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Several resistors are connected at a time to form a circuit without branches in the middle, which is called a resistor in a series circuit. The figure below shows a resistor in a series circuit composed of two resistors.
Resistors in Series Circuits
Features of the series circuit:
(1) The current in the series circuit is equal everywhere.
When n resistors are connected in series, then
(2) The total voltage across the circuit is equal to the sum of the divided voltages on the series resistors.
(3) The total resistance of the circuit is equal to the sum of the series resistance.
R is called the equivalent resistance of R1 and R2 in series. After R1 and R2 are replaced with R, it does not affect the current and voltage of the circuit.
In Figure 1, (b) is an equivalent circuit of (a).
When n resistors are connected in series, then
(4) The relationship between voltage distribution and power distribution in series circuits.
Since the current in the series circuit is equal everywhere, so
The above two formulas show that the voltage across each resistor in a series circuit is proportional to the resistance of each resistor. The power consumed by each resistor is also proportional to the resistance of each resistor. Therefore, when the series circuit is composed of n resistors, the voltage division formula of the series circuit can be obtained.
Tip: In practical applications, resistors are often connected in series to expand the measuring range of the voltmeter.
A circuit that connects two or more resistors between two points in a circuit with the same voltage at both ends of the resistor is called a resistor in a parallel circuit.
Resistors in Parallel Circuits
Features of the parallel circuit:
(1) The voltage across each resistor in the circuit is the same.
(2) The total current of the resistors in a parallel circuit is equal to the sum of the currents of the branches.
(3) The inverse of the total resistance of the parallel circuit is equal to the sum of the inverses of each parallel resistor.
(4) The relationship between current distribution and power distribution of resistors in parallel circuits.
In a parallel circuit, the voltages across the parallel resistors are the same, so
The above formula shows that the current of each branch in the parallel circuit is inversely proportional to the resistance. And the power consumed by the resistors of each branch is inversely proportional to the resistance.
When two resistors are connected in parallel, the current through each resistor can be calculated with the shunt equation. The shunt formula is:
The formula above shows that in a resistor in a parallel circuit, the current of the branch with a small resistance is large, while the current of the branch with a large resistance is small.
Note: Resistors in parallel circuits are widely used in daily life. For example, electrical appliances in lighting circuits are usually connected in parallel. Only when the electrical appliances are used in parallel, can other electrical appliances work normally when one of the electrical appliances is disconnected, closed, or has a broken fault.
One resistor can be connected together with numerous other resistors in series and parallel to form complex resistive circuits.
If we connect various resistors in both parallel and series in the same circuit, how do we calculate the sum of the resistance, current, and voltage of these resistors in the circuit?
Resistive circuits that combine series and parallel resistors are commonly referred to as resistor combination or hybrid resistor circuits. The method of calculating the equivalent resistance of a circuit is the same as that of any single series or parallel circuit. We now know that series resistors carry the same current, and parallel resistors have the same voltage.
Calculate the total current (IT) drawn from a 12v power supply in the following circuit.
At first glance, this may seem like a daunting task, but if we observe it carefully, we can see that the two resistors, R 2 and R 3, are actually connected together in series. So we can add the resistance together. Therefore, the combined resistance of this combination is:
R2+R3 = 8Ω+4Ω = 12Ω
So we can replace the resistors R2 and R3 with a resistor of 12Ω
And the circuit now has a resistor RA and a resistor R4 connected in parallel. Then, we can reduce this parallel combination to a single equivalent resistance value R (combination) by using the following resistance formula.
As a result, the resistive circuit now looks like this:
We can see that the two remaining resistances, R1 and R (comb) are connected in a series, and they can be added together again between points A and B.
R = R comb+R1 = 6Ω+6Ω = 12Ω
A single resistor of 12Ω can be used to replace the original four resistors connected in the original circuit.
Now by using Ohm's law, the current value(I) of the circuit is simply calculated as follows:
Therefore, by using the above steps to replace all the resistors connected in series or parallel, we can reduce any complex resistor circuit composed of several resistors into a simple single circuit with only one equivalent resistor.
We can also obtain two branch currents, I1 and I2 by using the further Ohm method:
V(R1) = I*R1 = 1*6 = 6V
V(RA) = VR4 = (12-VR1) = 6V
therefore:
I1 = 6V÷RA = 6÷12 = 0.5A or 500mA
I2 = 6V÷R4 = 6÷12 = 0.5A or 500mA
Because the resistances of the two branches are both 12Ω, I1 and I2 are both 0.5A (or 500mA). Therefore, the total supply current IT = 0.5+0.5 = 1.0A.
After these changes, it is sometimes easier to draw or redraw new circuits with complex resistor combinations and resistor networks, which becomes the visual aids of mathematics. Then continue to replace any series or a parallel combination until you find an equivalent resistance REQ. Let's try another more complicated resistor combination circuit.
