Summary of Input Impedance and Output Impedance

1. Input Impedance

Input impedance refers to the equivalent impedance of a circuit input terminal. Add a voltage source U to the input terminal and measure the current I at the input terminal. The input impedance Rin is U/I. You can imagine the input terminal as the two ends of a resistor. The resistance value of this resistor is the input impedance.

Input impedance is no different from an ordinary reactive element. It reflects the magnitude of the resistance to current. For voltage-driven circuits, the larger the input impedance, the lighter the load on the voltage source, making it easier to drive and having no effect on the signal source. For current-driven circuits, the smaller the input impedance, the lighter the load on the current source. Therefore, we can think of it this way: if it is driven by a voltage source, the larger the input impedance, the better; if it is driven by a current source, the smaller the impedance, the better (Note: This is only suitable for low-frequency circuits. In high-frequency circuits, impedance matching must also be considered. In addition, if you want to obtain the maximum output power, you must also consider impedance matching.)

2. Output Impedance

Whether it is a signal source, amplifier or power supply, there is a problem of output impedance. Output impedance is the internal resistance of a signal source. Originally, for an ideal voltage source (including power supply), the internal resistance should be 0, or the impedance of an ideal current source should be infinite. Output impedance is the most important thing to pay attention to in circuit design.

But the voltage source in reality cannot do this. We often use an ideal voltage source in series with a resistor r to be equivalent to an actual voltage source. The resistor r in series with the ideal voltage source is the internal resistance of the (signal source/amplifier output/power supply). When this voltage source supplies power to the load, a current I will flow through the load and generate a voltage drop of I×r on the resistor. This will cause the output voltage of the power supply to drop, thereby limiting the maximum output power (for why the maximum output power is limited, please see the question "Impedance Matching" below). Similarly, for an ideal current source, the output impedance should be infinite, but this is impossible in actual circuits.

3. Impedance Matching

Impedance matching refers to a suitable match between a signal source or transmission line and a load. Impedance matching is discussed in two cases: low frequency and high frequency.

Let's start with a DC voltage source driving a load. Since actual voltage sources always have internal resistance (please refer to the question about output impedance), we can make an actual voltage source equivalent to a model of an ideal voltage source connected in series with a resistor r. Assuming the load resistance is R, the power supply electromotive force is U, and the internal resistance is r, we can calculate the current flowing through resistor R as: I=U/(R+r). It can be seen that the smaller the load resistance R, the greater the output current. The voltage on load R is: Uo=IR=U/[1+(r/R)]. It can be seen that the larger the load resistance R, the higher the output voltage Uo. Let's calculate the power consumed by resistor R as:

P=I2×R=[U/(R+r)]2×R=U2×R/(R2+2×R×r+r2)

=U2×R/[(Rr)2+4×R×r]

=U2/{[(Rr)2/R]+4×r}

 (Note: 2 is the square)

For a given signal source, its internal resistance r is fixed, and the load resistance R is selected by us. Note that in the formula [(Rr)2/R], when R=r, [(Rr)2/R] can achieve the minimum value of 0, and the maximum output power Pmax=U2/(4×r) can be obtained on the load resistance R. That is, when the load resistance is equal to the internal resistance of the signal source, the load can obtain the maximum output power, which is one of the impedance matching we often say. For pure resistance circuits, this conclusion is also applicable to low-frequency circuits and high-frequency circuits. When the AC circuit contains capacitive or inductive impedance, the conclusion changes, that is, the real part of the signal source and load impedance must be equal, and the imaginary part must be opposite to each other, which is called conjugate matching. In low-frequency circuits, we generally do not consider the matching problem of transmission lines, but only consider the situation between the signal source and the load, because the wavelength of the low-frequency signal is very long relative to the transmission line, the transmission line can be regarded as a "short line", and reflection can be ignored (it can be understood this way: because the line is short, even if it is reflected back, it is still the same as the original signal). From the above analysis, we can conclude that if we need a large output current, we should choose a small load R; if we need a large output voltage, we should choose a large load R; if we need the maximum output power, we should choose a resistor R that matches the internal resistance of the signal source. Sometimes impedance mismatch has another meaning. For example, the output of some instruments is designed under specific load conditions. If the load conditions change, the original performance may not be achieved. This is also called impedance mismatch.

