The superposition theorem is a derivational result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem cases that for a linear system (notably combining the subcategory of time-invariant linear arrangements) the response (voltage or current) in any branch of a reciprocal linear circuit having more than one independent source corresponds to the algebraic sum of the responses generated by each independent source beginning alone, where all the other independent sources are replaced by their intramural impedances.
To determine the addition of each individual source, all of the other origins first must be “turned off” (set to zero) by:
- Changing all other independent voltage sources with a short circuit (thereby get rid of difference of potential i.e. V=0; internal impedance of ideal voltage source is zero (short circuit)).
- Restoring all other independent current sources with an open circuit (thereby excluding current i.e. I=0; internal impedance of the ideal current source is infinite (open circuit)).
This procedure is followed for each origin in turn, then the resultant responses are added to determine the true action of the circuit. The resultant circuit action is the superposition of the different voltage and current sources.
The superposition theorem is very essential in circuit analysis. It is used in modifying any circuit into its Norton correspondent or Thevenin equivalent.
The theorem is germane to linear networks (time-varying or time-invariant) subsisting of independent sources, linear dependent sources, linear passive elements (resistors, inductors, capacitors), and linear transformers.
Superposition pursues voltage and current but not power. In other words, the sum of the powers of each source with the other sources turned off is not the real absorbed power. To calculate power we first use superposition to find both the current and voltage of each linear element and then calculate the sum of the aggregated voltages and currents.
However, if the linear network is achieving in steady-state and each extraneous independent source has a distinctive frequency, then superposition can be applied to compute the average power or active power. If at least two independent sources have the clone frequency (for example in power systems, where many generators conduct at 50 Hz or 60 Hz), then superposition can’t be used to regulate average power.
What Is Superposition Theorem?
“If more than one source acts together in an electric circuit, then the current through any one of the arms of the circuit is the addition of currents which would flow through the branch for each source, keeping all the other sources dead.”
To calculate the personal contribution of each source in a circuit, the other source must be replaced or abolished without altering the final result. While abolishing a voltage source, its value is set to zero. This is done by changing the voltage source with a short circuit. When abolishing a current source, its value is set to zero. This is done by replacing the current source with an open circuit.
The superposition theorem is very critical in circuit analysis because it converts a complex circuit into a Norton or Thevenin equivalent circuit.
Guidelines to keep in mind while using the superposition theorem
When you sum the particular contributions of each source, you should be careful while accrediting signs to the quantities. It is suggested to assign an allusion direction to each unknown quantity. If a contribution from a source has the clone direction as the reference direction, it has a positive sign in the sum; if it has the differing direction, then a negative sign.
To use the superposition theorem with circuit currents and voltages, all the segments must be linear.
It should be noted that the superposition theorem does not apply to power, as power is not a linear abundance.
How to apply Superposition Theorem?
- The first step is to select one source among the multiple sources being in the bilateral network. Among the various sources in the circuit, any one of the sources can be contemplated first.
- Except for the selected source, all the sources must be replaced by their intramural impedance.
- Using a network simplification approach, appraise the current flowing through or the voltage drop across a particular aspect in the network.
- The same considering a single source is reiterated for all the other sources in the circuit.
- Upon accessing the respective responsibilities for the individual source, perform the summation of all responses to get the overall voltage drop or current through the circuit element.
Steps for Solving network by Superposition Theorem
Considering circuit diagram A, let us see the various steps to solve the superposition theorem:
- Step 1: Take only one self-sufficient source of voltage or current and deactivate the other sources.
- Step 2: In-circuit diagram B shown above, consider the source E1 and replace the other source E2 by its internal resistance. If its internal resistance is not given, then it is taken as zero and the source is short-circuited.
- Step 3: If there is a voltage source then short circuit it and if there is a current source then just open-circuit it.
- Step 4: Thus, by mobilizing one source and deactivating the other source find the current in each branch of the network. Taking the above example find the current I1’, I2’and I3’.
- Step 5: Now acknowledge the other source E2 and replace the source E1 with its internal resistance r1 as shown in circuit diagram C.
- Step 6: Determine the current in various sections, I1’’, I2’’, and I3’’.
- Step 7: Now to determine the net branch current utilizing the superposition theorem, add the currents accessed from each individual source for each branch.
- Step 8: If the current obtained by each branch is in the same direction then add them and if it is in the opposite direction, deduct them to obtain the net current in each branch.
The actual flow of current in the circuit C will be given by the equations shown below:
Prerequisites for the Superposition Theorem
Quite simple and delicate, don’t you think? It must be noted, though, that the Superposition Theorem works only for circuits that are reducible to series/parallel combinations for each of the power sources at a time (thus, this theorem is useless for acknowledging an unbalanced bridge circuit), and it only works where the elemental equations are linear (no mathematical powers or roots).
The precondition of linearity means that Superposition Theorem is only relevant for determining voltage and current, not power!!! Power dissipations, being nonlinear functions, do not algebraically add to an accurate total when only one source is considered at a time. The need for linearity also means this Theorem cannot be conducted in circuits where the resistance of a component changes with voltage or current. Hence, networks accommodating components like lamps (incandescent or gas-discharge) or varistors could not be analyzed.
Another essential for the Superposition Theorem is that all components must be “bilateral,” meaning that they behave the same with electrons flowing in either direction through them. Resistors have no polarity-specific behavior, and so the circuits we’ve been examining so far all meet this principle.
The Superposition Theorem encounters to use in the study of alternating current (AC) circuits, and semiconductor (amplifier) circuits, where sometimes AC is often blended (superimposed) with DC. Because AC voltage and current equations (Ohm’s Law) are linear just like DC, we can use Superposition to analyze the circuit with just the DC power source, then just the AC power source, associating the results to tell what will appear with both AC and DC sources in effect. For now, though, Superposition will suffice as a break from having to do concurrent equations to analyze a circuit.
Conclusion
This post examines the superposition theorem, another approach for circuit analysis. The superposition theorem states that a circuit with various voltage and current sources is equal to the sum of simplified circuits using just one of the sources. A circuit belonged to two voltage sources, for example, will be equal to the amount of two circuits, each one using one of the sources and having the other eliminated.
To facilitate a circuit using the superposition theorem, the following steps to be followed: identify all current and voltage sources in the circuit; create various versions of the circuit and the other sources must be removed using the following rule: voltage sources must be replaced with a short circuit and current sources just removed from the circuit, every version including just one of the sources; find the currents and voltages required, and sum the results obtained in all circuits.