To find the equivalent resistance REQ used in the following resistor combination circuit.
Again, this ladder resistor network may seem very complicated at first sight, but as before, it is just a combination of series and parallel resistors connected together. Starting from the right side and using a simplified formula of two parallel resistors, we can find the equivalent resistance/ combination of R8 and R10 and call it RA.
(Formula 4-1)
Therefore RA+R7 = 4+8 = 12Ω
The resistance of 12Ω is now parallel to R6 and the total resistance can be calculated as RB.
(Formula 4-2)
RB+R5 = 4+4 = 8Ω
The resistance value of 8Ω is now in parallel with R4 and can be calculated as RC as shown.
(Formula 4-3)
RC is connected in series with R3, so the total resistance is RC+R3 = 8Ω, which is shown in the figure.
The 8Ω resistance is in parallel with R2 and we can be calculated as RD:
RD is connected in series with R1, so the total resistance is RD+R1 = 4+6 = 10Ω as shown in the figure.
Ultimately, the initial complex resistor network that includes ten independent resistors connected in series and parallel can be replaced by an equivalent resistance REQ of 10Ω.
When we face any circuits consisting of resistors in series and parallel, first we need to identify the simple series and parallel resistance of each branch and then replace them with an equivalent resistance.
This will enable us to reduce the complexity of the circuit and help us convert complex combined resistance circuits into a single equivalent resistor.
However, the calculation of more complex T-pad attenuators and resistance bridge networks cannot be simplified to simple parallel or series circuits with an equivalent resistance. They need to be solved by using Kirchhoff's current law and Kirchhoff's voltage law.
The figure below shows the short circuit in a series circuit. In the circuit, resistor R1 and R2 were originally connected in series, but now the resistor R2 is shorted. At this time, the following changes will occur in the series circuit.
Short Circuit in a Series Circuit
1) After the resistor R2 is short-circuited, only the resistor R1 exists in the series circuit. At this time, the total resistance of the circuit decreases, which is equal to the resistance of the resistor R1.
2) Because the DC working voltage + V in the circuit has not changed, and the total resistance value of the series circuit has been reduced, the current of the series circuit will increase after the resistor R2 is shorted.
The increasing amount of current in the circuit is related to the resistance of the short-circuit resistance R2. If the resistance of R2 is relatively large, the increasing amount in the series circuit after the short circuit will be relatively large, which will cause an overcurrent. When the power supply cannot withstand the excessive current, it may be burned out. So short circuits in series circuits are very harmful.
3) At the same time, because the increased current also flows through other resistors (such as R1) in the series circuit, it will also cause overcurrent in other resistors, which will also damage them.
4) In a series circuit, if the current flowing through a certain component is increased, it indicates that there is a short circuit in the circuit. Because the current in a series circuit will increase after a short circuit, the current flowing through other resistors will also increase, which will also increase the voltage drop across the other resistors.
5) A short-circuit fault in a series circuit belongs to serious failure. It may cause damage to all components in the series circuit as the current flowing in the series circuit increases.
When an open circuit occurs in a resistor in a series circuit, there will be no current flowing in the circuit no matter which part of the series circuit is open.
An open circuit fault generally does little harm to the series circuits. However, sometimes due to the open circuit, the drive circuit voltage of the load rises, causing the drive circuit to break down.
Open Circuit in a Series Circuit
The following table is a summary of the fault analysis of the series circuits with resistor R1 and R2.
Component Name | Fault Type | Fault Analysis | Analyzing Thought |
R1 | Open Circuit | No current flows through R1 and R2, and no voltage can be measured across R1 and R2 | No current flows through the resistor, so there is no voltage. |
Short Circuit | The current flowing through R2 increases, which may burn out R2 | The total resistance decreases, which increases the total current. If the current flowing through R2 is too large, it will burn out R2. | |
Resistance Increased | The current flowing through R1 and R2 decreases, the voltage of R1 increases, and the voltage of R2 decreases. | After the resistance of R1 increases, the total resistance increases, the total current decreases, and the voltage drop of R2 decreases, so the voltage of R1 increases. | |
Resistance Decreased | The current flowing through R1 and R2 increases, the voltage of R1 decreases, and the voltage of R2 increases | After the resistance of R1 increases, the total resistance decreases, and the total current increases, so that the voltage on R1 decreases | |
Poor Contact | The circuit will work normally on one occasion, and work abnormally on another. | The circuit works normally when contact is normal, the circuit works abnormally when contact is abnormal | |
R2 | The Above Five Faults | Failure analysis is the same as above (replace R1 with R2) | The analytical thinking is the same as above (replace R1 with R2) |
There are many ways to check the failure of the resistors in the series circuit. For example, the resistance value of each resistor in the circuit can be measured by the ohm range of the multimeter. However, in troubleshooting, the inspection method is often flexibly selected.
