Introduction
Resistors are usually connected in a circuit in various ways, and the two most basic ways are series and parallel. This article will mainly introduce these two connection methods, including their definitions, formulas, circuit diagrams, examples and identification methods. In addition, the article also introduces Ohm's law and Kirchhoff's law, which are very important in understanding the series and parallel connections of resistors.
You may need these two calculators in reading this arrticle:
② Parallel and Series Resistance Calculator
The following video explains the basics of resistors in series and parallel, which can promote your understanding of this article. But it does not matter so much if you skip this video since the article explains in detail and is comprehensive.
Resistors in series and parallel  deriving the formula
Catalog
5.2 Kirchhoff's First Law (Nodal Current Law) 
I Series Connection of Resistors
(1) Circuit characteristics
Figure1. Resistors in series
The figure shows the series connection of n resistors, and the voltage and current reference directions are related. The circuit characteristics are derived from Kirchhoff’s law:
(A) The resistors are connected in sequence. According to KCL, the current flowing through the resistors is the same;
(B) According to KVL, the total voltage of the circuit is equal to the sum of the voltages of the series resistors, namely:
(2) Equivalent resistance
Figure2. Equivalent resistance circuit
Substituting Ohm's law into the voltage expression, we get:
The above formula illustrates that the series circuit of multiple resistors in Figure (a) and the circuit of single resistor in Figure (b) have the same VCR, which is equivalent to each other.
The equivalent resistance is:
In conclusion:
 The resistors are connected in series, and the equivalent resistance is equal to the sum of the subresistances;
 The equivalent resistance is greater than any one of the series resistance.
 The partial pressure of series resistance
If the total voltage across the series resistor is known, what is the divided voltage on each resistor?
From figure (a) and figure (b) we know:
Meet
In conclusion:
Resistors are connected in series, and the voltage on each subresistor is proportional to the resistance value. The higher the resistance value, the higher the voltage. Therefore, the series circuit can be used as a voltage divider circuit.
Example 1: Calculate the voltage across the two series resistors as shown in the figure.
Figure3. Circuit of Example1
Solution: From the partial pressure formula of series resistance:
(Note the direction of U_{2})
(3) Power
The power of each resistor is:
So
Total power:
Draw conclusions from the above formulas:
 When resistors are connected in series, the power consumed by each resistor is proportional to the size of the resistor, that is, the larger the resistance, the larger the power consumed;
 The power consumed by the equivalent resistance is equal to the sum of the power consumed by each series resistor.
II Parallel Connection of Resistors
(1) Circuit characteristics
Figure4. Parallel circuit characteristics
The figure shows the parallel connection of n resistors, and the voltage and current reference directions are related. The circuit characteristics are derived from Kirchhoff's law:
(a) The two ends of each resistor are connected together. According to KVL, the two ends of each resistor are at the same voltage;
(b) According to KCL, the total current of the circuit is equal to the sum of the currents flowing through the parallel resistors, namely:
(2) Equivalent resistance
Figure5. Equivalent resistance in parallel connection
Substituting Ohm's law into the current expression, we get:
G =1/R is the conductance
The above formula illustrates that the parallel circuit of multiple resistors in Figure (a) and the circuit of single resistor in Figure (b) have the same VCR, which is equivalent to each other.
The equivalent conductance is:
Therefore,
Namely,
The most commonly used formula to find the equivalent resistance when two resistors are connected in parallel:
In conclusion:
 The resistors are connected in parallel, and the equivalent conductance is equal to the sum of the conductances and greater than the partial conductance;
 The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the subresistances, and the equivalent resistance is less than any parallel subresistance.
 Current distribution of parallel resistance
If the total current of the parallel resistance circuit is known, find the current on each subresistance and call it a shunt. From figure (a) and figure (b) we know:
Namely,
Meet
For two resistors in parallel, there are:
Conclusion: When the resistors are connected in parallel, the current on each subresistor is inversely proportional to the resistance value, and the current divided by the larger resistance value is smaller. Therefore, the parallel resistor circuit can be used as a shunt circuit.
