capacitor) or RL (resistor-inductor) network or find the resonant frequency of an LC (inductor-capacitor) network. Why do we need such a tool? Sure, you can easily Google to get a calculator for almost any purpose but typically such online calculators give you a couple of fields to fill in with the known values, say, resistance or capacitance and frequency, and then you click the “Calculate” button to get the answer. But this does not allow you to get an overall picture of how passive components such as resistors, capacitors and inductors interact to determine the frequency behaviour of circuits. For example, if you look at a typical amplifier circuit, it is not the active components such as op amps, transistors or Mosfets which largely determine the frequency response, it is the interaction of the above mentioned passive components. For example, in the very simplified circuit of a complementary symmetry amplifier in Fig.1, the low frequency rolloff is determined by the interaction of resistor R1 and capacitor C1 in the input circuit and also in the negative feedback network, by R2 and C2. On the other hand, the high frequency performance is determined by the interaction of inductors, resistors and capacitors in the input and output of the amplifier. For example, ostensibly all that resistor R1 does is to provide input bias current to transistor Q1. It also sets the voltage at the output of the amplifier (to 0V). But just as importantly, those R1 and C1 values partly determine the low frequency rolloff of the amplifier. 88  Silicon Chip formulas but let us transfer the process to the reactance chart of Fig.2 (opposite), with a few examples. Say you want to know the impedance of a 100nF (100 nanofarads or 0.1µF) capacitor at a frequency of 1kHz. We have highlighted in red how you read the values off the chart, in Fig.7. The first step is to find the value of 100nF on the right-hand vertical axis. Then you trace down the line at 45° to where it intersects the horizontal line for 10kHz which again is marked on the right-hand vertical axis. You then take a vertical (red) line down from that “intersection” to the horizontal axis. The value shown where the red line intersects that horizontal axis is about 1.6kΩ (the calculated impedance is actually 1.592kΩ). So the three steps in this process are shown as red lines on the reduced chart of Fig.7. Note that all the axes on this chart are logarithmic and this means that when you are interpolating values between actual printed lines, the value you read off the respective axis is always a bit of a guesstimate. That’s By LEO SIMPSON And capacitor C1 can also determine the ultimate signal-to-ratio of the amplifier at very low frequencies, because we need it to have a low impedance. So there is more to these simple passive components than meets the eye. So let’s look at how you can determine the impedance of any capacitor or inductor from the wall chart. First, the impedance of a capacitor at any frequency can be calculated by the formula Z = 1/(2fC) where Z is the impedance in Ohms;  i is the constant 3.1415926...; f is the frequency in Hertz and C is the capacitance in Farads. Similarly, the impedance of an inductor at any frequency can be calculated by the formula Z = 2fL where L is the inductance in Henries and f is the frequency in Hertz. You can calculate impedances to your heart’s content using the above Fig.1: In this typical audio amplifier, the overall frequency response is mainly determined by R1 & C1 at the input and R2 & C2 in the feedback network. SIGNAL INPUT C1 B R1 +VCC B E E C C + – C E B OUTPUT R2 B C2 B C E C E −VEE siliconchip.com.au SILICON CHIP REACTANCE – INDUCTANCE – CAPACITANCE – FREQUENCY 1n F 10 0p F 10 pF 0. 1p F .0 1p F H 1 .0 H 1 0. H 1 H 10 H H 0 10 1m 1p F READY RECKONER .COM.AU H m 10 10 nF 100MHz 10MHz 10 0n F H 0m 10 1H 1 F 1MHz H 10 10 F 100kHz 0H 10 10 0 F 10kHz H 10 1k 00 F 1kHz 10 kH 10 00 0 F 100Hz siliconchip.com.au 1M 10 H 0k 10 00 0 0 F 10Hz 100k 10k 1k 100 January 2016  89 10 1Hz 1 L C OUT OUT R Fig.4 HIGH PASS F F pF 10 F 1p H 1p 1m .0 H 1m 1p Fig.6 0. H 1m H .0 C green lines on Fig.7.) You can use a similar process when working with “high pass” filters and in the simplest case, the positions of the resistor and capacitor in the circuit of Fig.3 are swapped to give the circuit in Fig.4. In this case, the circuit passes high frequencies and progressively blocks lower frequencies due to the impedance of the capacitor increasing as the frequency is reduced. Feeling adventurous? Let’s take a circuit example involving an inductor and