The circuit shown in Fig. 26–89 is a primitive 4-bit digital-t...

Question

The circuit shown in Fig. 26–89 is a primitive 4-bit digital-to-analog converter (DAC). In this circuit, to represent each digit (2ⁿ) of a binary number, a “1” has the nᵗʰ switch closed whereas zero (“0”) has the switch open. For example, 0010 is represented by closing switch n = 1, while all other switches are open. Show that the voltage V across the 1.0 - Ω resistor for the binary numbers 0001, 0010, 0100, and 1010 (which represent 1, 2, 4, 10) follows the pattern that you expect for a 4-bit DAC.

Accepted Answer

Welcome back everyone. In this problem. A three bit digital to analog converter or DC circuit converts binary inputs into analog voltages using switches and resistors. Each binary bit two to the end is linked to a switch that if one connects the corresponding resistor to the circuit while zero leaves it disconnected, the circuit has an input voltage of 10 volts and includes resistors of 10 5 and 2.5 kg ohms for bits 01 and two respectively. A two ohm resistor is positioned at the output to measure the resulting analog voltage. Calculate the voltage across the two ohm resistor for the binary inputs. 001010011 and 101 which represent decimal values 123 and five respectively. Our diagram shows our D AC circuit and for our answer choices. A says that for binary 001, the voltage is two millivolts for 010 is four millivolts 011 is six millivolts and 101 is 10 millivolts. B says there are 247 and 10 millivolts respectively. C 246 and nine millivolts and D 357 and nine millivolts respectively. Now, before we figure out the voltage across the two resistor, let's make sure we understand our binary input system here. So if we think about it, if we look at all of our digits, we have three digits in our binary number. OK? Where our first digit represents N equals zero. So that circuit at the top or that switch at the top, rather the second N equals one and the third N equals two. OK? So for example, if we're talking about the binary input 001, that means or one will be here. So only the circuit are not the, only the circuit where N equals zero will be connected. OK? If we're talking about 010, again, let's go back to the drawing board. So that's 010. We notice here that our one is at N equals one. So that means only our one is going to be connected. OK? And that's basically how we can interpret the rest. So we can use this to help us figure out the output voltage, the voltage across the tool or resistor. OK? Now let's take a high level look at this before we start to figure out what those voltages are. What, what's our plan here? What could we do? Well notice here that since the two ohm resistor is very small compared to RNR one and R two, which are all in the kilo ohm range, it does not significantly affect the overall circuit resistance. So we can calculate the current based on the larger resistor network and then use this current to find the voltage across the two ohm resistor. In other words, we could say that I or current is going to be equal to our input voltage divided by the E equivalent resistance of the circuit factoring in our larger resistors here. OK. And the voltage, the output voltage that we want is going to be equal to that current multiplied by the output resistance. In this terms our two ohm resistor. So if we substitute it here, then to figure out that output voltage, we could say it equals to the input voltage divided by the equivalent resistance multiplied by our output resistance. Now we know our input voltage remains the same 10 volts. Our output voltage. As a matter of fact, let's make a note of that. Our input voltage is 10 volts. OK? And for the purpose of this problem, we're referring to our output voltage as sorry, our output resistance as two ohms. So the only thing that's really changing here each time is our equivalent resistance. So let's go ahead for each of these and use this formula depending on which resistor is connected. So let's start with binary 001. OK. I know for binary 001 which is only R not OK. Then that means the voltage is going to be well before we calculate that first our equivalent resistance, rather equivalent resistance since only R knot will be connected will be R knot, which is 10 kg. So since that's the value of our equivalent resistance, then the voltage is going to be equal to 10 volts divided by 10 kg ohms. OK. Multiplied by two ohms. And when we go ahead and solve here, then we should get it OK to be equal to 2 million volts. So that would be the voltage for the binary 001 next for 010. Like we said earlier, this would be only, this is only our one. OK. So now when only R one is connected, then that tells us our equivalent resistance will be equal to R one which is 5 kg ohms. OK. And now that means the voltage then is going to be, it's gonna be 10 volts divided by 5 kg ohms multiplied by two ohms. OK. Which is going to be equal to where are my live woods? Now, in our third configuration, in our third binary input, it's going to be 011. OK. So now if we think about it here, if we change this to 011, what does this tell us? Well, no, this tells us that are not and R one are connected in parallel. In other words, by 011, these two are going to be connected. Thus, we're going to find the equivalent resistance a bit differently because no, OK. For 011 where R knot is parallel to R one. OK, then the equivalent resistance, we're gonna calculate it using what we know about two resistors in parallel. So that's going to be equal to one divided by RN plus one, divided by R one. And then we're going to find the reciprocal of our result. OK. So now in this case, that is gonna be equal to one divided by 10 kg ohms plus one divided by 5 kg. We're gonna reciprocate that result. I know when you wear the salt, we should get the equivalent resistance to be 3.33 kg ohms. Now that we have the equivalent resistance that tells us then that the output voltage here is gonna be equal to 10 volts divided by 3.33 kg ohms multiplied by two ohms. OK? And now when we go ahead and sulfate this one, we should get it to be six milliwatts. Now, finally, let's try to find it when our binary input is 101. Now let me clear my boxes here. OK? And no, for 101, think about it. 101, which ones are connected? Well, no, we can tell that N equals two is connected and N equals zero is connected. So R not is in parallel with R two. So now let's make some space here for binary binary input, 101, you're gonna have R not parallel to R two So again, that means to get the equivalent resistance, we are going to find one divided by 10 kg ohms. OK. Plus one divided by 2.5 kg ohms. And then we're gonna reciprocate that result. And now when we do that, OK, then we should get it to be equal to 2 kg ohms. So with an equivalent resistance of 2 kg ohms, that means the voltage is going to be equal to 10 volts divided by 2 kg ohms eh multiplied by two ohms which is going to be equal to 1010 million votes. Ok. So no, what do we know we know that for input 001, the voltage is two millivolts, 010 is four millivolts, 011 is six millivolts and 101 is 10 millivolts. When we look back at our answer choices that tells us a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Digital-to-Analog Conversion

Binary Number Representation

Voltage Division in Circuits

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