(3) Power
The power of each resistor is:
So
Total power:
Draw conclusions from the above formulas:
 When resistors are connected in parallel, the power consumed by each resistor is inversely proportional to the size of the resistor, that is, the larger the resistance, the smaller the power consumed;
 The power consumed by the equivalent resistor is equal to the sum of the power consumed by each parallel connected resistor.
consumed by each series resistor.
III Resistor Combination(Mixed Resistor Circuit)
A circuit with resistors connected in series and connected in parallel is called a resistor combination or mixed resistor circuit. The part where the resistors are connected in series has the characteristics of a resistor series circuit, and the part where the resistors are connected in parallel has the characteristics of a resistor parallel circuit.
Example 2: The circuit is shown in the figure, please calculate the voltage and current of each branch.
Figure6. Example circuit 2
Solution: This is a resistor series and parallel circuit. First find the equivalent resistance Reg = 11W, and the current and voltage of each branch are:
The general steps for solving series and parallel circuits can be obtained from the above examples:
⚫ Find the equivalent resistance or equivalent conductance;
⚫ Apply Ohm's law to find the total voltage or total current;
⚫ Apply Ohm's law or voltage division and shunt formula to find the current and voltage on each resistor.
Therefore, the key issue in analyzing seriesparallel circuits is to distinguish the relationship between series and parallel circuits.
To determine the seriesparallel relationship of the circuit, the following 4 points should be mastered:
⚫ Look at the structural characteristics of the circuit. If two resistors are connected endtoend, they are connected in series;
⚫ Look at the relationship between voltage and current. If the current flowing through the two resistors is the same current, it is connected in series; if the two electrical groups bear the same voltage, it is connected in parallel.
⚫ Equivalent to deformation of the circuit. For example, the left branch can be twisted to the right, the upper branch can be turned down, the curved branch can be straightened, etc.; the short circuit in the circuit can be compressed and extended at will; the multipoint grounding can be connected by a short circuit . Generally, if it is really a problem with a resistor series circuit, it can be distinguished.
⚫ Find the equipotential point. For circuits with symmetrical characteristics, if two points can be judged to be equipotential points, according to the concept of circuit equivalence, one is to use short wires to connect the equipotential points; the other is to break the branch that connects the equipotential points. Open (because there is no current in the branch), thus obtain the seriesparallel relationship of the resistance.
IV Ohm's Law
4.1 What is Ohm's Law?
(1) The content of Ohm's law
When there is a potential difference between the two ends of the conductor, an electric field appears inside the conductor, and the charge moves in a directional motion under the force of the electric field to generate current. German physicist Ohm summed up Ohm's law in 1826 through a large number of experiments: Under steady conditions, the intensity of the current passing through a section of conductor is proportional to the voltage across the conductor.
(2) Mathematical expression of Ohm's law
Note: The unit of the physical quantity in the formula: the unit of I is ampere (A), the unit of U is volt (V), and the unit of R is ohm (Ω).
The proportional coefficient R in the formula is determined by the properties of the conductor and is called the resistance of the conductor. Unit: Ohm (Ω). The reciprocal of resistance is called conductance and is represented by G, that is
Unit: Siemens (S).
(3) Understanding and explanation of Ohm's law
● Applicable conditions of Ohm's law: applicable to pure resistance circuits (that is, when working with electrical appliances, the consumed electrical energy is completely converted into internal energy.)
● I, U and R in the formula must correspond to the same conductor or the same circuit. If it is in different time, different conductor or different section of circuit, I, U, and R can not be mixed, therefore, the three physical quantities should be marked with angles in order to distinguish under normal circumstances.
● For the same conductor (that is, R does not change), I and U are proportional; for the same power source (that is, U does not change), I and R are inversely proportional.
● R=ρL/S is the definition of resistance, which means that the resistance of a conductor is determined by the material, length and crosssectional area of the conductor itself. In addition, resistance is also related to factors such as temperature.
● The formula transformed from Ohm's law is a measure of resistance. It indicates that the resistance of a conductor can be given by U/I, that is, the ratio of R to U and I is related, but the magnitude of R itself is related to the applied voltage U and the passing current Factors such as the size of I are irrelevant.
● Knowing any two quantities among I, U and R, you can find another quantity.
● Issues that need special attention and reemphasis: I, U and R in the formula must be in the same circuit; when using the formula to calculate, the unit of each physical quantity must be unified.
