Input impedance is the total opposition a circuit presents to the current flowing into it from a source. To calculate it, you apply Ohm’s Law (V/I) at the input terminals, measuring the voltage across the terminals and the current entering them. For simple circuits, this means combining resistors, capacitors, and inductors using series and parallel rules, often requiring complex numbers when reactive components are present.
What Exactly Is Input Impedance?
Think of a circuit as a box with two input terminals. The input impedance is what the source “sees” when it tries to push current into that box. It is a measure of how much the circuit resists or impedes the flow of alternating current (AC) or direct current (DC).
For DC circuits, impedance is simply resistance, measured in ohms (Ω). For AC circuits, impedance includes resistance and reactance. Reactance comes from capacitors and inductors, which store and release energy. This makes the impedance a complex quantity, expressed as Z = R + jX, where R is resistance and X is reactance.
The value matters because it determines how much current flows, how much power is transferred, and how a circuit interacts with what is connected to it. A mismatch between source impedance and input impedance can cause signal reflections, power loss, or distorted signals.
How To Calculate Input Impedance Of A Circuit: The Basic Method
The fundamental approach uses Ohm’s Law. The input impedance (Zin) equals the input voltage (Vin) divided by the input current (Iin).
Zin = Vin / Iin
You can measure these directly. Apply a known voltage to the input terminals. Measure the current flowing into the circuit. Divide the two. This works for any circuit, no matter how complex the internal components are.
For a purely resistive circuit, this is straightforward. If you apply 10 volts and measure 2 milliamps of current, the input impedance is 5,000 ohms. The calculation becomes more involved when capacitors and inductors are present because their opposition to current changes with frequency.
In AC analysis, you must account for the phase difference between voltage and current. The impedance is then a complex number, not a simple ratio of magnitudes. You need to use vector mathematics or complex arithmetic to get the correct result.
Calculating Input Impedance in Series and Parallel Circuits
For circuits made of discrete components, you combine impedances using the same rules you use for resistors.
Series combination: Add the impedances directly.
Ztotal = Z1 + Z2 + Z3 + …
Parallel combination: Add the reciprocals.
1/Ztotal = 1/Z1 + 1/Z2 + 1/Z3 + …
When combining complex impedances, you must treat each one as a complex number. A resistor has impedance R. An inductor has impedance jωL, where ω is the angular frequency (2π times frequency in hertz) and L is inductance in henries. A capacitor has impedance 1/(jωC), where C is capacitance in farads.
Consider a simple circuit with a resistor and capacitor in series. The input impedance is R + 1/(jωC). This is a complex number with a real part (R) and an imaginary part (-1/(ωC)). The negative imaginary part indicates capacitive behavior, meaning the current leads the voltage.
For a parallel RC circuit, you would use the reciprocal formula. The result is more complex algebraically, but the principle is the same. Combine the components according to how they are physically connected.
Why Frequency Changes Everything
Input impedance is not a fixed number for circuits with capacitors or inductors. It changes with the frequency of the applied signal.
A capacitor has very high impedance at low frequencies. At DC, it blocks current completely, acting like an open circuit. As frequency increases, its impedance drops. An inductor behaves in the opposite way. It has very low impedance at low frequencies and high impedance at high frequencies.
This frequency dependence is why you cannot quote a single input impedance value for many circuits. You must specify the frequency at which the measurement or calculation applies. A circuit might have an input impedance of 1,000 ohms at 1 kHz but only 100 ohms at 100 kHz.
In practice, engineers often characterize circuits over a range of frequencies. They plot impedance magnitude and phase angle versus frequency. This gives a complete picture of how the circuit behaves across its operating band.
Input Impedance in Amplifiers and Active Circuits
Active circuits, such as amplifiers, present a special case. The input impedance is not simply the combination of passive components. Feedback and active device characteristics affect it.