In high-frequency circuits, we must also consider the problem of reflection. When the frequency of the signal is very high, the wavelength of the signal is very short. When the wavelength is short enough to be comparable to the length of the transmission line, the reflected signal superimposed on the original signal will change the shape of the original signal . If the characteristic impedance of the transmission line is not equal to the load impedance (i.e., mismatched), reflection will occur at the load end. Why reflection occurs when the impedance is mismatched and how to solve the characteristic impedance involve solving the second-order partial differential equation. We will not go into details here. Those who are interested can refer to the transmission line theory in books on electromagnetic fields and microwaves. The characteristic impedance (also called characteristic impedance) of the transmission line is determined by the structure and material of the transmission line, and has nothing to do with the length of the transmission line, the amplitude of the signal, the frequency, etc.

For example, the characteristic impedance of the commonly used closed-circuit television coaxial cable is 75Ω, while some RF equipment often uses coaxial cables with a characteristic impedance of 50Ω. Another common transmission line is a flat parallel line with a characteristic impedance of 300Ω, which is more common on TV antenna racks used in rural areas and is used as a feeder for Yagi antennas. Because the input impedance of the RF input end of the TV is 75Ω, the 300Ω feeder will not match it. How is this problem solved in practice? I don’t know if you have noticed that there is a 300Ω to 75Ω impedance converter in the accessories of the TV (a plastic package with a round plug on one end, about the size of two thumbs). It is actually a transmission line transformer that transforms the 300Ω impedance into 75Ω, so that it can be matched. One thing that needs to be emphasized here is that characteristic impedance is not the same concept as resistance as we usually understand it. It has nothing to do with the length of the transmission line and cannot be measured by using an ohmmeter. In order to avoid reflection, the load impedance should be equal to the characteristic impedance of the transmission line. This is the impedance matching of the transmission line. What are the adverse consequences if the impedance does not match? If it does not match, reflection will be formed, energy cannot be transmitted, and efficiency will be reduced; standing waves will be formed on the transmission line (in simple terms, the signal is strong in some places and weak in others), resulting in a reduction in the effective power capacity of the transmission line; the power cannot be transmitted, and even the transmitting equipment will be damaged. If the high-speed signal line on the circuit board does not match the load impedance, oscillation and radiation interference will occur.

When the impedance does not match, what methods are there to make it match? First, you can consider using a transformer to do impedance conversion, just like the example of the TV mentioned above. Second, you can consider using the method of series/parallel capacitors or inductors, which is often used when debugging RF circuits. Third, you can consider using the method of series/parallel resistors. The impedance of some drivers is relatively low, and a suitable resistor can be connected in series to match the transmission line. For example, high-speed signal lines sometimes have a resistor of tens of ohms connected in series. The input impedance of some receivers is relatively high, and the parallel resistor method can be used to match the transmission line. For example, the 485 bus receiver often connects a 120-ohm matching resistor in parallel at the data line terminal.

To help you understand the reflection problem when impedance is not matched, let me give you two examples: suppose you are practicing boxing - hitting a sandbag. If it is a sandbag of appropriate weight and hardness, you will feel very comfortable when you hit it. However, if one day I tamper with the sandbag, for example, replace the inside with iron sand, and you still hit it with the same force as before, your hand may not be able to bear it - this is the case of excessive load, which will produce a large rebound force. On the contrary, if I replace the inside with something very light, when you punch, you may miss, and your hand may not be able to bear it - this is the case of too light load. Another example, I don’t know if you have ever had this experience: going up/down the stairs when you can’t see the stairs clearly, when you think there are stairs, you will feel that "load is not matched". Of course, this example may not be appropriate, but we can use it to understand the reflection situation when the load is not matched . For convenience, I will briefly explain it with pure resistance.

    The input signal source of an amplifier and the output voltage of this amplifier can both be equivalent to the part surrounded by the dotted line in the figure, that is, the series connection of a voltage source and an internal resistance; and the resistor R in the figure can be the input resistance of this amplifier or the equivalent load to which the amplifier is connected.

   If the voltage and internal resistance of the input signal source are constant, the larger the input resistance of the amplifier (i.e., high input impedance), the smaller the current obtained from the signal source, and the smaller the voltage drop on the internal resistance of the signal source, the signal voltage can be added to the input of the amplifier with the smallest possible loss. When the input resistance is very small, the situation is just the opposite . Of course, in general, we need the former.

    The same analytical approach, if the amplifier's output resistance is smaller, the signal source voltage (amplifier's output voltage) will lose less on the internal resistance, and the load will get the highest possible output voltage, which is often referred to as "strong load capacity" . This does not include the situation where the load needs to get the maximum power.

Therefore, when voltage amplification is required, an amplifier with high input resistance and low output resistance is needed .