The above explanations are all part of Ohm’s law, which only applies to pure resistance circuits.
(4) Pure resistance circuit
A pure resistance circuit is a circuit with only resistance elements in addition to the power supply, or inductance and capacitance elements, but their influence on the circuit is negligible. The voltage and current have the same frequency and phase.
The resistance converts all the energy obtained from the power supply into internal energy. This kind of circuit is called a pure resistance circuit. Here is a brief explanation from the energy point of view.
Basically, as long as there is no conversion of electric energy other than internal energy, this circuit is a pure resistance circuit.
4.2 What is Closed Circuit Ohm's Law?
In an AC circuit, Ohm's law also holds, but the resistance R should be changed to impedance Z, that is, I = U/Z. If the circuit is closed and contains a power supply, it is called a full circuit, as shown in the figure below. The dotted line in the figure is the power supply, which is called an internal circuit. The circuit outside the power supply is called an external circuit. Since the power supply has internal resistance, the current not only has a voltage drop when passing through an external circuit, but also has an internal voltage drop when passing through an internal circuit. In the whole circuit, the current intensity is proportional to the electromotive force E of the power supply, and inversely proportional to the resistance (R+r) of the whole circuit (including the inner circuit and the outer circuit). This is the Ohm's law of the whole circuit, expressed by the formula:
Where I the current in the circuit, A; E the electromotive force of the power supply, V; R the resistance of the external circuit, Ω; r the resistance of the internal circuit, Ω.
From the above formula, in the circuit shown in the figure below, E=IR+Ir=Uouter+Uinner.
Figure7. The simplest closed circuit
In the formula, U external = IRexternal circuit voltage; U internal = Irinternal circuit voltage.
It should be noted that, since the internal resistance of the power supply itself and the internal resistance of the connecting wires are generally not large, the calculation results that are ignored in the calculation are basically correct. But sometimes it is necessary to calculate the internal voltage drop of the power supply, and to accurately calculate the current of the whole circuit, it is necessary to use the whole circuit Ohm's law. For example, in the figure below, if E=10V, r=0.1Ω, R=1kΩ, then:
Figure8. An application example of Ohm's law of closed circuit
① When S is connected to the 1 position, the circuit is in the open state,
Ammeter reading
The reading of the voltmeter is U=IR=0.01×1000=10 (V), or U=EIr=100.01×0.1≈10 (V).
②When S is connected to the 2 position, the circuit is in an open state, so the reading of the ammeter is 0; the reading of the voltmeter is U=E=10(V).
③When S is connected to the 3 position, the circuit is in a shortcircuit state, the reading of the ammeter is I=E/r=10/0.1=100(A)A; the reading of the voltmeter U=0(V).
4.3 The Key Points of Studying Ohm's Law
Ohm's law is an important basic law in electricity. It is a law that is summarized and summarized through experiments. To master this law, we must pay attention to the following points:
(1) Ohm's law applies to the entire circuit or a part of the circuit from the positive pole to the negative pole of the power supply, and it is a pure resistance circuit.
(2) The current I "passing through" in Ohm's law, the voltage U at "both ends" and the resistance R of the "conductor" are all corresponding physical quantities on the same conductor or the same circuit. The above relationship does not exist between the current, voltage, and resistance of different conductors. Therefore, when using the formula I=U/R, the current, voltage, and resistance of the same conductor or the same circuit must be substituted into the calculation, and the three correspond one to one.
(3) There is simultaneity among the three physical quantities in Ohm’s law. Even on the same part of the circuit, the closing or opening of the switch and the movement of the sliding position of the sliding varistor will cause the change of the circuit, which will lead to the current in the circuit. , Voltage, resistance changes, so the three quantities in the formula I=U/R are the same time value.
(4) The difference between I=U/R and R=U/I:
Ohm's law expression I=U/R means that the current in the conductor is related to the voltage across the conductor and the resistance in the conductor. When the resistance R is constant, the current I in the conductor is proportional to the voltage U across the conductor; when the voltage U across the conductor is constant, the current I in the conductor is inversely proportional to the resistance R of the conductor.