For an operational amplifier (op-amp) in a non-inverting configuration, the input impedance is very high, often in the megaohm range. This is because the input connects directly to the op-amp’s high-impedance input terminal. In an inverting configuration, the input impedance is approximately equal to the input resistor value, because the inverting input is held at virtual ground.
Transistor amplifiers have input impedances that depend on the biasing network and the transistor’s internal parameters. The base-emitter junction of a bipolar junction transistor (BJT) looks like a diode, giving a relatively low input impedance. A field-effect transistor (FET) has a very high input impedance because its gate is insulated from the channel.
When measuring input impedance of an active circuit, you must be careful. The circuit may oscillate or behave unpredictably if the test setup is not appropriate. Use a known series resistor and measure the voltage across it to determine the current drawn by the circuit.
Matching Impedances for Maximum Power Transfer
Impedance matching is a practical application of input impedance calculations. The maximum power transfer theorem states that maximum power is delivered to a load when the load impedance equals the source impedance.
For purely resistive circuits, this means the load resistance equals the source resistance. For reactive circuits, the load impedance must be the complex conjugate of the source impedance. That means equal resistance and opposite reactance.
This principle is critical in audio systems, radio frequency transmission, and telecommunications. Mismatched impedances cause signal reflections. In audio, this can alter frequency response. In RF systems, it can cause power loss and standing waves on transmission lines.
Common impedance standards include 50 ohms for RF systems, 75 ohms for video and cable television, and 600 ohms for professional audio. These values were chosen as compromises between power handling, loss, and practical component values.
Measuring Input Impedance in Practice
You do not always need to calculate input impedance from component values. You can measure it directly with the right equipment.
The simplest method uses a signal generator, a known resistor, and a voltmeter. Place the known resistor in series with the circuit input. Measure the voltage across the resistor and the voltage at the circuit input. Calculate the current from the resistor voltage and resistance value. Then divide the input voltage by the current.
For more accurate measurements, especially at high frequencies, you would use a network analyzer or an impedance analyzer. These instruments sweep frequency and measure the complex impedance automatically. They display magnitude and phase, giving you a complete impedance profile.
One practical caution: test equipment has its own impedance. A standard voltmeter has high input impedance, typically 10 megohms, so it does not load the circuit significantly. A network analyzer typically has a 50-ohm input, which can affect measurements of high-impedance circuits. Account for this when interpreting results.
Common Mistakes When Calculating Input Impedance
Several errors appear frequently when people calculate input impedance by hand.
Forgetting the frequency: Calculating impedance of a capacitor or inductor without specifying frequency gives a meaningless result. Always include ω in the calculation.
Ignoring component tolerances: Resistors have tolerances of 5% or 10%. Capacitors often have even wider tolerances. Your calculated value is an estimate, not an exact figure.
Neglecting parasitic elements: At high frequencies, wires have inductance and component leads have capacitance. These parasitics can significantly alter the actual input impedance compared to your ideal calculation.
Applying DC rules to AC circuits: You cannot simply add resistances and reactances as scalar numbers. They must be combined as complex quantities, accounting for phase.
Forgetting the source impedance: The input impedance of a circuit is measured at its terminals, independent of the source. Do not include the source resistance in your calculation of the circuit’s input impedance.
Frequently Asked Questions
What is the formula for input impedance?
Input impedance equals input voltage divided by input current, expressed as Zin = Vin / Iin. For AC circuits, this is a complex quantity that includes resistance and reactance.
Why does input impedance matter in a circuit?
It determines how much current the circuit draws from the source and how much power is transferred. A mismatch between source and input impedance causes signal loss and reflections.
How do I measure input impedance with a multimeter?
You cannot measure impedance directly with a standard multimeter, but you can measure resistance for DC circuits. For AC circuits, use a signal generator and a known series resistor, then calculate impedance from voltage and current measurements.
Does input impedance change with frequency?
Yes, for any circuit containing capacitors or inductors, input impedance varies with frequency. Capacitors have high impedance at low frequencies, while inductors have high impedance at high frequencies.