Structure of a Multimeter
If the device works in a DC circuit, use the DC voltage range of the multimeter to measure the voltage across R1 (two test leads are respectively connected to the two pins of R1), then you can find out whether the circuit has an open circuit fault or not.
If the device works in an AC circuit, the AC voltage of the multimeter and the AC voltage range of the digital meter can be used to measure the AC voltage across R1.
In theory, you can also use the above method to measure the voltage across R1 when checking the short circuit fault of the resistors in the series circuit. If the voltage across R1 is higher than the normal value, it can be explained that there is a short circuit fault in the circuit. Because only the short circuit in the series circuit will increase the current and the voltage across R1.
However, there is a problem with the above-mentioned short-circuit fault inspection method. It is necessary to know what the normal voltage across R1 is, otherwise it cannot be determined whether the current in the circuit has increased.
When the circuit is powered off, use a resistance gear of the multimeter to measure the total resistance of the parallel circuit. Under normal circumstances, the measured total resistance value should be <R1 <R2.
Detection of Open Circuit in Parallel Circuits
If the measured resistance value is greater than either R1 and R2, it means that R1 or R2 is open in the circuit. Specifically, to judge which circuit is open, we could measure the current of each resistance branch.
If the measured total resistance is zero, it means that there is a short circuit in the parallel circuit. If you need to understand the detailed Failure position and cause, you need to do the further measurement. This is of great significance for troubleshooting, which determines the scope of the faulty circuit and the direction of inspection.
ESR is short for equivalent series resistance. It is an equivalent "series" resistance, meaning that connecting two resistors in series will increase this value while connecting them in parallel will decrease it.
The emergence of ESR caused the behavior of capacitors to deviate from its original definition. In theory, a perfect capacitor does not generate any energy loss by itself, but, because the material used to make the capacitor has resistance and the insulation medium has energy loss, the capacitor becomes imperfect. This loss appears externally as if a resistor is connected in series with the capacitor, so it is named "equivalent series resistance".
We believe that the voltage on the capacitor cannot be abruptly changed. When a current is suddenly applied to the capacitor, the voltage of the capacitor will rise from 0 due to its own charging. But with ESR, the resistor itself generates a voltage drop, which causes a sudden change in the voltage across the capacitor. Undoubtedly, this will reduce the filtering effect of the capacitor, so many high-quality power supplies use low ESR capacitors.
Similarly, in the case of oscillating circuits, ESR will change the function of the circuit, causing serious consequences such as circuit failure or even damage.
So in most cases, low ESR capacitors tend to perform better than high ESR capacitors. But there are exceptions, and sometimes ESR is used to do something useful.
For example, in the voltage stabilization circuit, when the load is transient, a capacitor with a certain ESR will immediately generate fluctuations and trigger the feedback circuit. This fast response is obtained at the expense of certain transient performance. especially when the response speed of the power tube is slow, and the volume or capacity of the capacitor is strictly limited. This situation is seen in some three-terminal voltage regulators with mos tubes as regulators or similar circuits. At this time, low ESR will reduce overall performance.
In fact, there are more occasions that require lower ESR, and low-ESR bulk capacitors are relatively expensive. Therefore, parallel connection is adopted in many switching power supplies. People connect multiple aluminum electrolytic capacitors with relatively high ESR in parallel to form a low-ESR capacitor. It is often cost-effective to sacrifice a certain amount of PCB space in exchange for a reduction in device costs.
Another concept similar to ESR is ESL, which is Equivalent Series Inductance. Early rolled capacitors often had very high ESL, and the larger the capacitance, the larger the ESL. ESL often becomes part of ESR, and also causes some circuit failures, such as series resonance. However, in terms of capacity, the proportion of ESL is too small, and the probability of problems is very small. In addition to the advancement of the capacitor production process, ESL has been gradually ignored, and ESR is used as the main reference factor besides capacity.
ESL and ESR Cancellation for Capacitors
By the way, the capacitance also has a quality factor Q similar to that of inductance. This coefficient is inversely proportional to ESR and is related to frequency. It is also rarely used.
Circuit failures caused by ESR are often difficult to detect, and the effects of ESR are easily overlooked during the design process. The simple way is that, during the simulation, if you cannot choose the specific parameters of the capacitor, you can try to artificially connect a small resistor in series with the capacitor to simulate the effect of ESR. Generally, the ESR of Tantalum Capacitors is usually below 100 milliohms, and aluminum electrolytic capacitors are higher than this value, and the ESR of some types of capacitors can even reach several ohms.
Typical relationship between ESR and Frequency for Tantalum Capacitors
The relationship between ESR and the ripple voltage can be expressed by the formula:
V = R(ESR)×I
In this formula, V is the ripple voltage, R is the ESR of the capacitor, and I is the current. It can be seen that when the current is increased, the ripple voltage will be doubled even when the ESR remains unchanged.
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