R=U/I is derived from Ohm’s law expression. It means that the resistance value of a certain section of conductor is equal to the ratio of the voltage across the section of the conductor to the current passing through it. This ratio R is the property of the conductor itself and cannot be understood as R is directly proportional to U and inversely proportional to I. This is also the difference between physics and mathematics.
(5) Ohm's law reflects the causal relationship between current intensity and voltage, and the restrictive relationship between current intensity and resistance under certain conditions. That is, when the resistance is constant, the current intensity is proportional to the voltage across the conductor; when the voltage is constant, the current intensity is inversely proportional to the resistance of the conductor. When establishing a proportional relationship, we must pay attention to its conditions. Ohm's law states that the current intensity through a conductor is determined by two factors, the voltage across the conductor and the resistance of the conductor.
V Kirchhoff's Law
Kirchhoff's law includes the first law and the second law. They are the basic laws that are indispensable for the analysis and calculation of complex circuits.
5.1 Concepts
• Branch
A twoterminal element connected in a circuit is a branch. Usually a certain current flows through the branch. (This definition is not universal. For example, if two components are connected in series and then connected in a circuit, it can only be regarded as a branch.)
• Node
The connection point between the branch and the branch is called a node. Usually the current diverges at the junction.
• Loop loop
A closed path formed by branches is called a loop.
Figure9. 6 elements, 6 branches, 4 nodes, 3 independent circuits
5.2 Kirchhoff's First Law (Nodal Current Law)
The textual expression of KCL: For any node, the algebraic sum of the current flowing into (or out of) the node is equal to zero.
Its mathematical expression:
The regulation of current positive and negative: Generally, the current flowing into the node is positive, and the current flowing out of the node is negative.
The physical meaning of KCL: conservation of charge
Note: KCL is not only applicable to a node, but also to a part of the circuit, as shown in the shaded part of the above figure:i_{3}＝i_{6}
5.3 Kirchhoff's Second Law (Law of Loop Voltage)
KVL’s literal expression: In any closed loop of the circuit, go around a circle in a certain direction, and the algebraic sum of the voltage of each segment is zero.
That is: or. When applying the law of loop voltage, the electromotive force is often written on the left side of the equation, and the voltage is written on the right side of the equation.
The method for determining the sign of each electromotive force and voltage in the second expression is as follows:
① First select the current direction of each branch.
② Any choice of the detour direction along the loop (clockwise or counterclockwise).
③ If the direction of the current flowing through the resistor is the same as the detour direction, the voltage drop on the resistor is positive, otherwise, it is negative.
④ If the direction of the electromotive force is the same as the direction of the orbit, the electromotive force is positive, otherwise, it is negative.
The physical meaning of KVL: energy conservation.
5.4 Application Note of Kirchhoff's Law
• Kirchhoff’s law is a general law that the circuit should satisfy, and has nothing to do with the specific properties of the components;
• Kirchhoff’s law applies to any lumped circuit, that is, nonlinear, timevarying circuits, etc.;
• Application steps:
A. Divide the branch roads and number them;
B. Specify the branch current and voltage reference direction, and generally need to be associated;
C. Select the appropriate node according to the meaning of the question, and apply KCL;
D. Or choose the appropriate circuit according to the meaning of the question, apply KVL, and pay attention to independence.
Example: Use KVL to derive the relationship between the total resistance and the subresistance and the voltage division formula in the series resistance circuit.
Apply KVL according to the current and voltage reference direction and the detour direction of the loop calibrated in the figure:
u+u_{1}+u_{2}+…+u_{n} = 0 or u=u_{1}+u_{2}+…+u_{n}
Because the voltage and current of each resistor obey Ohm's law: u_{k}=iR_{k}, there are:
u = i × R_{1} + i× R_{2} +...... +i× R_{n}
= i× ( R_{1}+R_{2}+…+R_{n})
= i R_{e}
among them:
R_{e}=R_{1}+R_{2}+…+R_{n}, which is the total resistance or equivalent resistance.
