1.9: Sign conventions
- Page ID
- 98533
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In electrical circuits, sign conventions are established to maintain consistency in analyzing and interpreting circuit parameters such as current, voltage, and power. These conventions help ensure that the results obtained from circuit analysis are meaningful and can be compared across different components. Here's a description of the sign conventions for current, voltage, and power:
Figure 1.9.1: The direction of electric current is opposite to the flow of electrons.
Current Sign Convention:
The assumed direction of current flow is typically from the positive terminal to the negative terminal. This convention aligns with the flow of positive electric charge. The positive charge flows from the positive to the negative terminal is considered as positive current. If the actual flow is opposite to the assumed direction, it is considered negative. In Figure 1.9.1, the current I is aligned with the direction from the positive to the negative, so I am positive.
Voltage Sign Convention:
Voltage is considered positive when it is assumed higher at the terminal where the current enters a passive component (like a resistor, capacitor, or inductor). Negative Voltage represents a voltage rise. This happens when the current flows from lower voltage to higher voltage. Vba is negative because the current flows from the negative to the positive terminal. Positive Voltage represents a voltage drop. This happens when the current flows from higher voltage to lower voltage. Vab is positive because the light bulb causes a voltage drop.
Power Sign Convention:
The power absorbed or supplied by an element results from multiplying the voltage across the element by the current through it. A positive sign in power indicates that the element delivers or absorbs power. Conversely, a negative sign implies that the element is supplying power. To determine the sign of power, it is essential to consider the direction of the current and the polarity of the voltage. According to this convention,
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Power Absorption (Load or Power Sink):
If the current enters the positive terminal of a device, and the voltage is positive at that terminal, the power is absorbed or consumed by the device. Power is positive for devices such as resistors, indicating that they absorb power.
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Power Generation (Source or Generator):
If the current enters the negative terminal of a device, and the voltage is positive at that terminal, the power is generated or supplied by the device. Power is negative for active devices like generators, indicating they supply power to the circuit.
Power Balancing:
Power balancing in a circuit refers to the equilibrium or equality between the power supplied to the circuit by sources and the power consumed or absorbed by various elements within the circuit. In an ideal situation, the total power entering the circuit from sources should equal the total power dissipated or consumed by the circuit elements. This concept is derived from the principle of conservation of energy.
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Power Supplied by Sources:
- Power sources within a circuit, such as batteries or generators, provide electrical energy to the system.
- The power supplied by these sources is usually expressed as a product of voltage and current and is measured in watts (W).
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Power Consumed by Circuit Elements:
- Circuit elements, such as resistors, capacitors, and inductors, consume electrical power.
- The power consumed or dissipated by a resistor, for example, is given by the product of current and voltage across the resistor, according to the formula P=VI.
- Power consumption in circuit elements can also be expressed in watts.
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Conservation of Energy:
- According to the principle of conservation of energy, the total power entering a closed circuit should equal the total power leaving the circuit.
- This implies that the sum of power supplied by sources should equal the sum of power consumed by circuit elements.
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Mathematical Representation:
- Mathematically, power balancing in a circuit can be expressed using the following equation: ∑Psources=∑Pconsumed
- The summation includes all power sources and all power-consuming elements within the circuit.
Examples
Example 1:
The circuit contains four elements: a voltage source, a current source, and two resistors. The arrows or + and - signs show the current directions and voltage polarity. Use the sign convention to determine the currents i1, i2, and i3 and voltages v1, v2, v3, and v4. Calculate the power of the DC current element. Is it a power source or a power sink?
The current i1 is +3A which is given. The current i3 is -1A because it is the same current as the DC current source. The direction is opposite to the DC current so we need to add the negative sign. The Ohm's law can determine i2:
i2 = V2/10 = +2A
This is a parallel circuit with three wires. The DC power on the left-hand side shows the top voltage is higher than the bottom voltage by V1 which is +20V. The middle element must have the same voltage drop, so V2 is +20V. The right-hand wire crosses one resistor and one current element. The total voltage drop is (V3 - V4) and is also equal to +20V. The voltage drop V3 is i3*30 = +30V because the current flows from positive to negative. The V4 is a voltage increase because the current flows from negative to positive and the voltage V4 is -10V from the following relationship.
V2 = V3 + V4 = 30V + V4
The DC current is a power source because the current enters into the negative terminal and leaves from the positive terminal.
Example 2:
Find the power of each element in the above example. Use positive to represent the power sink and negative to represent the power source. What is the net power of the above circuit? We use P = I * V to determine the power. If the current flows into the positive terminal, we use positive, otherwise, use the negative voltage.
P1 = 3A * -20V = -60W (power source)
P2 = 2A * 20V = 40W (power sink)
P3 = 1A * 30V = 30W (power sink)
P4 = 1A * -10V = -10W (power source)
The total power of the whole circuit is zero. This agrees with the conservation of energy.
Considerations
Give some considerations such as system requirements or "gotchas" for this setting or control or programming syntax.