resistor, an RL network set up as a low pass filter. You will often see examples of this sort of network at the input of a preamplifier where we want to block extremely high frequencies by using a ferrite bead inductor. In this case, if you look at the formula for the reactance of an inductor, you will realise that it rises in a linear fashion with increasing frequencies, eg, a doubling a frequency will double the reactance. By the way, for the purpose of using this chart, the terms reactance and impedance mean the same thing. In fact, some readers would regard the term mH 10 0m 10 H 1m F OK though because if you had used the formula to calculate the precise value, you would always round it off when selecting an actual component value for a circuit. Which brings us to the next example. Say you need to come up with a simple RC filter which will roll off frequencies above 20kHz (the -3dB point) and then roll off at -6dB octave above that point. This is the simplest possible “low pass” filter, meaning that it passes low frequencies and attenuates (rolls off) higher frequencies. The circuit is shown in Fig.3. So if the resistor value R is known to be 8kΩ and the wanted cut-off frequency is 20kHz, you take a vertical line (green) up from the 8kΩ mark on the horizontal axis until it meets the horizontal line corresponding to a frequency of 20kHz on the right-hand vertical axis. You then take a line up at 45° until it meets the top horizontal axis which corresponds to a value of a whisker over 1nF. (The calculated value is 992pF or almost exactly one nanofarard. We have shown three steps in this process with L Fig.5 0. Fig.3 LOW PASS R 100MHz 10 10 H m nF 1n C IN F IN 0p OUT 10 R IN 10MHz H 10 0m 0n F 10 1H 1m F 1MHz H 10 10 10 10 0m F 10kHz 0H H 10 1k 00 mF 1kHz 10 10 00 0m F 100Hz kH H 10 0k 00 10 00 mF 10Hz 1MW 90  Silicon Chip mF 100kHz Fig.7: the coloured lines on this example of the reactance chart demonstrate examples (see text) of how you can find the impedance of a capacitor or inductor, the cut-off frequency of a simple RC or RL network or the resonant frequency of a series or paralleltuned LC circuit. Many other impedance calculation can by done by a similar two or 3-step process. “reactance” as being obsolete. OK, so now we have a simple RL low pass filter, as shown in the circuit of Fig.5. Let’s say the value of the inductor is 500 microhenries (500µH). You can find where the 500µH line on the chart intersects the top horizontal axis – it is marked in blue and is at an angle of 45° (sloping up to the left) on the chart of Fig.7. In fact all the inductance lines slope up to the left in the same way, just as all the capacitance lines slope up the to right. If we project that line down to the horizontal line for a frequency of 10MHz and then project down from the intersection of those two lines down to the bottom line of the chart and the impedance can be read off as just over 30kΩ (actually 31.4kΩ). That’s fine, but what would be the result if the circuit of Fig.5 used a 500µH inductor and a resistor value of 1kΩ? What would be the cut-off frequency. In this case, we take the same 500µH sloping line and intersect it with the 1kΩ vertical line. In this case, the two lines intersect at a point corresponding to a frequency of just over 300kHz (actually, 318kHz). Finally, let’s find the resonant frequency of a parallel LC network, as shown in Fig.6. In this case, we will use an inductor of 200 millihenries (200mH) and a capacitor of 2 microfarads (2µF). In this case we need to find the intersection of the sloping line for a value of 200mH with the sloping line for a value of 2µF. Both lines are shown in pink and you will see that if you project across to the right from their intersection, you can read the resonant frequency from the vertical right-hand axis as 250Hz (on Fig.7). As you can see, this chart enables you make many thousands of impedance, resistance, capacitance or frequency calculations, all without resorting to SC formulas or calculators. 100kW 10kW 1kW 100W 10W GIANT A2 CHART NOW AVAILABLE! The chart overleaf is great . . . but imagine how much easier it would be to use if it was larger! SILICON CHIP now has available HUGE A2 (420 x 590mm) charts, printed on heavy art paper, ready for your lab, workshop or office! Price is just $10.00 each inc GST + P&P, mailed folded, or $20 each inc mailed unfolded (in a protective tube). Order from the SILICON CHIP Online Shop (siliconchip.com.au/shop) or call SILICON CHIP during office hours (9-4.30, NSW Time Mon-Fri) to obtain your copy. 1Hz 1W siliconchip.com.au