u_{k} = iR_{k}=( u/R_{e} ) R_{k, }which is the voltage division formula of the series circuit
VI Series and Parallel Circuit Identification Methods
Method 1: Current flow method
(1) Starting from the positive pole of the power supply, use arrows to mark the path of the current along the connected wires, and finally return to the negative pole of the power supply;
(2) Observe whether the current has a shunt and confluence point:
If there is only one path for the current in the circuit, the components are connected in series (as shown in Figure a below);
If there is a shunt point and a confluence point in the circuit, that is, the direction of the current is greater than one path, the components between the shunt point and the confluence point are connected in parallel (as shown in Figure b below)
Figure10. Current flow method
Method 2: Demolition method
Remove any electrical appliances:
If the other electrical appliance cannot work, the two electrical appliances are connected in series (as shown in Figure a below)
If the other consumer still works without being affected, the two consumers are connected in parallel (as shown in Figure b below)
Figure11. Demolition method
Method 3: Node Method
For other nonintuitive nonseries circuits, the situation is more complicated and needs to be judged according to several steps:
The first step is to mark nodes. That is, use different letters (or symbols) to mark all nodes of the circuit. As shown in the following figure (a), the four points A, B, C, and D are all nodes in the circuit.
The second step is to merge the nodes. According to the characteristics of the nodes, some of the nodes you have marked may be equivalent to the same node. The letters (or symbols) belonging to the same node must be changed to the same letter (or symbol), as shown in the circuit shown in Figure (a) Point A and point C are the same node, C should be changed to A, point B and point D should be the same node, D should be rewritten as B, that is to say, the circuit shown in Figure (a) essentially has two nodes A and B .
Figure12. Node Method
The third step is to determine the connection mode of the circuit. There are usually two ways to judge:
Method one:
Direct judgment: as shown in the figure (a) above, both ends of the resistors R1, R2 and R3 are independently connected to nodes A and B, so R1, R2 and R3 are connected in parallel.
Method Two:
Drawing judgment: that is, draw the intuitive equivalent circuit diagram of the original diagram. The specific drawing method of the intuitive equivalent circuit diagram of the circuit diagram in Figure (a) is: first determine the two points A and B on the paper, and then combine the original diagrams A and B. The components between the two points B are independently connected to the newly determined points A and B, as shown in the above figure (b), that is, the equivalent circuit diagram of figure (a) is figure (b).
Warm reminder: The "node method" is generally used to identify irregular and more complex circuits, which has certain difficulties. There are many ways to identify series and parallel circuits, but you can choose the most suitable method according to your own understanding of the method when using it.
VII Quiz
The voltage dropped across the 300 ohm resistor is
A. 6V
B.9V
C.2V
D.30V
Answer: A
Ⅷ FAQ
1. What is the difference between two resistors connected in series, and two resistors connected in parallel?
When resistors are in series then net resistance is the sum of individual resistances whereas in parallel it is the sum of the reciprocal of individual resistances.
When a resistor is in series the current is the same through all resistors but the voltage is different. The sum of the voltage drop across each resistor is equal to the voltage across a resistor connected in series.
When the resistor is in parallel the voltage across each resistor is the same while the current through each resistor is different.
In series, the net resistance is higher (sum of each resistance) while in parallel net resistance is lower (net resistance is lower than smallest resistance connected in parallel).
2. Why are resistors connected in series and parallel?
Connecting resistors in series increase their total resistance and the power they can handle by distributing the applied voltage. The current flow is the same for each resistor regardless of its resistance.
Connecting resistors in parallel reduce their total resistance while at the same time increasing their power they can handle by sharing the current flow in the circuit. The voltage drop across each resistor is the same regardless of its resistance.
3. What is the difference between resistors in parallel and resistors in a series?
For resistors in parallel, the voltage across them is the same while the current is the sum let take a case of two resistors connected in parallel the formula 1/Req=1/R1+1/R2 further simplify Req=R1*R2/(R1+R2)
While for resistors in series their current is the same but the voltage is the sum and let still take the case of two resistors connected in series to obtain their equivalent Req= R1+R2.
4. How are resistors added in series and parallel?
When resistors are connected one after each other this is called connecting in series. This is shown below. To calculate the total overall resistance of a number of resistors connected in this way you add up the individual resistances. This is done using the following formula: Rtotal = R1 + R2 +R3 and so on.
5. Why is resistance different in series and parallel?
When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, so the total resistance is lower. Each resistor in parallel has the same full voltage of the source applied to it, but divide the total current amongst them.
