The peak current is proportional to both the angular frequency and the inductance. Where X L = ωL is the effective resistance of the inductor and is known as the inductive reactance. The current follows a negative cosine graph while the voltage follows a sine graph, explaining why the current lags the voltage by 90 o Integrate to find the current as a function of time: To understand why this is, apply Kirchoff's loop rule: This time theĬurrent lags the voltage by 90 o in a purely inductive circuit. In a circuit with only an inductor and an AC power source, there is also a 90 o phase difference between the current and voltage. The higher the frequency of the voltage, the shorter the time available to change the voltage, so the larger the current has to be. The larger the capacitance of the capacitor, the more charge has to flow to build up a particular voltage on the plates, and the higher the current will be. We can understand this by realizing that the capacitor voltage has to match the source voltage at all times. Note that the peak current is inversely proportional to both the angular frequency and the capacitance. Where X C = 1/ωC is the effective resistance of the capacitor and is known as the capacitive reactance.
We often write the peak current in the form: The current follows a cosine graph while the voltage follows a sine graph, explaining why the current leads the voltage by 90 o
To understand why this is, simply apply Kirchoff's loop rule: Put another way, the current leads the voltage by 90 o in a purely capacitive circuit. It turns out that there is a 90 o phase difference between the current and voltage, with the current reaching its peak 90 o (1/4 cycle) before the voltage reaches its peak. A capacitor is a device for storing charging. In addition, the current is in phase with the voltage.Ĭonsider now a circuit which has only a capacitor and an AC power source (such as a wall outlet). The peak current is the peak voltage over the resistance, and does not depend on frequency. The relationship $Delta V = IR applies for resistors in an AC circuit, so the current is:
Rms value of a sine wave is 1/2 1/2 = 0.707 The integral of cos(2θ) over the interval 0 to π is zero, so we get:Īverage value of sin 2(θ) = 1/2π ∫ dθ = 1/2 Where the limits on the integral are from 0 to π This can be done by integrating:Īverage value of sin 2(θ) = /π To determine the rms average of sin(θ) over one period, first square it and then find the mean. Root mean square averaging turns everything positive, and weights larger numbers more than smaller numbers. The average of the numbers -1, 1, 3, and 5 is 2, while their rms average is 3. For a sine wave, the relationship between the peak and the rms average is: Voltages and currents for AC circuits are generally expressed as rms values. The particular averaging method used is called root mean square (square the voltage to make everything positive, find the average, take the square root), or rms. We talk of a household voltage of 120 volts, though this number is a kind of average value of the voltage. V o represents the maximum voltage, which in a household circuit in North America is about 170 volts. The angular frequency is related to the frequency, f, by: In a household circuit, the frequency is 60 Hz. In alternating current (AC) circuits the voltage oscillates in a sine wave pattern: Direct current (DC) circuits involve current flowing in one direction.