6. How do you calculate resistors in parallel?
Parallel Resistor Equation
If the two resistances or impedances in parallel are equal and of the same value, then the total or equivalent resistance, RT is equal to half the value of one resistor. That is equal to R/2 and for three equal resistors in parallel, R/3, etc.
7. Why is resistance less in parallel?
When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, so the total resistance is lower.
8. How do you sum resistors in parallel?
The sum of the currents through each path is equal to the total current that flows from the source. You can find total resistance in a Parallel circuit with the following formula: 1/Rt = 1/R1 + 1/R2 + 1/R3 +... If one of the parallel paths is broken, the current will continue to flow in all the other paths.
9. What happens when you add a resistor in series?
When resistors are connected in series, the total voltage (or potential difference) across all the resistors is equal to the sum of the voltages across each resistor. ... In other words, the voltages around the circuit add up to the voltage of the supply.
10. What is the difference between series connection and parallel connection?
A parallel circuit refers to a circuit with two or more two paths for the current to flow. ... In a series circuit, all the components are arranged in a single line. In a parallel circuit, all the components are arranged parallel to each other.
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9 comments
Is it better to wire LEDs in series or parallel?
author
Re:
Series components have the same current through them but fluctuating voltages. Generally speaking, most LED lighting uses a seriesparallel combination. Ideally, for reliability and lighting consistency, it would be best to have one strip of LEDs all wired in series to a constant current driver.
I would like to know for batteries, which is safer, a series connection or a parallel connection?
author
Re:
In general, parallel batteries will provide more current while parallel resisters will pass more current without being destroyed (conditions apply about wattage and heat dissipation…etc). Recharging batteries in parallel can be tricky due to heat dissipation and the internal resistance of each battery type (NiCads are different from Lithium Oxide as they charge differently and require a different circuit.) Please do not mix them. The results can be explosive…. literally.
Batteries in series will provide a higher voltage and will be able to provide a higher voltage output. Resisters in series will reduce current flow. Recharging batteries can also be tricky as the voltage differential across each battery has to be monitored and the power calculated to prevent them from bursting.
Do batteries last longer in parallel or series?
author
Re:
When batteries are hooked up in series, the voltage increase. For example, two  6 Volt batteries connected in series produce 12 Volts. When batteries are hooked up in parallel, the voltage remains the same, but the power (or available current) will increase. This means that the batteries would last longer.
Could you please explain Ohm's law in simple terms?
author
Re:
Ohm's law says that in an electrical circuit, the current passing through a resistor between two points, is related to the voltage difference between the two points, and are related to the electrical resistance between the two points.
Ohm's law is vitally important to describing electric circuits, so just be more patient to read the article. You can benefit from it.
What is a closed switch's resistance?
author
Re:
If you are asking about short circuit and open circuit then “short circuit” is usually equivalent to “closed switch” where as “open circuit” is equivalent to “open switch”
So by following that context,
● The resistance of a closed switch is considered to be zero as current will flow without any opposition.
● Whereas, the resistance of an open switch is considered to be infinity as no current will flow.
What happens if resistance is too high?
author
Re:
If resistance is too high, current will be low if voltage is okay.
NOTE: When the voltage stays the same, such as in an Automotive Circuit... current goes up as resistance goes down, and current goes down as resistance goes up. Bypassed devices reduce resistance, causing high current.
I'm wondering what happens if a circuit has no resistance...
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If there really was no resistance in the circuit, the electrons would go around the circuit, and arrive back at the beginning of the circuit with as much energy as the potential difference (the voltage). Current will flow, and since the resistance is low  though not zero  you will just get a really large current.
Is the current constant in parallel?
author
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Components connected in parallel are connected along multiple paths so that the current can split up; the same voltage is applied to each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents flowing through each component.
How should I do a nodal analysis?
author
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Follow these steps while solving any electrical network or circuit using Nodal analysis.
Step 1 − Identify the principal nodes and choose one of them as reference node. We will treat that reference node as the Ground.
Step 2 − Label the node voltages with respect to Ground from all the principal nodes except the reference node.
Step 3 − Write nodal equations at all the principal nodes except the reference node. Nodal equation is obtained by applying KCL first and then Ohm’s law.
Step 4 − Solve the nodal equations obtained in Step 3 in order to get the node voltages.